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Journal of Mathematical Imaging and Vision 15: 127–168, 2001c© 2001 Kluwer Academic Publishers. Manufactured in The Netherlands.

A Dynamic Scale–Space Paradigm

ALFONS H. SALDEN∗, BART M. TER HAAR ROMENY AND MAX A. VIERGEVERImage Sciences Institute, Utrecht University Hospital, Heidelberglaan 100, 3584 CX Utrecht, The Netherlands

[email protected]

[email protected]

[email protected]

Abstract. We present a novel mathematical, physical and logical framework for describing an input image ofthe dynamics of physical fields, in particular the optic field dynamics. Our framework is required to be invariantunder a particular gauge group, i.e., a group or set of transformations consistent with the symmetries of thatphysical field dynamics enveloping renormalisation groups. It has to yield a most concise field description in termsof a complete and irreducible set of equivalences or invariants. Furthermore, it should be robust to noise, i.e.,unresolvable perturbations (morphisms) of the physical field dynamics present below a specific dynamic scale,possibly not covered by the gauge group, do not affect Lyapunov or structural stability measures expressed inequivalences above that dynamic scale. The related dynamic scale symmetry encompasses then a gauge invariantsimilarity operator with which similarly prepared ensembles of physical field dynamics are probed and searchedfor partial equivalences coming about at higher scales.

The framework of our dynamic scale-space paradigm is partly based on the initialisation of joint (non)localequivalences for the physical field dynamics external to, induced on and stored in a vision system and represented byan image, possibly at various scales. These equivalences are consistent with the scale-space paradigm considered andpermit a faithful segmentation and interpretation of the dynamic scale-space at initial scale. Among the equivalencesare differential invariants, integral invariants and topological invariants not affected by the considered gauge group.These equivalences form a quantisation of the external, induced and stored physical field dynamics, and are associatedto a frame field, co-frame field, metric and/or connection invariant under the gauge group. Examples of theseequivalences are the curvature and torsion two-forms of general relativity, the Burgers and Frank vector densityfields of crystal theory (in both disciplines these equivalences measure the inhomogeneity of translational and(affine) rotation groups over space-time), and the winding numbers and other topological charges popping up inelectromagnetism and chromodynamics.

Besides based on a gauge invariant initialisation of equivalences the framework of our dynamic scale-spaceparadigm assumes that a robust, i.e. stable and reproducible, partially equivalent representation of the physicalfield dynamics is acquired by a multi-scale filtering technique adapted to those initial equivalences. Effectively, thehierarchy of nested structures of equivalences, by definition too invariant under the gauge group, is obtained byapplying an exchange principle for a free energy of the physical field dynamics (represented through the equivalences)that in turn is linked to a statistical partition function. This principle is operationalised as a topological current offree energy between different regions of the physical field dynamics. It translates for each equivalence into aprocess governed by a system of integral and/or partial differential equations (PDES) with local and global initial-boundary conditions (IBC). The scaled physical field dynamics is concisely classified in terms of local and non-local equivalences, conserved densities or curvatures of the dynamic scale-space paradigm that in generally are notcoinciding with all initial equivalences. Our dynamic scale-space paradigm distinguishes itself intrinsically from thestandard ones that are mainly developed for scalar fields. A dynamic scale-space paradigm is also operationalised

∗To whom correspondence should be addressed.

128 Salden, ter Haar Romeny and Viergever

for non-scalar fields like curvature and torsion tensor fields and even more complex nonlocal and global topologicalfields supported by the physical field dynamics. The description of the dynamic scale-spaces are given in terms ofagain equivalences, and the paradigms in terms of symmetries, curvatures and conservation laws. The topologicalcharacteristics of the paradigm form then a representation of the logical framework.

A simple example of a dynamic scale-space paradigm is presented for a time-sequence of two-dimensionalsatellite images in the visual spectrum. The segmentation of the sequence in fore- and background dynamics atvarious scales is demonstrated together with a detection of ridges, courses and inflection lines allowing a concisetriangulation of the image. Furthermore, the segmentation procedure of a dynamic scale-space is made explicitallowing a true hierarchically description in terms of nested equivalences.

How to unify all the existing scale-space paradigms using our frame work is illustrated. This unification comesabout by a choice of gauge and renormalisation group, and setting up a suitable scale-space paradigm that might beuser-defined.

How to extend and to generalise the existing scale-space paradigm is elaborated on. This is illustrated by pointingout how to retain a pure topological or covariant scale-space paradigm from an initially segmented image that insteadof a scalar field also can represent a density field coinciding with dislocation and disclination fields capturing thecutting and pasting procedures underlying the image formation.

Keywords: gauge group, frame field, co-frame field, metric, connection, equivalence, statistical partition function,free energy, topological current, exchange principle, conserved density, curvature

1. Introduction

In scale-space theory [237] there exist several imageanalysis and processing methods that yield compara-ble results on images. Most of the methods are basedon a set of axiomatics related to invariance principlesassumed to be underlying the image analysis and pro-cessing techniques given the inflicted physics in the im-age formation. Among those principles considered arescale invariance and invariance under Euclidean move-ments. Obviously, user-defined scale-space paradigmsare more natural as the stored dynamics about the vi-sual fields and also others inflict an environment andautonomous system appropriate interaction with the in-duced external physical fields.

In relation to image understanding up to now onlydescription schemes for linear scale-spaces [68, 109,110, 113, 195, 196, 204, 206, 235, 236] are devel-oped. These schemes are based on perturbative andnon-perturbative methods engendered by invariancetheory [197, 201], Morse theory [68, 196], modern ge-ometry [195, 196, 199] and differential topology [109,110, 113, 199, 206]. The perturbative method is lim-ited to a continuous setting, whereas the nonperturba-tive method one can also apply within a semi-discreteand discrete realm. Sub-voxel refinement of the non-perturbative method is not desirable in order to ob-tain a parametrisation of “critical curves” in the linearscale-space. The sub-voxel dynamics determining thatparametrisation is namely obsolete and obstructs any

local perturbative analysis in a continuous setting. Inorder to obtain superresolution and valuable hyperac-curacy results on a sub-voxel scale one has to determinethe renormalisation group active at those scales [9, 206,268].

In nonlinear scale-space theories the studied multi-scale physical objects are quite often not chosen consis-tent with the considered paradigm. E.g., the physicalobjects invariant under affine curve shortening flowsthat are solutions of the nonlinear Cauchy problem arenot equal to the standard affine objects related to theinitial curve itself. Normally those objects are no so-lutions of the considered affine flow problem. Equallyso, considering the ordinary image gradient in any ofthe nonlinear flow paradigms is not the proper iamgefeature to detect edgels.

In general an image consists of a finite number ofspatio-temporal samplings of external physical fields,such as the electromagnetic and gravitational fields. Incomputer vision and pattern recognition these fieldsconcern mainly the optic dynamics of the electro-magnetic field that in turn involves dynamic spatio-temporal variations of the source illumination, andreflection and absorption properties over surfaces ofobjects, the vision system and intermediate media. Thevision system operationalises those dynamics in termsof electro-chemical currents. How this system, subse-quently, analyses and processes those currents depends

A Dynamic Scale–Space Paradigm 129

on the tasks to be performed by the system, the architec-ture and functionality of the system given the acquiredand operationalised statistics concerning the inducedphysical field dynamics. In low level vision such cur-rents are normally modeled by pixels or voxels. Sucha pixel/voxel is the smallest detail, supported by cur-rents, that is just noticeable, that can hopefully be re-trieved in an invariant manner and that determines thespatio-temporal dynamic resolution of the vision sys-tem. This aggregate of currents can then be used as afilter or structuring element. Such a filter or elementrealised in the vision system is not necessarily intrin-sically coupled to the projected or correlated with theexternal physical field dynamics.

Resolution is normally expressed as the just perceiv-able angle and time difference in the optic array be-tween two events. As one speaks of events one should,of course, also include the sensitivity of the vision sys-tem to the external physical field dynamics.

Very sophisticated resolution and filter operators canbe constructed and operationalised. Encoding or ac-tivating the currents in the input image can be donethrough the read out of spatio-temporal positions andthe energy levels acquired by the sensors measuringdifferent aspects of the field dynamics such as spectralpolarisation and chirality [87, 134].

This process yields already a significant amount ofdata compression and reduction. The information aboutthe distribution of the energy over the spatio-temporaldomain of the sensor is lost, although sometimes thesensitivity profile of the sensor is known. Many fielddynamics are mapped to the same resolution elementor current (metamerism). Thus one is confronted, al-ready during the acquisition phase, with the problem ofconstructing the proper sensors for initialising such cur-rents. A natural adjustment to physical field dynamics isasked for, i.e. gauging to the macroscopic physical fielddynamics. For example, the existence of a Lorentzianmetric and non-flat and twisted connection on space-time dynamics is a macroscopically determinedentity.

In order to further analyse and process the input im-age a biological or artificial vision system establishes aso-called scale operator, i.e., a procedure, essentially aredistributing integral operator, to be applied to the pix-els/voxels of the input image of the external, inducedand stored physical field dynamics. One advantage ofsuch an operator is that it is nonperturbative with re-spect to the input image: pixels are combined, mergedor split without ever partitioning the input image nor

introducing new information than latent in the fielddynamics. Sub-pixel measurements can be carried outthough never yield higher qualitative assessments ofthe content of an input image. However, if one has amodel of the dynamics at sub-pixel scale, then an anal-ysis and processing at that scale may be worthwhile.One could think of setting up functorial schemes fortraversing from one category of tasks to another onthe basis of empirical data. With decreasing and in-creasing scale symmetry breaking might occur that in-duces a-similarity throughout the observation process.Statistical justification of such non-equivalent scalingbehaviour of physical field dynamics (well-known inmorphogenesis) can be found in [154].

The reasons for analysing and processing an inputimage by means of a particular set of (scaled) recursionoperators1 , i.e., generalised scale operators by defini-tion also invariant under the considered gauge group,are based on:

• Properties of (scaled) recursion operators in retriev-ing a concise set of equivalences, i.e., invariants un-der the gauge group, for the external physical fielddynamics stored by a vision system,

• Properties of (scaled) recursion operators in the ro-bust removal (sieving) of noise, detection, conser-vation, grouping, restoration and/or enhancement offield dynamics like that involved in the realisation ofwatersheds, crest-lines, edges, plateaus, divides andridges, channels and courses, and defect lines (see fordefinitions of these physical objects [51, 132, 179,196, 223]). These properties can be expressed intoequivalences, local and nonlocal conserved densitiesor curvatures of the system of recursion operators.This can yield a particular desired perceptual group-ing and image understanding.

In the context of image analysis and processing noiseand features can be discriminated, if a model for ei-ther the noise, the features, or both are known underthe (scaled) recursion operators. Besides that these op-erators are useful in image compression, data reduc-tion, image reconstruction and image enhancement,they may also be applied to segment the input imagein meaningful parts, i.e., local and non-local conserveddensities or currents operationalising such notions likeedges, junctions and other entanglements. Non-localityof the densities in this context obviously not only con-cerns multi-locality but in addition topological charac-teristics of the supports of the densities. This induced


dynamics may couple to or resonate with the fielddynamics stored in a vision system. As the physicalfield dynamics is subjected to various morphologicaltransitions, the least requirement would be that the in-put image/entanglements is slightly affected by thesechanges and that the spatio-temporal dynamic resolu-tion and the scale operator (and by definition the re-cursion operators) are invariant under a certain gaugegroup. For example, the group of Euclidean movementson a vision system in relation to an external physicalfield dynamics may already induce (diffeo)morphismsof its input image. This fact implies that the gauge groupto be considered is not the group of Euclidean move-ments but that of (diffeo)morphisms of the input imagein a spatio-temporal as well as dynamical sense. Thereader may object that the latter is a little bit overdoneas soon as one accepts the active exploration of the vi-sual field and the induced time-sequence of views asa scene coinciding with a single external, induced andstored field dynamics. In this case a classical group onspace-time like the Galilean group might already suf-fice to cover the gauge group. However, an ensembleof different scenes does require previously mentionedgauge group.

Applying recursively the set of (scaled) recursionoperators yields a stack of images of the input imagecoined “dynamic scale-space”. Dynamic refers to thetemporal adjustment of a scale operator to the input im-age of the external, induced and stored field dynamics.It’s obvious that the updated induced external physicalfield dynamics together with the statistically and dy-namically activated stored currents control the actual(scaled) recursion operators at a given time through theentanglement or coupling of the field dynamics. Theseoperators have to be designed such that strongly corre-lated currents in the field dynamics are enhanced andbecome coherent at the cost of small scale noise contri-butions in the dynamics. This grouping property leadsto recombinations of the induced field dynamics sat-isfying certain dissipative, splitting and superpositionprinciples. Note that this recombinatorial property doesnot prohibit the splitting or de-coherence of physicalfields, say objects.

In order to assess the structure of a dynamic scale-space more advanced syntactic and/or statistical op-erations primarily retained from the correspondingrecursion operators have to be applied. An exampleof such a syntactic operation is counting how manyfaces are coming together at a vertex point and howthis number changes upon applying the scale operator.

Another operation could be a quantification of thescaled physical field dynamics, i.e., the derivation ofequivalences, local and non-local conserved densitiesor curvatures of the evolution of the fields equations.This operation gives insight into the formation pro-cess, i.e., the cutting, pasting and entanglement pro-cedures, involved in the physical field dynamics. Thelatter implies that on changing view, e.g., the (scaled)recursion operators and other syntactic operators willgradually reveal (partial) correspondences between theviews, in particular of conjugated subimages. Statisti-cal operators, i.e., decision rules, in combination withthe dynamic scale-space can be used to discern betweendifferent input images, and qualify and quantify theirdifferences and probabilities for a scene in a categor-ical sense through the needed morphisms to changethe (scaled) field dynamics. Summarising, a dynamicscale-space paradigm is a model for processing an in-put image of physical fields such that a particular visiontask can be accomplished that is consistent with a scaleoperator clinging to the essential external and storedphysical field dynamics, an appropriately encoded in-put image of induced dynamic currents and also anarsenal of syntactic or statistical operators to describethe dynamic scale-space.

Our aim in this article is to substantiate, unify andextend existing scale-space paradigms [196, 237]. Wepresent a physical framework of a dynamic scale-space paradigm that can unify and extend the exist-ing ones. Our paradigm should allow reproducible andstable measurements on a dynamic scale-space de-spite small scale perturbations over an ensemble ofrealisations of the input image, for which the fluctu-ations of the induced or external field dynamics andthe internal noise cannot be resolved by the visionsystem during the acquisition phase of the input im-age. Perturbations are normally associated with a cer-tain type of noise. But the type of noise can be quitedifferent over an input image caused by, e.g., chang-ing interactions between illumination and inhomoge-neous surface properties over an object. The frame-work of our paradigm should be capable to incorporateor capture the spatio-temporal and dynamic inhomo-geneity in the field dynamics. The latter requirementmeans that the cutting, pasting and entanglements ofthe external and induced field dynamics is read out andquantised at different scales, fed back into the system,and coupled to the stored field dynamics. One couldspeak of a partial operationalising of the gauge grouppotentials.


In trying to reach our goals only physical (inductiveor field driven) and geometric (deductive or conceptdriven) reasoning is followed. For example, in theanisotropic scale-space paradigm the discontinuitiesin the input image are considered to be the essentialphysical objects, whereas in the geometric scale-spaceparadigms the homogeneous grey-valued regions ho-mogeneous are. It should be clear that the differencesbetween the various paradigms can be expressed interms of the non-commutativity of the corresponding(scaled) recursion operators and resides in the prod-uct topology and geometry of the observation spacesgenerated by them. All the latter implies that the struc-ture of the dynamic scale-space retained by means ofrecursion operators deviates quite significantly fromparadigm to paradigm.

In constructing the appropriate paradigm the opera-tionalisation of an induced field dynamics faithful to theexternal and stored one is crucial. For example, the elec-tromagnetic dynamics on the detector arrays engendersa certain dynamics of the vision system. The goal of avision system is to unravel the induced radiation signalssuch that the dynamics resulting in the system stronglycorrelates with and mimics that of the radiation fieldoutside. The question then arises how, in general, tocling to the dynamics of the electro-magnetic radiationfield dynamics.

We address this issue by making explicit the phys-ical principles involved in our dynamic scale-spaceparadigm [196] yielding multi-scale representations ofthe input image of the induced field dynamics. Mod-ern geometric and statistical physical concepts will beused in formalising our procedural approach. Our dy-namic scale-space paradigm is based on two insightsacquired in biology [92], computer vision [195, 196,199], physics [125] and mathematics [38, 173]:

• Directed and oriented linked or branching circuitsor path integrals of physical observations on the de-tector array (concerning their self- or mutual inter-actions between external, induced and stored fielddynamics) can reveal the equivalences involved inthe input image of the field dynamics that is invari-ant under a gauge group,

• Dynamic scale-spaces of an ensemble of similarlyprepared input images of the external field dy-namics contain stable and reproducible representa-tions above some scale or throughout the dynamicfiltering process as stored dynamics. The dynamicprinciple involved in the filtering process is empir-ically and theoretically modeled by a topological

interaction mechanism analogous to those for elec-tromagnetism and gravitation. The representationsare equivalences, local and non-local conserved den-sities or curvatures associated to the exchange or fil-ter principle which in turn is captured in a systemof integral or PDES provided with a suitable set ofIBC.

The applied path integrals, yielding equivalencesof image formation, and the dynamic scale-spaceparadigm are stipulated to be invariant under a cer-tain gauge group. Among the gauge groups consideredin low level vision are those of Euclidean, affine, pro-jective, Galilean or Lorentzian transformations, that ofanamorphoses, homotopies or contrast transformations(monotonic and local dynamic order preserving grey-value transformations—normally grey-value transfor-mations are monotonic and preserving global dynamicorder). On the basis of data and concept driven groundsthen frame fields, co-frame fields, connections and met-rics on observation space, associated (gauge group in-variant) equivalences, statistical partition functions and(scaled) recursion operators have to be derived. Theselow level vision approaches lead to two types of dy-namic scale-space paradigms:

• Conceptual paradigms that are merely conceptdriven, i.e., the topology, geometry and dynamics ofthe vision system, a-priori postulated by an end-user,determine the gauge group,

• Data driven paradigms that are besides concept alsodata driven, i.e., the topology, geometry and dynam-ics of the external, induced and stored physical fieldsdetermine the gauge group.

In this classification the difference between passiveand active transformations is eminent [64, 66]. Pas-sive transformations (conceptual) have to be thoughtto be living on the level of the mathematical modeland thus solely on that of the gauge group, whereas theactive transformations (data driven) besides the pas-sive ones include also transformations not capturedby the gauge group such as morphisms due to, e.g.,changes in the relative resolution characteristics of thevision system with respect to the observed outsideworld and perturbations by noise. Having an image ofa physical field, e.g., a radiance field, one is confrontedwith an intensive physical observable. Increasing, e.g.,the resolution properties may cause non-trivial mor-phisms of the the physical observable not being cap-tured by the gauge group assumed to be living on the


higher resolution input image. However, scale-spaceparadigms do provide mechanisms allowing us to for-get about the morphisms involved in certain activetransformations ensuring (partial) equivalence of im-ages [196, 201, 206]. In order to explain and predictcertain high-energetic phenomena like the existenceof the top quark one proposes in chromodynamics aconsistent renormalisation theory [9]. Contrary to thescale-space paradigms such a theory permits us to re-veal physical objects that are normally hidden to oursenses. Nevertheless, the scale-space paradigms can beadjusted to the latter theory [206]. Therefore, we ad-dress in this paper the following question:

• What is the mathematical, physical and logicalframework underlying a scale-space paradigm ingeneral to retain a robust and concise image descrip-tion and understanding of the exteral, induced andstored physical field dynamics, in particular of theoptic field dynamics?

Another question, as crucial as the preference for acertain dynamic scale-space paradigm, is the follow-ing:

• How to describe the deep structure of dynamic scale-spaces?

This question is a fundamental one in pattern recog-nition and image processing, because it boils down toa model consistent segmentation and interpretation ofimages in terms of interacting coherent image struc-tures. For the linear scale-space paradigm this problemhas been satisfactory solved by means of various dis-ciplines among which modern geometry [195], homo-topy theory [109, 199, 206], cohomology theory [199,206, 235, 236], topology [130, 196] and Morse the-ory [54, 68, 196]. In mathematical morphology [223]similar techniques are used as in the abovementioneddisciplines [205].

Besides the above fundamental question an equallyimportant question concerns the classification of thedynamic scale-space paradigm:

• What are the equivalences of a dynamic scale-spaceparadigm?

This questions not relates to the description of thedynamic scale spaces but to a classification of theparticular paradigms activated as PDES with IBC.

The latter question is hardly ever addressed in com-puter vision except in [206] and gives insight in thelogical framework underlying a dynamic scale-spaceparadigm.

Having a framework for deriving a scale-spaceparadigm another issue to be dealt with is the following:

• How could we unify scale-space paradigms?

This presupposes that it is possible to lay bare aprocedure to follow in order to set up any dynamicscale-space paradigm or to go from one paradigm toanother.

One might wonder about or doubt the uniquenessor the existence of so-called fundamental equations ofimage processing [2, 3, 4, 5]. In this context it’s worth-while to emphasise that in computer vision especiallyscalar fields (zero-forms) of the induced physical dy-namics are studied. Mainly isophotes, i.e. space-timeregions of constant intensity, or derived scalar fieldsconceived as graphs of ordinary functions are consid-ered for the derivation of scale-space paradigms. In par-ticular in optic flow and stereo vision, in which suchconditions as the optical flow constraint are advocated[67, 142], such fields seem to be essential. But real-ising that the scalar fields are caused by an intensivephysical entity [196] subjected to various non-trivialtransformations with change of view it’s clear that sucha constancy assumption does not apply in real visionand has only a very limited practical value. The moststandard transformation is covered by the group of(diffeo)morphisms of the input image, i.c. an energy-valued density field, in a spatio-temporal but also in adynamical sense [195, 196 198, 205]. The above funda-mental paradigms are developed for scalar fields. Whatabout dynamic scale-space paradigms for the cutting,pasting and entanglements in constructing the inputimage of the physical field dynamics as noted first in[196]. In this case the physical object of interest is nota scalar field but a density field that operationalisesthe inhomogeneity of the group actions involved in theimage formation. The latter operationalisation may bewith respect to inhomogeneities in a grey-valued im-age or to those in a space-time configuration such as arandom lattice endowed with a specific metric and/oraffine connection defined by edges, ridges, courses andinflection lines. Summarising, we address in the sequelalso the following questions:

• How could we extend scale-space paradigms? Whatare dynamic scale-space paradigms for equivalences


of intensive physical entities represented by vectordensity fields like curvature and torsion two-formsand topological invariants well known from moderngeometry and differential topology? How to incorpo-rate invariance under the group of (diffeo)morphismsinto a scale-space paradigm, thereby, extending ex-isting mathematical morphological and regularisedanisotropic scale-space paradigms to covariant ortopological scale-space paradigms?

The extension in this context partly concerns the trans-formation under which the paradigm remains invari-ant. This question about extensibility is also relatedto issues appearing in renormalisation theory [9, 206,268].

Our paper is organised as follows. In Section 2 wegive an overview of existing scale-space paradigms em-phasising their motivation and applications in low levelvision. It’s not intended as a review, but rather as areference frame to illustrate the differences and cor-respondences between our paradigm and the existingones.

In Section 3 we will reveal the basic ingredients ofour framework for obtaining a dynamic scale-spaceparadigm. Firstly, in Section 3.1 the initialisation ofa frame field, co-frame field, metric and connection,all supported by a gauge group, is treated for external,induced and stored physical field dynamics. On thebasis of those mathematically, geometrically and sta-tistically modelled physical machines joint (non)localequivalences of the physical field dynamics are pre-sented. Conservation laws or superposition principlesfor those equivalences are subsequently retrieved. Ex-amples of equivalences are given and their use in edge,ridge, course and inflection line detection is demon-strated. Secondly, our dynamic scale-space paradigmis presented in Section 3.2. It is based on the derivationof a statistical partition function for the found equiv-alences in Section 3.1, and on the application of anexchange principle for a free energy related to the par-tition function that ensures a conservation law just forthat energy. Next a paradigm consistent segmentationof the dynamic scale-space of the image and a paradigmclassification is carried out.

In Section 4 the unification of the existing scale-space paradigm through our paradigm is demonstrated.

Finally, in Section 5 it’s shown how our dynamicscale-space paradigm extends existing scale-spaceparadigms. In particular we show how covariant ortopologically invariant scale-space paradigms can beretained.

2. State-of-the-Art

In the sequel a survey of existing (dynamic) scale-spaceparadigms is presented, including:

• Quad tree scale-space paradigm• Pyramid scale-space paradigm• Linear scale-space paradigm• Wavelet scale-space paradigm• Anisotropic scale-space paradigm• Total variational scale-space paradigm• Morphological scale-space paradigms• Lie-theoretic scale-space paradigms

All these paradigms will appear to be covered by ourdynamic scale-space paradigm (see Section 4). Notethat we do not intend to give an extensive overview ofthe whole field of scale-space theories.

2.1. Quad Tree Scale-Space Paradigm

The quad tree paradigm [234], the seemingly simplestscale-space paradigm, splits an image into subimagesand merges subimages according to a user-defined rulebased on, e.g., the standard deviation of the grey-level values within a region in the image domain.This paradigm is motivated by the observation that thesubimages can be characterised by relatively uniformgrey-level values over certain regions of the image do-main. The degree of the tree, i.e., the number of subim-ages, and the fineness of the segmentation of the imageheavily depend on the chosen pre-specified thresholdfor the standard deviation and on the spatio-temporaldynamic resolution properties of the vision system. Ifthe thresholding is conceived as a scale operator, thenchanging the threshold will yield a multi-scale tree rep-resentation of the image.

2.2. Pyramid Scale-Space Paradigm

The pyramid paradigm [29, 44, 50, 93, 107] reduces theimage size by subsampling. This paradigm is motivatedby the fact that an image is the union of subimages,which consist themselves of subimages and these inthe end of one pixel. The pixels at higher scale are ob-tained by attributing a weight to the information contentof the initial pixels within a user-defined space-time re-gion of the image domain and summing these weightedpixels to quantise the coarser scale pixel information.An ensemble of such weights determine the coefficients


of a filter, i.e., a scale operator. This filter is normallysubjected to several constraints: the coefficients shouldbe positive-definite, unimodal (they are equal or de-creasing upon increasing the distance from the filter’scenter), symmetric (they are point symmetric with re-spect to the center of the filter), normalised (they sumup to unity) and ensure that all pixels contribute equallyto all levels. Such a filter can in turn be used for per-forming image algebra over several levels of scale inorder to achieve feature detection and data compres-sion. E.g., computation of the difference between twolevels of a pyramid yields a bandpass pyramid that ex-hibits the morphology of the input image under thepyramid paradigm.

2.3. Linear Scale-Space Paradigm

The linear paradigm [2, 13, 45, 63, 94–102, 127, 128,138, 182, 200, 257, 258, 263] is governed by the or-dinary linear isotropic diffusion equation (or a (semi)-discrete version of it) satisfying specific IBC. It em-beds the initial input image in a continuous or (semi-)discrete) multi-parameter family of smoothed imageslike the pyramid scale-space paradigm does, but retainsthe same spatio-temporal resolution at all scales, i.e.,down sampling does not take place as in the quad treeparadigm. The information content of the input imageis redistributed over the image domain at every level ofscale in the same manner (scale invariance), such thatthe scaling of the input image (smoothing or relaxation)performed by the vision system is spatio-temporallyhomogeneous (independent of position in the image do-main), isotropic (independent of orientation in the im-age domain with respect to a chosen reference frame),linear or grey-level shift invariant (smoothing not influ-enced by input image; weights of filter are only a func-tion of the spatio-temporal image domain and scales),causal or deterministic (a smoothed version of the in-put image can be obtained immediately from the inputimage by applying a filter appropriately scaled), satis-fies a maximum-minimum principle ensuring the one-parameter family of the input image and its derivativesto be between their infima and suprema, and last but notleast preserves the total flux within the image domain.In order to find the Green’s functions for the above dif-fusion equation with IBC, that are the filters to applyto the input image in order to realise the one-parameterfamily of smoothed images and their derivatives, themost immediate approach is by imposing a conserva-tion law for the total flux within a region of the image

domain and by imposing a similarity constraint [195].Note that, applying the backward diffusion equationwith initial-boundary condition is ill-posed [162]. Thelatter remark does not mean that one cannot relate theimages of a linear scale-space [130], for it is possibleto store what’s happening to the individual pixel in-formation upon de-focusing or increasing scale. Thebackward equation, besides not satisfying a conserva-tion law, introduces locally high frequency oscillationsof the input image upon increasing scale, because theweights of the corresponding filter in the Fourier do-main are emphasising instead of suppressing the highfrequency fluctuations in the image. It’s clear that forinfinite high scales, together with the numerical insta-bilities in the imaging device such a filtering processyields nothing but the highest frequency content of theinput image in a non-measurable way.

A physically sensible realisation of a linear paradigmcomes about by a discrete formulation of the linearscaling, for then the number of operations to be per-formed by a vision system and needed for smoothingremain finite [138, 195], and analysis of the geom-etry and topology of the input image under the lin-ear scaling becomes computationally feasible. Further-more, the linear scaling in that case is not instantaneousand does certainly not influence objects infinitely faraway: the transport velocity of information over theimage domain intrinsic to the scale operator realised inthe imaging device is clearly finite. However, becauseof the discretisation the above isotropy and alike re-quirements have to be adjusted to the symmetries ofthe underlying lattice structure. Taking the appropriatecontinuum limits yield still the related semi-discreteand continuous theory.

The main reason for applying a linear scaling likefor the other paradigms is, as already mentioned inthe introduction, the desire for robustness of measure-ments with respect to large dynamic scale structuresin the input image in a quantitative and qualitativemanner above some level of scale, irrespective smallscale perturbations of the input image. Normally onedoes not know anything about the actual image for-mation process in advance. Assume the linear scale-space paradigm to hold for the scaling behaviour ofthe input image and that of the external field. Now itis more than reasonable to allow the pixels to interactaccording to that paradigm in order to establish corre-lations and connections between the images over scale.These correlations or connections, subsequently, indi-cate to what extend and how images can be attached


over scale. Assessment of these correlations and con-nections is assured by the fact that the spatio-temporalsupport of the Green’s functions at different positionsin space-time and over scale can effectively partly orcompletely overlap.a In order to describe now a lin-ear scale-space of an input image in terms of its mostessential intrinsic symmetries (building blocks) underthis linear scaling use is made of singularity theory [45,126, 138, 182], logical filtering [130], algebraic invari-ance theory [191, 197, 201], differential and integralgeometry [63, 106, 192, 195, 197] and algebraic ge-ometry [135]. These disciplines deliver the tools forconstructing syntactic operators for reading out thecontent of the linear scale-space. Nowadays the lin-ear scale-space paradigm is applied in various areasof computer vision, e.g., pattern recognition [20, 191,220, 231, 232], optical flow and stereo vision [61, 67,70, 71, 137, 139, 140, 142, 158]. Because the quad tree,the pyramid and the linear scale-space paradigm yielddifferent scale-spaces under spatio-temporal grey-value transformations and non-Euclidean transforma-tions of the input image other scale-space paradigmswere developed to counteract these shortcomings (seeSection 2.4–2.8).

2.4. Wavelet Scale-Space Paradigm

The wavelet paradigm [103, 144, 221, 222] yields amulti-scale analysis of the input image in terms of aset of wavelets for a given Lie group action on theimage domain, such as the Euclidean, the similarity,the affine, the Lorentzian, the projective group actions.In order to realise this decomposition an appropriatescaling function (kernel) should be derived that formstogether with its transformed versions under the Liegroup action a so-called partition of the unity opera-tor. Moreover, a measure invariant under the Lie groupaction is needed as integration measure. The decompo-sition is obtained by computing the cross-correlations(weighting) between the input image and the trans-formed and scaled wavelets. Data compression can bestraightforwardly achieved by storing only those cross-correlations that are predominant at those positionsand scales where the most important events in the in-put image occur. Only recently in [42, 47] nonlinearwavelet-like paradigms were introduced that captureall scales, locations, and orientations for near-optimalcompression and de-noising of objects with discontinu-ities along curves. However, the lifting procedure canbe clearly conceived as a lifting in an integral geometric

manner that will be covered by our paradigm (see Sec-tion 3.2).

Although overlap of the wavelets in general does notoccur this does not mean that the correspondence prob-lem is resolved. Furthermore, the wavelet paradigmsmentioned above except for [47] do not directly coupleto the induced field dynamics which implies that thecross-correlations depend on the group actions and onthe wavelets constructed with respect to the referenceframe adopted for the spatio-temporal image domain.Finally, it’s doubtful whether a Lie-group action is suf-ficient to describe the typical non-integrable processesinvolved in the field dynamics that obviously ask formorphisms to jump over subimages.

2.5. Anisotropic Scale-Space Paradigm

The anisotropic paradigm [7, 41, 47, 75, 159, 171, 219,252–256, 260, 261] yields a scale-space that is gener-ated by a nonlinear diffusion equation with IBC. Herethe redistribution of the image content over the im-age domain is influenced by the local image informa-tion, in particular the local image gradient field andthe second order differential structure of the input im-age. The image gradient field and second order imagestructure can determine a diffusion tensor which in turncan control the flux of the information through an areaelement. This controlled diffusion yields enhancementof edges, corners and textures, and piecewise smooth-ing of coherent structures such as grey-values in theinput image. In order to ensure well-posedness of theparadigm, i.e., there exists a unique and stable solutionfor the initial-boundary value problem, a regularisationof the diffusion tensor is normally carried out [41, 251,261], which introduces additional free parameters. Asthe paradigm is applied to the input image, normally adensity field, one ends up with a constant image after in-finite scaling; this property also holds for the quad tree,pyramid, linear and wavelet scale-space paradigms. Toallow for non-trivial infinite scale images constraintsare normally imposed (see Section 2.6–2.8).

The anisotropic scale-space paradigm is applied, forinstance, in texture segmentation [261] and medicalimage analysis [41].

2.6. Variational Scale-Space Paradigm

The total variational paradigm [151–153, 160, 161,180, 181, 189, 190, 224, 225] yields a scale-space of


reconstructed images of the input image through min-imising an energy functional consisting of, e.g., threecost terms, viz. a deviation cost, a stabilising cost andan edge cost with respect to the desired reconstructedimage, and the diffusivity or the total edge length. Thecontrol parameters in this variational paradigm are as-sumed to be free parameters independent of the inputimage. The goal is to stay in some sense as close as pos-sible to the input image and the edge information, andto obtain a desirable representation for further analy-sis. As the paradigm involves also the specification ofLangrange multipliers, this specification can be usedas scale operators to obtain a multi-parameter scale-space. The variational paradigm is applied in order toachieve, e.g., edge detection [181], matching stereo im-ages [226], and image sequence analysis [180, 226].

2.7. Morphological Scale-Space Paradigms

The morphological scale-space paradigm [2–6, 8, 25–27, 30, 31, 55–57, 65, 84, 85, 115, 120–122, 124,131, 145, 149, 164, 165, 167–169, 185, 186–188,194, 195, 199, 212, 213, 214–218, 223, 241–245,265] yield scale-spaces that are governed by specificclasses of (non)linear PDES with IBC. The morpho-logical paradigms found in [76, 84, 86, 145, 223] arebased on size density estimators [227, 228], statisti-cal and Bayesian techniques [73, 266], parabolic di-lations [241, 246] and watershed methods [17, 18,148, 179, 249]. Up to recently these filtering tech-niques did not make explicit the used gauge groupdespite the intimate relationship with integral geom-etry [135, 145, 211]. It is assumed, e.g., in the caseof the size density estimators that the geometry livingon the image domain is Euclidean, whereas the imageitself coupled to the external physical field dynamicsmay require the induction of a far more complex dy-namics on the induced and stored physical field dy-namics. Furthermore, the morphological filtering tech-niques based on watersheds and parabolic dilations donot take full advantage of the dynamics latent in theimage formation as the more geometric paradigms do[195, 196, 198, 205, 237]. As in the linear paradigmthe specificity of the latter paradigms is determined bythe postulates in Section 2.3 concerning the inducedfield dynamics. These postulates are causality, invari-ance under spatio-temporally homogeneous contrast(luminance) changes (often confused with homotopicinvariance or invariance under the group of anamor-phoses which preserve the spatio-temporal inclusion

relations together with the dynamic ordering relations),Euclidean invariance, affine invariance, projective in-variance, and conservation of integral invariants, suchas lengths and areas, or even topological invariants. Onthe basis of these features conservation laws and globalconstraints can be stated as in the variational paradigmin Section 2.6.

On the basis of these requirements the classes offiltering can be formulated in terms of grey-values orin terms of metric relations between a set of events of in-terest in the image domain extracted from a grey-valuedinput image. Contrary to the linear and wavelet scale-space paradigms, but in analogy with the regularisedanisotropic and variational scale-space paradigm, ittries to couple to the input image in an immediate man-ner, i.e., the scale operator inherits some informationabout the field dynamics in terms of a frame field, co-frame field, metric, connection and equivalences. As inthe linear paradigm these nonlinear scale-spaces can bedescribed by means of algebraic invariance theory [191,197], differential and integral geometry [195, 199], sin-gularity theory [182] and topology [123, 130, 199]. Theabove mentioned nonlinear scale-space paradigms areapplied to image sequence analysis [3] and image seg-mentation and reconstruction [40, 143].

2.8. Lie-Theoretic Scale-Space Paradigm

The Lie-theoretic scale-space paradigm [203] is basedon the requirement to transform a vision task into amore suitable one by incorporating diffusive objectsinto the scale operator, such that certain sub-tasks inthe original task are immediately accomplished with-out applying additional syntactical operators. Theseparadigms are essentially nonlocal and characterisedby so-called potential symmetries [22]. The morpho-logical scale-space paradigms could be extended con-siderably in this manner.

3. Physical Framework

We treat our dynamic scale-space paradigm that is(un)committed to the physical field dynamics. A properdescription of this dynamics is indispensable before awell substantiated image analysis and processing canbe carried out that retrieves stable and reproducibleequivalence relations of the input image, i.e., proper-ties of the image invariant under a specific group or setof transformations.


3.1. Initialisation

The first setup of our framework consists of an initiali-sation of the field dynamics in terms of so-called equiv-alences. This initialisation requires a thorough knowl-edge of the underlying physics and a field correlatedmeasuring device, i.e. the output of the detectors is,despite the losses, a reasonable abstraction, i.e. rep-resentation and quantification, of the physical fields.This requirement boils down to the construction of aframe of reference adapted to a particular gauge group.In order to arrive at a field dynamics consistent set ofequivalences in the sequel we elaborate on the notionof an image, a gauge group, vector bundle, tangent bun-dle, bundle metric, fiber transformations, frame field,co-frame field, metric tensor, connection, torsion andcurvature [46].

3.1.1. Image. Let us mathematically model an im-age I as a mapping of a vector-valued energy-densityfield (current) of the external field dynamics M onto avector-valued density field of the induced field dynam-ics N of the vision system:

Definition 1. An image I is defined by a mapping:

I : M → N ,

where M is an external field dynamics and N an in-duced field dynamics on the vision system.

In the above definition M can be the external electro-magnetic field dynamics fallen onto the vision system,whereas N can be conceived as the induced vision sys-tem dynamics given the induction operator I whichhides the interaction or entanglements of external, in-duced and stored physical field dynamics. Note that I ,M and N form a mathematical physical model for thespace-time and the external, induced and stored phys-ical field dynamics.

The image I can be a superposition of several otherimages of physical observables. The camera systemcould analyse in addition the fine structures of the en-ergies as function of the frequency of the light fallenonto the set of detector arrays [82], as function of thechirality (left- or right-handedness) [87, 134] and/oras function of their polarisation states spectrally, en-ergetically and angular-temporally [264]. It could per-form such measurements for a temporal sequence ofstereo-images IL and IR . In this context the bi-reflectivedistribution function of object surfaces [264] does not

need to satisfy any symmetry relations. Furthermore,the perceived equivalences of these fine structures ofan image, shortly to be defined, depend on the illu-mination field, reflection and absorption properties ofsurfaces and media between sources and surfaces, andbetween surfaces and camera system.

3.1.2. Vector and Tangent Bundle. The notion of avector bundle over an image (Definition 1) will shortlyplay a crucial role in establishing an image consistenttangent bundle.

Definition 2. A r -times continuous differentiable Cr

vector bundle E of rank k over a differential manifoldIM,N is a manifold together with a projection π : E →IM,N satisfying:

• Existence of k-dimensional vector space V , such that∀p ∈ IM,N , the fiber over p is E p = π−1(p) is areal vector space isomorphic to V .

• Each point p ∈ IM,N is contained in some open setU ⊆∈ IM,N , such that there is a Cr diffeomorphism�U : π−1(U ) → U × V with the property that �U

restricted to the fiber E p maps E p onto {p} × V .• For any two such open sets U , U ′ with U ∩ U ′ �=, the

map �U · �−1U ′ : (U ∩ U ′) × V → (U ∩ U ′) × V

is a Cr local vector bundle isomorphism over theidentity.

Note that r can be infinite in particular if the imageformation is the result of a testing in a distributionalsense, because image, (Definition 2), is modelled as adifferential manifold, i.e. PDES with IBC.

Definition 3. The tangent bundle π : TIM,N → IM,N

is a C∞ differential structure on TIM,N i.e. the unionof all tangent spaces to IM,N .

Again the explicit form of the tangent bundle de-pends on the particular image, (Definition 1). For ex-ample, if it is a two-dimensional grey-valued image notaffected by the group of Euclidean movements the lo-cal tangent spaces are most concisely represented bytranslation and rotation vectors and intensity scalings.

3.1.3. Gauge Group. Now our dynamic scale-spaceparadigm is based, first of all, on the derivation of aset of equivalences of the image (Definition 1) thatare invariant under a gauge group, a group or set oftransformations freely acting on the underlying physi-cal objects being the external and stored physical field


dynamics, and the induction I . E.g., the sensitivity andresolution properties together with the view of an en-semble of vision systems may be different asking foradaptive analysis and processing of the input data suchthat weakly equivalent input data representations can berealised. The latter representations can be connected byLie group, Lie pseudo group, semi-group or monoids.

Definition 4. A gauge group G consistent with image(Definition 1) is a group or set of active or passive self-transformations of the external physical field dynamicsM , the induction I and the induced physical field dy-namics N that leaves invariant the corresponding vectorbundle’s fibers (see (Definition 2)) associated a specificbundle metric g and/or connection � over the image,(Definition 1).

Here the specific bundle metric and connection (seefor definitions next sections) are still left open, as tomake them explicit asks for an explicit definition ofimage, (Definition 1). For example, a projective metricand connection in a Euclidean setting for the imageis always permitted but so are affine ones. Note thatin our definition of the gauge group we specify it asa group or set of transformations, whereas the gaugetransformations can be the solution of PDES with IBCor even monoids. The latter implies as in the case ofa renormalisation or filtering process that invertibilityof the group action need not to be satisfied. Shortlywe return to these complications due to gauging andrenormalisation (for references on this issue see also[9, 206, 268]).

A discretisation of image (Definition 1), i.e., a sam-pling of the vector-valued energy-density field N ,which is invariant under a gauge group, can be ob-tained by aggregates of detectors satisfying related dis-crete gauge symmetry group requirements [30, 150,196]. Among the gauge groups considered in com-puter vision and mathematical morphology are [195,196, 199, 223, 237]

• Groups of Euclidean movements: the semi-directproduct of the translation group and the rotationgroup on Euclidean space,

• Groups of (unimodular) affine movements: the semi-direct product of the translation group and the gen-eral linear group on (unimodular) affine space,

• Groups of projective movements: central perspectivetransformations of planar objects onto planar imag-ing domain can be covered by the projective groupon the plane,

• Groups of Galilean movements: the semi-directproduct of the group of temporal shifts and the groupof Euclidean movements,

• Groups of Lorentzian movements: the group ofspatio-temporal transformations preserving theMinkovski metric ds2 = dx2 − dt2,

• Groups of conformal transformations: metric gchanges under the group as γ = f 2g where f isa definite positive or negative function on the do-main of the input image (angle relations are beingpreserved),

• Groups of anamorphoses, homotopies, contrasttransformations, diffeomorphisms: groups of imagedeformations in a spatio-temporal as well as a dy-namical sense preserving spatio-temporal inclusionand dynamic order relations (classical length andangle relations are not preserved but the inclusionand dynamic order relations can be used to define agauge invariant metric and/or connection); Groups ofhomeomorphisms: spatio-temporally continuous in-vertible dynamic order preserving grey-value trans-formations, in particular, of spatio-temporally anddynamically discrete grey-valued images,

• Groups of similarity transformations or scale trans-formations: groups defined by spatio-temporal ex-change principles for image equivalences of M , Nand I , that are captured in terms of PDES with IBCgoverning the macroscopic scaling behaviour of thephysical field dynamics (see Section 3.2),

• Renormalisation groups defining the breaking ofsymmetries explaining the popping up of new topo-logical or geometric charges in observations [9, 206,268] below inner scales and above critical physicaldynamics (such a group of transformations under-lies the physical field dynamics behaviour at superresolutions).

The action of gauge group G on M , N and I thencan either yield, as discussed in the introduction, apassive or active transformation of the induced stateof the vision system N and of the derived observa-tions. Passive transformations concern mainly defor-mations of the mathematical representation of N , Mand I , whereas active transformations include non-diffeomorphic transformations of them. Except forthe two last gauge groups above all the other groupsgenerate passive transformations. However, most ofthese groups are also active but can readily be trans-formed into passive ones. The similarity as well asrenormalisation group consist of active transformations(morphisms) of the image that cannot be viewed as


deformations of an initial image, i.e., M , N and I .Thus active transformations can be caused by a partic-ular image analysis and processing paradigm carriedout by the vision system through a particular similaritygroup action. Besides by the above paradigm they canalso come about by an (discontinuous) inhomogeneousLie group action on M , N and I due to morphisms ofthe resolution and sensitivity characteristics and posi-tion in space-time of the vision system, and due to per-turbations of the external physical field dynamics thatcan be covered by renormalisation groups. Groups ofsimilarity transformations, however, allow us to forgetpartially about morphisms caused by such perturbativeactive transformations, and to cling to the relevant andsemantically meaningful (for the autonomous system)induced meso-scopic and macroscopic physical fielddynamics. The relevant dynamics may then be capturedby so-called equivalences persistent above a particularscale, local and non-local conserved densities or cur-vatures consistent with the similarity group invariantunder the corresponding gauge group.

In order to retain such equivalences for above-mentioned gauge groups the reader is referred to [28,33–39, 58–60, 72, 80, 104, 163, 166, 170, 175, 238].The goal of these methods is to retain physical objectsthat are invariant under the gauge group. For specificapplications of such methods in computer vision see[62, 129, 150, 165, 191, 192, 196, 199].

The above-mentioned equivalences of the (induced)external physical field dynamics come about after set-ting up a (co)frame field, metric and/or connection in-variant under a gauge group (Definition 1) of interest toa specific vision task. In the sequel we formally presentthese geometric attributes to arrive at gauge invariantphysical objects, i.e., equivalences, represented by theinduced physical field dynamics N that we use in a dy-namic exchange principle to be treated in Section 3.2.

3.1.4. Frame Field. The product space, MI × N ofimage (Definition 1) can be associated a frame field[195, 196, 199]:

Definition 5. A frame field (vp) = (xi , e j , lk) is asection of the tangent bundle T (MI × N ) of image(Definition 1).

By exponentiating the frame vector fields xi , e j andlk one obtains a parametrisation of space-time occupiedby the vision system and a parametrisation of the exter-nal, induced and stored optic field dynamics. The firsttwo sets of frame vector fields form, e.g., a section of the

Galilean group on space-time. Another section encoun-tered in scale-space theories are the local Euclideangroup of movements along isophotes and flowlines ina two-dimensional grey-valued image. The third setvector fields generates the intensity field.

3.1.5. Co-Frame Field. Besides a frame field there’salso realised a co-frame field to operationalise the ob-servations related to the frame fields.

Definition 6. A co-frame field (dv p) = (dxi , de j ,dlk) of frame field (Definition 4) is a section of thecotangent bundle T ∗(MI × N ) on image (Definition 1).

The co-frame fields accepting, for example, theframe fields of the previous section allow measurementof time shifts, translations and rotations, and dynamicalvariations. Frame fields you can conceive as the sticksthat allow you to measure entities like heights throughthe use of corresponding co-frame fields.

A frame field (Definition 4) and co-frame field(Definition 5) then satisfy not necessarily a dualityconstraint:

Definition 7. A frame field (Definition 5) and the co-frame field (Definition 6) are their duals, if and onlyif:

dv p(vq) = δ pq ,

where δ is the Kronecker delta-function.

All possible frame fields and their duals define nowin principle gauge group (Definition 1). The specificgauge group, operationalised by the vision system, isinduced by the external physical fields fallen onto thedetector arrays, and is coupled to the statistics of thestored dynamics of the vision system. For examplesof gauge groups being the direct product of a classi-cal group and the group of anamorphoses the reader isreferred to [16–18, 148, 179, 196, 223, 237, 241, 246,249]. Another important gauge group is the group ofdiffeomorphisms of the image [195, 196, 205]. Notethat relaxing the invariance conditions involved in agauge group reduces the number of possible identi-fiable and distinguishable physical objects in the in-put image considerably. E.g., if the gauge group coin-cides with a spatio-temporal volume preserving groupof diffeomorphisms, then the only objects retrievablein a gauge invariant manner are those spatio-temporalregions bordered by discontinuity sets, (non)isolated


singularity sets, ridges and courses [132, 195] (see also(Example 2)).

In practical problems, however, like that for stereoand optic flow, even such a large gauge group asthe group of diffeomorphisms is not sufficient, be-cause consecutive images in time or in view can onlybe brought in correspondence through a morphismbecause of different scaling properties of the externalphysical field due to e.g. inhomogeneous surface prop-erties. But addressing the problem on the product spaceof images one can do away with these nuisances [195,196, 199, 205]. Only for different scenes, i.e., physi-cal field dynamics, such a categorical and morphologi-cal view is really indispensable and applicable. Thus agauge group asks for solving an equivalence problemthat concerns gauge invariance, to be defined shortly,of a physical object F in the input image that mightconsist of a stereo-image or even a time-sequence ofstereo-images. Clearly occlusions cannot be alleviatedby a group of fiber-preserving transformations [166]between two views. Moreover, such a group, as al-ready pointed out above, is not physical as the pairor sequences of stereo-views are different instances ofthe external physical field dynamics.

3.1.6. Metric and Connection. In order to relate andcompare local or nonlocal states of the vision systembesides a frame field (Definition 5) and the co-framefield (Definition 6) also a metric g and a connection� are necessary [195, 196, 199]. The latter geometricmachines allow measurements of distances, angles andenergies.

Definition 8. A metric tensor g is a (non-degenerate)bilinear form on T (M × N ) such that

g(vp, vq) = δpq ,

where δ is the Kronecker delta function.

Definition 9. A connection � on image (Definition 1)is defined by one-forms oq

p on the tangent bundleT (M × N ):

∇�vp = oqp ⊗ vq; oq

p(vr ) ∈ K,

where ⊗ denotes the tensor product, and ∇� is the co-variant derivative on image (Definition 1), and K a fieldof scalar numbers representing physical observationsrelated to Lie group actions.

Note that in a particular reference frame, e.g., aglobal Galilean coordinate frame the components ofthe metric may still be functions of coordinates (ex-ternal observer), whereas on the level of the framefield (Definition 4) and the co-frame field (Definition 5)these system dependencies are not communicated, i.e.one assumes the viewpoint of an internal observer[197].

In general the metric (Definition 8) and the connec-tion (Definition 9) are assumed to be compatible witheach other [195, 196, 199].

Definition 10. The metric (Definition 8) and the con-nection (Definition 7) are compatible if and only if:

∇�g = 0.

This means that, e.g., the angles between and lengthsof vectors measured by the metric tensor under the par-allel transport associated with the connection are pre-served. The advantage of having no compatibility is thatwe can detect mass-creation or disspation in terms ofdilation, shearing and rotational curvature currents (tobe encountered shortly) [195, 196, 199]. These energycreation currents can be derived by setting up a physicalfield dependent classical metric-connection on space-time and consider the evolution of the curvatures ofa physical field dependent connection not necessarilycompatible with the former metric and compute the en-ergy creation currents by means of equivalences, to beintroduced shortly, with time running. Note that com-patibility does not imply that observation space is tor-sion free; in general a connection on a Cartan–Einsteinmanifold is curved and twisted [195, 196, 199].

At discontinuity, singularity and bifurcation sets ofan image (Definition 1) physical observations are path-dependent. Let us clarify this path-dependency by firstselecting a two-dimensional surface S, parametrisedby uk = uk(d, δ), on the set of energy-states of thedetector array, i.e., the jet bundle j N of order N ofimage (Definition 1) consistent with gauge group (Def-inition 4), and by taking g and h as vector fields gen-erating an infinitesimally small simply-closed circuitC = (p0 p1 p2 p3 p0) around p in S. Next let us study theframe field V = (ei ) consisting of a set of independentphysical observations ei , e.g., the image I itself or theimage gradient ∇ I . Now let us quantify the variation offrame field V at point p0 along path C = (p0 p1 p2 p3 p0)

with respect to the local frame Vp0 . In general the localframe Vp2 is not equal to local frame Vp2 . The change of


the frame field along the upper part C+ = (p0 p3 p2) ofthe circuit C generated by (dg)(δh) and its lower partC− = (p0 p1 p2) generated by (δh)(dg) defines curva-tures �i of the physical observations ei at the point pon S [38, 125, 196]. In order to quantify the physicalfield dynamics, in our dynamic scale-space paradigm,curvatures Vi of frame field (Definition 5) are read out.

Definition 11. The curvature Vi of a frame vector fieldin (Definition 5) at point p on a two-dimensional sur-face S parametrised by frame field (Definition 5) isdefined by:

Vi (p, S) =∮

C∇�vi ,

where the sense of traversing circuit C is chosen suchthat the interior of the circuit C on surface S is to itsleft.

Note that on S at point p Cartan’s affine transportis actually directed [38, 195, 196, 199]. The latter di-rectional aspect is in many textbooks on differentialand integral geometry neglected or obscured; forget-ting about it leads naturally to an unwanted averagingof morphisms. Furthermore, that the vision system isin general gauged for his own stored topological, geo-metric and dynamical intricacies. Whether the systemis initially in any sense curved or twisted only partly in-fluences the analysis, processing and interpretation ofthe physical field dynamics. Clinging to the latter dy-namics suffices to forget about or to correct for thosesystem characteristics.

Using Stokes’ theorem curvature (Definition 11) canbe expressed as:

Vi (p, S) =∫

Co⊂S∇ � ∧ ∇�vi =

∫Co⊂S

O ji v j ,

where Co is the interior of the circuit C in S, ∇�∧ thecovariant exterior derivative in which ∧ is the wedgeproduct consistent with metric (Definition 8) and/orconnection (Definition 9), and O j

i represent the cur-vature two-forms. These curvature two-forms also popup in the (Cartan) structure equations in case of a dif-ferential geometric treatment of the curvature of a con-nection [199].

In physics the above notions for connection and cur-vature forms are known as gauge potentials and fieldstrengths, respectively. They are on an equal footingwith the curvature and torsion tensors one encoun-ters in general relativity [83, 125]. In defect theory

[108, 125] they are known as so-called Burgers andFrank vector density fields for the inhomogeneity of thetranslation group action and that of the rotation groupaction, respectively. They should not be confused withthe curvatures that appear as invariant functions in aconnection or metric in equivalence problems as for,e.g., planar curves under the group of Euclidean move-ments [35, 59, 60, 166]. We advocate the former quan-tities as they are of an integral form and quite stableunder finite perturbations, whereas the invariant func-tions are very sensitive in particular to infinitesimallysmall scale perturbations. In (Example 2) we demon-strate that those curvatures allow us to retain meaning-ful and robust fore- and background dynamics corre-sponding to a region-based definition of edges, ridges,courses and inflection lines. In Section 3.2 the curva-tures and higher order equivalences become partiallyequivalent for an ensemble of input images finitely per-turbed by noise and even connected, above some scaletypical for the noise, through a pseudogroup of (dif-feo)morphisms. A pseudo-group in this context is aset of transformations governed by a system of par-tial differential equations or integral equations that cancover renormalisation groups. Note that we do not lookdown on differential invariants; they will be shown inSection 3.2 to be crucial in order to derive sensibleintegral operators reading out the equivalences of theinduced physical field dynamics.

From these curvatures we can in turn derive higherorder curvatures Vi; j1... jk by taking successively co-variant derivatives ∇v jl

with respect to frame vectorfields v jl . Together they form locally and directionallyequivalence relations that quantify the physical fielddynamics.

Equivalence 1. The local and directional equiva-lences of the physical field dynamics are given by:

Vi; j1... jk = ∇�v jk

· · · ∇�v j1

Vi ,

where; indicates taking a covariant derivative (see also(Definition 9)).

If there are symmetries, such as those of rigid Eu-clidean movements, underlying the physical field dy-namics, then it can be worthwhile to try to find theirreducible equivalences [38]. (Equivalence 1) allowsthe quantification of the homogeneity or coherence ofthe physical field dynamics [195, 196, 199]. They canbe used to locate coherent structures, as we’ll demon-strate in (Example 2), in the physical field dynamics.


If we consider a set of circuits, {C}, on a set of relatedsurfaces, {S}, through point p, then (Equivalence 1) atp satisfies obviously a local conservation law (superpo-sition principle) such that the directional informationis obsolescent.

Equivalence 2. The local equivalences of the physi-cal field dynamics are given by:

Vi; j1... jk (p, {S}) =∑{S}

Vi; j1... jk (p, S),

being total curvatures of the vector fields vi in framefield (Definition 5) over the set of surfaces, {S}, each ofwhich contains one corresponding circuit C, throughpoint p.

These local equivalences explain the problem of de-scribing, e.g., junctions solely on the basis of a localanalysis. Such a local analysis maps a sequence oftopological and geometric quantum numbers onto one[195].

The integral geometric conservation laws (Equiva-lence 1) and (Equivalence 2) appearing in differentialgeometry as Bianchi identities (curvature and torsiontwo-forms form together with the metric, connectionand frame field a closed system sufficient and neces-sary to capture any local equivalence) applies in partic-ular for dislocations and disclinations, i.e., Volterra pro-cesses inserting or removing coherent induced physicalfield dynamics, in which one averages Cartan’s affinetransport over all possible patches S to quantise thephysical field dynamics at interfaces between regions.In the case of dislocations and disclinations a super-position principle holds that coincides with the well-known law of Kirchhoff for electric currents in a circuit[196].

(Equivalence 2) can be complemented by a globalconservation law for a region U on the vision system.

Equivalence 3. The global equivalences of the phys-ical field dynamics are given by:

Vi; j1... jk ({S}, U ) =∫

UVi; j1... jk (p, {S}) dU,

in which U is a region on N not necessarily of constantdimension nor simply connected to point p.

Up to now we unravelled only local density currentsintegrated over either simply or multiply connected

regions. The question arises what the possible inter-actions are between such currents? First of all therewould be a gauge group transformation(s) needed tobring the (co)frame fields and connections at two posi-tions (more than two positions) in line. The latter gaug-ing will be noticeable for an external observer [108].An internal observer, the point of view chosen in ourframework, will not become aware of this action as heor she is falling freely along some kind of geodesic.Now we are in the position to carry out the compar-ison by establishing joint equivalences, i.e., the ana-logues of semi-differential, multi-local, simultaneousor joint (differential) invariants [59, 60, 166, 192]. As-suming the considered gauge group to be living on thewhole image (if different gauge groups would be appli-cable over subimages the analysis does not deviate) themost immediate construction follows upon computingthe structure functions for a multiple point set of mor-phisms. The latter set should not be confused with thegauge group.

Definition 12. The elements of morphisms are gen-erated by a set of gauge invariant propagators definedby:

�pj1... jk = V pi; j1... jk

∇�(p)

vi (p),

where p labels selected points.

The Lie algebra of this set of propagators defines thesought joint equivalences.

Equivalence 4. The joint equivalences of the physi-cal field dynamics are given by the structure functionsV r

pq given by:

[�p, �q ] = V rpq�r ,

where � are one of the aforementioned propagators.

Note that ratios or differences of the components ofthe aforementioned equivalences are also joint equiv-alences but can be retained by simple algebra on theset of local equivalences. Furthermore, that wheneverone expects additional constraining symmetries to bepresent the search for irreducible joint equivalences isworthwhile and one should take advantage of meth-ods from both geometry and invariance theory [192,197, 201]. In particular the joint differential invariantscan then be retained readily by applying suitable Aron-hold recursion operators to the set of local equivalences


[91], and by applying Wick’s theorem that reduces mul-tiple point interactions to two-point interactions [125].These joint equivalences can be quite important mea-sures in image matching, stereo and optic flow [196].

The frame field, metric and connection, curvaturespresented in this paper determine equivalences of theobserved external physical field dynamics. They wereup to now acquired by performing just circuit integralson a two-dimensional surface and integrating them sub-sequently over other physical objects. Characteristic forthese equivalences and thus also for related paradigms,is that they exclude yet the operationalisation of collec-tive, cooperative and/or antigonistic, functional inter-actions between subprocesses in the external physicalfield dynamics. It should be possible in the line of Biot-Savart’s law for the force between two conducting cur-rents to formulate attractive and repulsive currents andrelated equivalences. In [204–206] we already notedthat among the desired equivalences are (self)-linkingnumbers, (generalised) Vassiliev invariants and Mobiusenergies for knots, links, braids and more general CW-complexes [14, 15, 24, 32, 69, 174, 229, 240, 247, 248,262]. These topological numbers, invariants and ener-gies are true generalisations of the presented equiva-lences, because they can capture the (self)-interactionsof collective physical processes in the induced physi-cal field dynamics. The existence of chirality2 indicatesthat the electromagnetic field in media and in particularin the human visual system supports the idea that mech-anisms grouping electromagnetic dynamics by observ-ing (self-)linking and branching aspects in the opticfield dynamics should belong to the standard machin-ery of such a system. Such mechanisms then also per-mit a self-organisation of the vision system consistentwith the external and stored physical field dynamics.Although the mentioned features possess a very highcomputational complexity they may play a major roleas constraints or granulometric characteristics in filter-ing schemes restoring, enhancing and/or simplifyingcoherent dynamical processes (see Section 3.2).

Equivalence 5. The functional equivalences V of thefield dynamics are generalised Vassiliev invariants andMobius energies related to the induced field dynamics.

Note that these functional equivalences distinguishthemselves from global equivalences as appearing inEinstein’s theory of general relativity concerning thenumber of holes in space-time through their non-locality [196]. However, it is false to conclude from

Einstein’s work that the Gauss–Bonnet number comesfrom purely local behaviours. The above functionalequivalences quantify, however, other nonlocal inter-actions between physical objects in space–time. Un-derstanding and assessing the transitions of all theseequivalences within a pseudogroup or groupoid likeframework will be one of the biggest challenges in fu-ture theoretical and experimental research in physicsand may help boost not only physical sciences but alsoinformation and learning sciences [183]. Anticipatoryevolution operators for equivalences in the context ofgenetic learning processes may be retained in a statis-tically or empirically manner [48, 49, 205]. The latterfunctorial operations are in the next section appearingas topological currents quantifying the morphologicalchanges in state and dynamics.

Let us give an example of those functional equiv-alences and point out how they directly relate to theequivalences already found.

Example 1. Consider, e.g., the so-called self-linkingnumber SL(K ) for knots K belonging to a class K ofthe same knot type [32, 174] (see for similar propertiesof higher dimensional (sub)manifolds [240, 262]). Fora knot the self-linking number is an invariant of theimbedding of the circle S1 into three-space R

3 or three-sphere S3.

Now, we can define a potentialV through an orderingand inclusion relations on R

3 or S3 as follows:

V = log (g(x2(s2) − x1(s1), x2(s2) − x1(s1)))G2 ,

with xi ∈ Ki ∈ K, i = 1, 2, G is a fundamental physi-cal constant such as those for gravitational and electro-magnetic interactions. Here Ki is a knot with arbitraryparametrisation by si , and g is a flat and torsion-freemetric such as that for Euclidean three-space. In thesequel we put for convenience G = 1.

Next take along knot K1 and knot K2 the variationof the potential V:

dV = gqr ds2dxq(s2)

ds2

×(

xr (s2) − xr (s1)

g(x(s2) − x(s1), x(s2) − x(s1))12

),

− gqr ds1dxq(s1)

ds1

×(

xr (s2) − xr (s1)

g(x(s2) − x(s1), x(s2) − x(s1))12

).


Finally, let us map the knots K1 and K2 onto eachother. This mapping can be either such that x1 = x2

or x1 �= x2 despite the fact that K1 = K2. Ap-plying Stokes’ theorem, assuming x(s1) and x(s2) toparametrise a two-surface in R

3 or S3, dividing themapping in the above cases and normalising by meansof the total solid angle for a sphere the so-called self-linking number of knot K = K1 = K2 reads [32]:

SL(K ) = 1

4π

∫S1×S1

d ∧ dV

= 1

4π

∫S1×S1

εpqr ds1dx p(s1)

ds1ds2

dxq(s2)

ds2

×(

xr (s2) − xr (s1)

g(x(s2) − x(s1), x(s2) − x(s1))32

)

+ 1

2π

∫K

tds

where the last term represents the total torsion with t theEuclidean torsion of K and s the Euclidean arclengthalong K .

Summarising, defining a connection on the basis ofa two-point potential for a knot K we computed itsassociated curvature SL(K ), which is an invariant ofthe imbedding of S1 into R

3 or S3.

Using other potential functions related to multiplepoint semi-differential invariants [150], multi-local orsimultaneous invariants [91, 196, 197] or joint differ-ential invariants [59, 60, 166, 192] we may wonderwhether we can derive straightforwardly all type as-pects of knots, links, braids and even more general sys-tems of CW-complexes in integral form as in [206,239]. Another related question concerns the problemof chirality of such systems [87, 134]. Can chiralitymeasures retained in an analogous scheme? For bothproblems we refer the reader to [206].

If the imposed gauge group does not coincide with apseudo-group of diffeomorphisms and reflections, thenwe can retrieve other than pure topological quantumnumbers, e.g., the Mobius energies (compare for ex-ample the latter energies [69] with the Lyapunov func-tionals as information measures introduced in [232]). Inthis context it is also worthwhile to contemplate othergeometric invariants similar to those derived in [32,174, 240, 262].

To solve a higher order equivalence problem, suchas carried out in [59, 60, 165] seems to us not tomake so much sense, unless the noise causing a mor-phism of the image can be covered in advance by

a sensible pseudogroup action. Furthermore, the factthat observations concern mainly intensive entities pre-cludes us from considering differential invariants andlocal equivalences right from the start. Nevertheless,the joint differential invariants can, as shown in (Exam-ple 1), generate nonlocal conserved densities like self-linking numbers of nonsingular CW-complexes that arerobust even when subjected to transformations that arenot fiber-preserving such as noise. E.g., the total inten-sity and other moments invariant under classical gaugegroups remain also very robust to noise despite thefact that the noise generates strong perturbations of thelocal differential invariants. Therefore, joint integralmeasures have our preference and so have nonlocalconserved densities for describing the characteristicsof the scaled physical field dynamics (see Section 3.2).

An important property of a physical object or pro-cess F related to a gauge group (Definition 1) and theinduced dynamics N is its invariance under the gaugegroup.

Definition 13. A physical object or process F is in-variant under gauge group (Definition 4) if and onlyif:

GF = F.

All above mentioned geometric objects are by defi-nition unaffected by the corresponding gauge group.

Theorem 1. The frame field (Definition 5), co-framefield (Definition 6), metric (Definition 7), connec-tion (Definition 8), and equivalences (Equivalence1), (Equivalence 2), (Equivalence 3), (Equivalence 4)and (Equivalence 5) are invariant under gauge group(Definition 4).

Proof: Proofs follow simply by applying gaugegroup to the induced physical field dynamics, comput-ing the physical objects and comparing with the objectsthat are not transformed. ✷

Recall that we have constructed gauge invariant ge-ometric and physical objects. In [46] one talks aboutgauge transformations not affecting the vector bun-dle metric. But because the Koszul connection on acomplex matrix valued vector bundle does depend onisometries or nonsingular linear transformations pre-serving the metric, the curvatures of that connectionchanges under gauge transformations. The norm of thecurvature as well as the Yang–Mills action, however,are gauge invariant. But it’s clear that it is in this case


still possible to define frame and co-frame fields, met-rics and connections on the basis of physical objectstruely invariant under the gauge group right from thestart. Furthermore, still one can try to find Koszul con-nections that minimise the Yang–Mills action. Havingfound such connections one can subsequently derivethe curvature of one of the Koszul connections that iscalled a Yang–Mills field.

Unfortunately, a large part of the computer visioncommunity is still unconvinced of the usability of theabove machines. However, in disciplines as crystal the-ory and condensed matter the presence of dislocationsand disclinations are just described by above actionprinciple [108, 196]. Symmetry breaking in the dy-namics of physical fields, also in the image formation,is a very normal phenomenon in daily life: hairdressingis what matters and boldness does not.

In the following we give an example in which all theabove geometric objects and processes are of interestin the detection of edges, ridges, courses and inflectionlines. Furthermore, we emphasise that these objectsand the geometries defined on them are invariant underthe group of homeomorphisms, i.e. the edges, ridges,rust and inflection lines are deformed but the local dy-namic ordering and spatio-temporal inclusion relationsare being preserved under adding a density field that iscaused by an integrable vector field on the space-timedomain of the image.

Example 2. Let us assume for simplicity that thespatio-temporal image domain can be modelled as aGalilean space. We partition a time-sequence of two-dimensional Meteosat images in the visual spectrum[53] by extracting spatial and temporal edges, and spa-tial ridges, courses, inflection lines and other curves.Thereto, we choose the following frame field ε:

(εi , εt ) = (∇ s ∧ R, ∇

s R, ∇ t R

),

co-frame field ω:

(ωα) = (dsi , dt),

metric:

γ = ωα ⊗ ωα,

and metric connection determined by the frame andco-frame field [199]:

(ωij , ω

tt ) =

((0 −κi

κi 0

)dsi , 0

),

ωtj = ωi

t = 0.

Here R is an image (Definition 1) representing a radi-ance field, i, j denote the isophotes and flowlines, dsand κ refers to the Euclidean differential arclength andcurvature field on those curves, and t denotes the timevariable. We could also adapt the local spatial geometryto the perspective transformation caused by the visionsystem. Doing so we could also associate to each visualray and perceived pixel radiance a surface area of theplanetary boundary layer. However, the latter geomet-ric adaptation we will not pursue any further. It’s clearthat the above frame field generates cells of space-timeregions of an extent fully complying to the radiancefield dynamics present in the Meteosat image.

Studying the changes of the frame vector field ε2 inthe direction of this field itself, we can nicely locate thespatial edges by means of the signature of ∇

ε2ε2 pro-

jected onto ε2, whereˆdenotes normalisation by meansof metric γ :

σ sedge = sign

(γ(∇

ε2ε2, ε2

)).

Actually we could equally well measure the signatureof the second order variation of the radiance field alongthe flowline in the direction of increasing radiance, i.e.sign(∇

ε2(x0)(∇

ε2(x0)R). The latter argument of the sig-

nature can readily be derived as the second derivativeof R with respect to the flowline arclength parameters2 realising that image gradient is perpendicular to theflowline curvature vector field. These edges coincidewith the watersheds encountered in mathematical mor-phology [223, 205].

A similar signature can be given for the temporaledges:

σ tedge = sign

(γ(∇

εtεt , εt

)).

Again this signature can be derived from the signa-ture of the numerator of the above expression, i.e.sign(∇

ε3(x0)(∇

ε3(x0)R).

In Fig. 1 a spatial input image is displayed at noonof a time-sequence of images in the visual spectrumacquired by Meteosat between t = 9 h and t = 18 hconsisting of 18 temporal views of spatial slices com-posed of 200×200 voxels each with a dynamic resolu-tion of 8 bits. In Figs. 2 and 3 the spatial and temporaledges in the time-sequence of Meteosat images in thevisual spectrum around noon occur there where σ s

e andσ t

e , respectively, change sign. The sought edges are re-trieved by simple binary operations on the voxels; oneobtains a dynamic cellularisation of the time-sequence


Figure 1. Spatial slice of Meteosat image in the visual spectrum atnoon.

Figure 2. Spatial back- and foreground dynamics of Meteosat im-age in the visual spectrum at noon.

of image in different types of fore- and backgrounddynamics.

For the spatial slices of the image sequence in theneighbourhood of ridges, courses and inflection linesthe normalised flowline curvature vector field κ2 re-verses orientation. Following the ridge, course or in-flection line and continuing on the tangent inscribingcircle, we traverse this circle clock- or counter clock-

Figure 3. Temporal back- and foreground dynamics of Meteosatimage in the visual spectrum at noon.

wise. In order to retrieve all these special curves at oncewe advocate the signature of ∇

ε2ε2 projected onto ε1:

σphase = sign(γ(∇

ε2ε2, ε1

)).

Again this signature can be derived from the signa-ture of the numerator of the above expression, i.e.sign(∇

ε2(x0)(∇

ε1(x0)R) which is the opposite signature

of κ2(∇ ε2(x0)

R)2.On the basis of the sign of the isophote curvature:

σrr = sign(γ(∇

ε1ε1, ε2

)).

we can subsequently distinguish between ridges andcourses, since κ1 > 0 for a ridge and κ1 < 0 for a course.Again it suffices to study the signature of the numeratorof the above expression, i.e. sign(∇

ε1(x0)(∇

ε1(x0)R), to

determine the requested signature. Note that signatureσrr is not sufficient to make the distinction betweenridges or courses, and inflection lines. Furthermore,that the classification criterion for these special curvesare latent in the pure Euclidean geometric propertiesof isophotes and flowlines within the spatial image do-main, whereas the edges are obviously associated tothe radiance field pattern along the flowlines.

In Figs. 4 and 5 a segmentation is shown of the spa-tial image in Fig. 1 by means of the signatures σphase

and σrr , respectively. Sign changes in σphase indicatevoxels neighbouring ridges, courses and inflection


Figure 4. Partitioning of spatial slice of a Meteosat image in thevisual spectrum on the basis of σphase at noon.

Figure 5. Partitioning of spatial slice of a Meteosat image in thevisual spectrum on the basis of σrr at noon.

lines, whereas σrr determine whether the voxel are partof convex or concave segments of isophotes.

In order to distinguish between ridges or courses andinflection lines the following signature:

σ sinflection = sign

(γ(∇

ε2(∇

ε2ε2), ε2

)),

which is equal to the third order derivative of the radi-ance field R with respect to the Euclidean flowline ar-clength parameter s2, sign(∇

ε2(∇

ε2(∇

ε2R))), has to be

applied. It preserves sign at ridges and courses awayfrom the spatial edges, whereas at inflection lines itchanges sign. Note that here directional derivatives arenot taken as there’s now a clear difference betweenthose and covariant derivatives at location x0. On com-putation this signature we will find the third order di-rectional derivative along the flowline compensated bya term depending on radiance gradient and flowlinecurvature. Furthermore, that the direction of ε2 is ofimportance.

Subsequently, by applying simple morphologicaloperations the ridges and courses can be retained bylooking for edgels where only σ s

phase changes sign.Whenever both σ s

phase and σ sinflection change sign we

have encountered an inflection point. Note that the re-covered inflection lines can be ordered going from acourse up to a ridge; there exist apparently tilted andslanted plateaus. Furthermore, walking along differentridges and courses we can also induce a meaningfulordering on these dual physical objects.

In [132] Koenderink defines a ridge and a courseof a topographic image of a height function, follow-ing Rothe [184], as special curves on which tributaries(flowlines) approach, join (in one or more junctionspossibly at infinity) and contribute to a rivulet. In orderto be able to find these kind of ridges and courses onehas to rely on a nonlocal analysis op the radiance field.Having a smooth surface the equation for the flowlineswill have hopefully a computable general integral andintegrating divisor. If so, then the vanishing of this di-visor locates ridges and courses.

In our approach the ridges and courses coincidewith other characteristics of the radiance field R thatclearly highlight the image formation processes at ei-ther side of those physical objects. Instead that ourridges and courses form end-points of flowlines theyform natural boundaries of so-called influence zonesbut for Euclidean geometric properties of the flowlines.They are defined by the change of σ s

phase, i.e. the DeSaint–Venant’s condition [132], and the preservationof σ s

inflection. Our method, therefore, would fail to de-tect Koenderink’s course of a helicoidal gutter [132].His ridge and course definition is actually equivalent tothat of a watershed in mathematical morphology [223]which obviously involves only zero-th and first orderinformation in the neighbourhood near such specialcurves. At ridges and courses the image gradient field


is at least three-fold degenerate: two patches of constantimage gradient and one strip or patch of the geometricmean of both. Convolving the discretised input imagewith a discrete pixel-shaped and normalised block test-function ridges and courses appear as locations wheretwo image gradient fields say t1 and t2 change acrosstheir interfaces in such a manner that the inner productof the gradients projected onto the unit normal vectorfield n to that interface are negative.

In order to solve this problem in [196] we proposed,alternatively, a topological method based on Morsetheory by cutting an image in one-dimensional strips.Studying subsequently the grey-valued image living onthese strips we observed that the ridge and course pointsin a two-dimensional grey-valued image are coincidingwith local maxima and minima, respectively. The latterextrema can be determined as simple sign changes forthe image gradient along these strips. The inflections inthe radiance field on the strips then also coincide withthe edges and inflections in the two-dimensional image.The inflections can be retained as the sign changes ofthe second derivatives of the radiance field along thestrips. However, a strip may just be coinciding with anedge for the radiance field in the direction perpendicu-lar to that strip. This coincidence is rare and thereforeasks seldom for a solution as in the previous paragraph.

Our ridges and courses in (Example 2) are like thoseof Koenderink invariant under homeomorphic transfor-mations of the radiance field. This gauge invariance ofthe spatio-temporal inclusion and local dynamic order-ing relations is most straightforwardly demonstrated byadding a radiance field being the consequence of an in-tegrable radiance gradient field. Because of the integra-bility of the latter field no other topological charges areinserted than those already latent in the present field ofridges, courses and inflection lines (singularities beingpart of the ridges and courses). Note, however, that al-though the nets and segments of these special curves arebeing deformed their local dynamic ordering and theirspatial inclusion relations are being preserved. Thusdislocational and disclinational currents, and connec-tivity measures based on these relations are in generalnot affected at all by such severe active transformations[195, 196].

As pointed out time is oriented and so can thetwo-dimensional space adjoined a direction of circu-lation. These two orientations permit in addition tothe above descriptors to distinguish in radiance sourcesor sinks along the time-axis, and spatial upslopes anddownslopes along the sense of circulation. The same

observation can be made when applying the topologi-cal method to describe images [196]: the strips can beall traversed in a particular direction allowing us to dis-tinguish left from right and whether the radiance field isdecreasing or increasing with respect to that direction.

From (Example 2) we can draw the following con-clusions:

• Automatic segmentation of spatio-temporal grey-valued images can be acquired by a partitioning ofthe image into different types of fore- and back-ground dynamics on the basis of a geometric or topo-logical method. The fore and background dynamicssegmentation can subsequently be refined by deter-mining the ridges and courses and their dynamicalorder and spatio-temporal inclusion relations. In or-der to find a figure ground segmentation togetherwith a refinement by using the dynamic orderingand spatio-temporal inclusion relations for the in-flections/edges, ridges and courses in agreement withKoenderink’s work [132] we can take advantage ofour topological method.

For higher dimensional spatio-temporal grey-valued images similar remarks hold. Fore- and back-ground segmentation is obtained on the basis of sim-ilar signatures as in the two-dimensional case. Forour ridges, courses and higher dimensional inter-faces one can use again the flowline curvature vectorfields. Those interfaces do not necessarily have to beof constant co-dimensionality or of a simple topolog-ical structure; the closest mathematical descriptionis that of a measure for a CW-complex [195, 196,198]. The co-dimensionality can readily be ascer-tained by counting the number of signature changesin the neighbourhood of interfaces.

• Adding an integrable frame field to the consideredframe field by superimposing a dummy image tothe original image does not affect the segmenta-tion, i.e. the spatio-temporal inclusion relation andthe local dynamic ordering relations. Of course, thedummy image should not give rise to other topolog-ical charges than those that were detectable in theoriginal image. Note that a spatio-temporally inho-mogeneous affine transformation of the singularityset and the net of edges, inflection lines, ridges andcourses still may occur. E.g., adding a slope inten-sity function to a continuous scalar image does, e.g.,translate extrema in the image.

• Applying the group of (volume preserving) diffeo-morphisms or homeomorphisms to the image does


not affect our segmentation. However, this is only thecase for passive transformations. In the case of activetransformations, e.g., due to increasing resolutions ofthe vision system, such an active transformation can-not be on an equal footing as a passive transforma-tion. The gauge group has definitively to be extendedto and modelled by a particular Lie pseudo-group in-corporating morphological nonlocal dynamics. Suchissues play a crucial role in general relativity andrenormalisation theories [9, 206, 268].

• Applying the group of homotopic transformationsthe global dynamic ordering relations are not pre-served but the local ones and the spatial inclusion re-lations of the figure ground segmentations and thosefor ridges, courses and inflection lines are.

• Topological, geometric and algebraic interpretationsof the segmentation in cells have become feasiblethat are invariant under the postulated gauge group[68, 109–113, 130, 135, 138, 199, 201, 233, 235,236] (see Section 3.2.3). E.g., in the case of the groupof diffeomorphisms the images can be described interms of the connectivity relations (topological di-mensions) on discontinuity and singularity sets, andthe inclusion relations for the cells enclosed. Also theother equivalences introduced in this section suchas the knot invariants come into play. Note, how-ever, that the discrete formulations of the invariantsfound in [62, 129, 150, 165, 191, 192, 196, 199] canbe equally well valuable in algebraic descriptions ofthese segmentations. Similar remarks hold wheneverthe above diffeomorphisms are volume preservingmeaning that the canonical volume of the total radi-ance field within figure or ground and ridges, coursesand inflection lines are being conserved. Readilycurves and alike can be defined in a rather robustmanner and semi-differential local, multi-local andglobal properties among which moments [196] canbe computed.

Unfortunately any image is degraded by unwantedperturbations due to noise in the vision system. Further-more, over images of an ensemble of similarly preparedphysical field dynamics there occur discrepancies dueto fluctuations, unresolvable by the vision system, thatcannot be covered by any conceivable gauge group inadvance. In order to reduce also the computational load(data compression) but simultaneously to retain a setof stable and reproducible equivalences we propose inSection 3.2 to take advantage of the above analysisas a first initialisation of an induced image analysis

and processing principle. We hope to retain thus par-tial equivalences for a whole set of similarly preparedphysical field dynamics.

3.2. Exchange Principle

Considering an ensemble of images one notices thatthey are in a modern geometric, topological and dy-namical sense perturbed versions of each other. Thisperturbation consists of non-integrable and integrabledeformations of the frame fields, co-frame fields, met-rics and/or connections resulting in a change of thecurvatures and even of the physical field dynamics.The non-integrable deformations due to noise and rela-tive resolution differences over the images not coveredby gauge transformations and renormalisation groupactions cause changes in (Equivalence 1), (Equiva-lence 2), (Equivalence 3), (Equivalence 4) and (Equiv-alence 5). The integrable deformations consistent withthe postulated gauge group on the contrary do notcause curvature changes or changes in the norm ofcurvature. However, in practice images are the resultof a density field requiring the integrable deforma-tions to belong to a certain class in order not aftermeasurement to give rise to non-integrable deforma-tion of a discrete frame field, co-frame field, metric,connection and corresponding equivalences. In orderto extract from an image a stable and reproducibleset of physical field dynamics despite these pertur-bations we proposed in [195, 196] a dynamic scale-space paradigm controlled by the equivalences them-selves. Essential in the context of this paradigm isthe derivation of a proper induced exchange princi-ple for equivalences between a region and its sur-roundings. This region and its surrounding either ad-jacent or not, and being currents and processes insteadof rigid and fixated spatio-temporal compact spacesare commonly operationalised by the equivalencesthemselves.

In order to extract from image (Definition 1) a sta-ble and reproducible set of equivalences despite per-turbations in the external, induced and stored phys-ical field dynamics there exist a possibility to de-rive a dynamic scale-space paradigm committed to theconnection, metric and dynamics living on the im-age of the physical field dynamics [195, 196, 199].This can be achieved by coupling the exchange prin-ciples intrinsically to the induced and stored physi-cal field dynamics. Essential in these principles arethe assessment of the topological interactions activated


in the vision system. These interactions can be op-erationalised by the vision system as a topologicalcurrent.

For a complete and irreducible set of equivalences a(un)committed, i.e., conceptual or data driven, orderingof the physical field dynamics can be succinctly formu-lated through the use of a statistical partition functionZ (comparable to the information function mentionedin [116]) related to free energy F for (Equivalence 1),(Equivalence 2), (Equivalence 3), (Equivalence 4) and(Equivalence 5).

Equivalence 6. The statistical partition function Zrelated to free energy F for a concise set of equiva-lences (Equivalence 1), (Equivalence 2), (Equivalence3), (Equivalence 4) and (Equivalence 5) of the for-mation of the vision system and those of the inducedexternal electromagnetic field dynamics is defined by:

Z = exp[−F[Vi (x)]],

with

F[Vi (x)] = − log Z

=∑i,k,p

dv p(Vi;πk (g1...gk )

(x, τi;πk (g1...gk )

)),

where x labels any state, πk a permutation of a se-quence of k ≥ 0 integers (g1 . . . gk) with k = 0 forlabeling frame vector fields vgk and τα;πk (g1...gk ) (inner)dynamic scales consistent with the gauge group G andthe equivalences Vi;πk (g1...gk ).

Note that V is intended also to encapture joint aswell as functional equivalences.

Of course, the partition function and the free energysatisfy a gauge invariance principle.

Theorem 2. (Equivalence 6) is invariant undergauge group (Definition 1)

Proof: Follows immediately from the gauge invari-ance of (Equivalence 1), (Equivalence 2), (Equiva-lence 3), (Equivalence 4) and (Equivalence 5) (see also(Theorem 1)). ✷

The partition function can be conceived as a mea-sure of the topological, geometric and dynamical com-plexity of the physical field dynamics. The advantage

of our measure is that it readily substantiates and ex-tends information theoretical measures as proposed in[230, 232] based on classical works as [1, 119, 178,259]. The new measures of complexity are to be pre-ferred for their conciseness and their straightforwardextension to measures of much higher complexity likethe functional equivalences. Moreover, the dynamicscale-space paradigms will therewith fall nicely in thesame realm as the modern theory of dynamical systems[116].

All equivalences are incorporated in the partitionfunction Z by taking all products of equivalencesVi;πk (g1...gk ) that reside in (Equivalence 1), (Equiva-lence 2), (Equivalence 3), (Equivalence 4) and (Equiv-alence 5). Thus we ensure that curvature aspectsof Volterra processes [195, 196] are operationalisedthrough an additional weight in free energy F viaVi;πk (g1...gk ). That each component of each density vec-tor field (a finite sequence of topological and geo-metric numbers) in the local partition function Z(x)

forms a factor in the total statistical partition func-tion Z . Last but not least, that for each polarisa-tion direction and path we have an associated fac-tor in the partition function. Each component of thetensor Vi;πk (g1...gk ) determines a factor in this func-tion (nonuniqueness of semantics supported throughcombinatorics).

One could equally well apply a probability argu-ment for yielding dynamics V . E.g., in the case thatcurvature Vi is parallel to vi , this curvature has a lowerprobability than that it is antiparallel to vi (recall thatZ and Z(x) can be used to define probability measuresfor physical field dynamics V ). Moreover, note that, forexample, functionals like the renormalised Yang–MillsLagrangian can form an integral part in our partitionfunction.

In statistical physics and simulated annealing [16,73, 147, 266] the Helmholtz free energy of a spin sys-tem coincides with our free energy concept. However,instead of one temperature times the Boltzmann con-stant a set of scales τi;g1...gk pop up that define the dy-namic scales corresponding to particular equivalences.Furthermore, we don’t assume the detectors to be de-generate nor identical. Last but not least, in statisticalphysics the change in free energy, dF, is expressed asvariations in temperature, dT , and variations in canoni-cal volume, dV , consistent with a metric or connection,as follows:

dF = −SdT − PdV,


where the entropy S and pressure P are given by:

S = −(

∂F

∂T

)V

,

P = −(

∂F

∂V

)T

.

The latter entities in our case can only be opera-tionalised by increasing scales τi;g1...gk . This increaseis essential to cause mixing or interactions betweenstates in the physical field dynamics. In the interac-tion or the entropy of mixing resides then the direc-tions in and paths along which an ensemble of physicalfield dynamics possibly evolves. Each direction, pathand evolution can be given a probability measure de-termined by the strength of the external, induced andstored physical field dynamics.

A variation of our free energy can be expressed asfollows:

dF = ∂F

∂τdτ + ∇vq jqdV,

where τ is a scale parameter, dV a canonical volumeelement and j a topological current to be specifiedshortly. Upon postulating dτ ≈ dT one can make thefollowing identifications:

S ≈ −∂F

∂τ,

P = −∇vq jq .

As the free energy F available for external, inducedand stored physical field dynamics will normally notchange, contrary to simulated annealing in which itdoes, it’s reasonable to assume that:

dF = 0,

or

SdT = −PdV.

The second law of thermodynamics says that for anyinteraction or mixing process dS/dt ≥ 0. We will ob-serve, shortly, that this law is just a manifestation ofthe maximum or comparison principle.

Because the redistribution of states/currents of a vi-sion system should be independent of the topological,geometric and dynamic intricacies of the vision systemat ground state determined by the stored physical field

dynamics the standard approach is just to operationalisea gauge invariant frame field, co-frame field, metricand connection supported by the vision system but in-duced by the external physical field dynamics. Notethat in these parametrisations it becomes for certaingauge groups still indispensable to apply the relatedLie group action to the frame fields to obtain the de-sired dynamic scale-space operator. Doing so and usingrelated integral measures, allows us to retrieve the nec-essary equivalences. This situation occurs in wavelettheory as well as mathematical morphology [237]. It’sclear that this approach has a computational drawbackresiding in the fact that an overcomplete set of ker-nels is needed to analyse and process images. Onlyin a globally Euclidean, Galilean or Lorentzian settingthose scale-space paradigms can be of equal compu-tational costs as the induced dynamic ones [195, 196,199, 237]. E.g., in the full globally affine case a suitablemeasure for identifying equivalent grey-valued imagedetails is obtained by considering affine invariant struc-tures in the two-jet. An analysis and processing of thisprimal image data is then carried out most effectivelyand in a concise manner by applying an affine geomet-ric flow with respect to this data [195, 196, 199]. Notethat this does not mean that higher order affine flowsare unfeasible.

Besides the induced gauge invariant canonicalparametrisation of space-time and dynamics as pro-posed also a topological interaction is needed to ensurean evolution towards a hierarchy of partially equivalentstates of the vision system for an ensemble of inducedphysical field dynamics that are slight perturbationsof each other. This topological current is in our dy-namic scale-space paradigm [195, 196, 199] broughtabout by the statistical partition function (Equivalence6). Studying two local factors Z(p1) and Z(p2) in thestatistical partition function going from state pi to statep j involves a factor k(i, j) to generate Z(p j ) fromZ(pi ), whereas going from p j to pi requires a factorK (i, j) (assume k ≤ K ) to generate Z(pi ) from Z(p j ),such that kK = 1, i.e., the notable Artin-Whaples for-mula in disguise [116] (see [21] for a remarkable con-nection between linear and a nonlinear diffusion pro-cess on the level of the density fields, namely throughFF = 1).

The interaction can only be brought about bythe transport of physical field dynamics through in-terfaces or wires between states. Therefore, it ismore than reasonable to let a topological currentbetween states pi and p j to be controlled by the


partition function Zr for two-state interactions. Thispartition function captures all the possible cou-plings between the states of all pairs of detectors asfollows:

Zr =∏i �= j

Zrij =

∏i �= j

(K (i, j) + K −1(i, j)

2

)

=∏i �= j

cosh(F(pi ) − F(p j ).

Note that these interactions between two states do notexclude long range forcings: Z(pi ) can incorporateinstantaneously physical field properties such as theequivalences in Section 3.1 where lengths, area, vol-umes and actual topological invariants such as knot,links and braid invariants popped up. The limited trans-port velocity need not be in contradiction with the in-stantaneity of an action if this action is primordiallypresent in the evolution (there’s no need for a changein the speed of light to explain the changes in physicsafter the Big Bang)!

With this two-state coupling partition function, Zr ,there’s associated also a two-state coupling free energyFr :

Fr = − log Zr .

Assuming the vision system to be a closed systemfor a particular region of space-time realised on thevision system, then the change in the state of thatpart of the vision system can be realised by a changein the two-state coupling free energy Fr . Keepingin mind that the free energy, see (Equivalence 6),should be preserved, i.e., dF(pi ) = −dF(p j ), thischange in the two-state coupling free energy, dFr , isgiven by:

dFr = −∑i �= j

tanh(F(pi ) − F(p j ))dF(pi ).

Thus the geometric or topological charges have becomethe generators of the induced physical field dynamicson the vision system. Now let us consider again theinteraction mechanisms between pairs of states pi andp j , and define the topological current to be the spatio-temporal curl of the induced connection on the two-state coupling free energy:

Definition 14. The topological current for the free en-ergy on activated vision system is defined by:

j F = ∇� ∧ dFr

= − ∇�vs

F

cosh2

(√g(∇�

vsF, ∇�

vsF

)dvs ∧ dF,

where vs is connecting free energy states F(pi ) andF(p j ) of the vision system.

Note that the topological current is steered by (Equiv-alence 1), (Equivalence 2), (Equivalence 3), (Equiv-alence 4) and (Equivalence 5) [195, 196, 199]. E.g.,the fact that a pixel belongs to a long spatial edge-segment can be used as some kind of stopping crite-rion or local reflective boundary condition during thedynamic exchange principle stated below [195]. Suchstopping criteria can also be conceived on an equal foot-ing with Lagrange multipliers appearing in the integralconstraints imposed in, e.g., anisotropic or variationalscale-space paradigms (see references Section 2).

As the free energy (Equivalence 6) should be pre-served, the dynamic exchange principle for free energyis in our dynamic scale-space paradigm made manifestthrough a physical law involving topological current(Definition 14):

Law 1. The dynamic exchange principle for free en-ergy says that the change per unit scale τ in the freeenergy (Equivalence 6) in a region � of the visionsystem is equal to the exchange of free energy F be-tween this region and its surrounding across their (com-mon) boundary S = ∂� quantised by topological cur-rent (Definition 14):

δτ F = − j F ,

with suitable initial and boundary conditions.

Theorem 3. The dynamic exchange principle resid-ing in (Law 1) is gauge invariant.

Proof: Proof follows those of (Theorem 1) and(Theorem 2). ✷

This law is in perfect agreement with the secondlaw of thermodynamics that states that the entropy of asystem with time is only increasing if elementary sub-systems not all in their ground states are permitted to


interact. Here the equivalences living on the cellular re-gions are recombined causing a substantial simplifica-tion of the image interpretation possibilities (multipleequilibrium states and consequently interpretations ofthe image formation processes due to the external phys-ical field and vision system dynamics are still feasible).The latter simplification property can also be viewed asa manifestation of the maximum or comparison prin-ciple postulated in scale-space theories. Extensions tothis scheme in the form of a feed- and backward andmean-field like scheme was proposed in [196] in or-der to take advantage of a full dynamic scale-space tosteer the flow at the initial scale. Similar but not so gen-eral approaches based on statistical pattern theory areproposed in [81, 154, 155].

Note that convective diffusion of the free energy isnot excluded as treated in [22, 141]: the conductive den-sity forms an integral part in our formulation. Theseprinciples can be succinctly qualified as a controlleddistribution after a weighted canonical reparametrisa-tion of the equivalences. For (Equivalence 1), (Equiva-lence 2), (Equivalence 3), (Equivalence 4) and (Equiva-lence 5) we can subsequently derive similar laws keep-ing in mind their tensorial character and effects of thegauge symmetries in taking covariant derivatives ofthem [195, 196, 199].

In Section 3.2.1 and Section 3.2.2 we give an in-tegral and differential formulation of the dynamicscale-space paradigm for the equivalences presentedin Section 3.1. Next, in Section 3.2.3 we present a seg-mentation method for a dynamic scale-space. Finally,in Section 3.2.4 we point out how to classify the variousdynamic scale-space paradigms.

3.2.1. Integral Formulation. Assume that each phys-ical observation with respect to the detector arrays,i.e., (Equivalence 1), (Equivalence 2), (Equivalence 3),(Equivalence 4) and (Equivalence 5), is independent.Now vary the free energy F by changing an equiva-lence Vi;g1...gk related to the observed dynamics of theexternal electromagnetic field dynamics with respect toscale τi;g1...gk . And do this according to the topologicalinteraction mechanism quantified by topological cur-rent (Definition 14) The dynamic exchange principlesthen boil down to:

Law 2.. The dynamic scale-space paradigms in inte-gral form based on frame field (Definition 5), co-framefield (Definition 6), metric (Definition 8), connection(Definition 9) and topological current (Definition 14)

for (Equivalence 1), (Equivalence 2), (Equivalence 3),(Equivalence 4) and (Equivalence 5) are defined by:

∫U

∂ Vi;g1...gk

∂τi;g1...gk

dU = −∫

Sg( j Vi;g1 ...gk , n) dS,

Vi;g1...gk ( , 0) = V 0i;g1...gk

,∫S∗

g( j Vi;g1 ...gk , n)dS = 0,

where dU is a volume measure, dS a surface measure, na normal vector field to boundary S = ∂U determinedby the gauge invariant metric g, equivalence V 0

i;g1...gkis

an initial condition and the last equation is a reflectiveboundary condition.

As topological current j Vi;g1 ...gk inherits all topologi-cal and geometric information about the physical fielddynamics, it is justified to assume that the scales τ

represent the ticks of a clock. Consequently, the localexchange of free energy in a particular direction is con-trolled by the topological and geometric charges felt inrelation to the total charges.

(Equivalence 1), (Equivalence 2), (Equivalence 3),(Equivalence 4) and (Equivalence 5) can control cur-vature current j Vi;g1 ...gk , such that this current is limitedto or suppressed in the neighbourhood of boundariesof regions on the detector arrays contained by disconti-nuity, singularity and bifurcation sets, respectively, de-spite the possible non-orientability of those sets. Thesimplification of the physical field dynamics is conse-quently spatio-temporally inhomogeneous, anisotropicand a-symmetric across physical objects with normalfields n and across dynamic features expressed in termsof equivalences. It can be restricted to regions enclosedby such objects and features before iteratively contin-ued. Note that it might happen that the scaled recursionoperator in each region is different because the physi-cal field dynamics in those regions dictate different ex-change principles among their constituents. This latterfreedom implies, for instance, that near the blind spotin the human visual system the stored physical fieldsmay be subjected to their own optimal filtering scheme.How the diffusion operator is splitting along the bound-aries of such a region is then controlled by other equiv-alences related to other physical field dynamics, e.g.,the gravity field, fed into the vision system.

3.2.2. Differential Formulation. Applying the di-vergence theorem the integral formulation of (Law2) can be rewritten in terms of a system of partial


differential equations with corresponding initial-boundary conditions:

Law 3. The dynamic scale-space paradigm in dif-ferential form based on frame field (Definition 5),co-frame field (Definition 6), metric (Definition 8),connection (Definition 9) and topological current(Definition 14) for (Equivalence 1), (Equivalence 2),(Equivalence 3), (Equivalence 4) and (Equivalence 5)are defined by:

∂ Vi;g1...gk

∂τi;g1...gk

= −gpq∇�vp

jVi;g1 ...gkq ,

Vi;g1...gk ( , 0) = V 0i;g1...gk

,

gpq jVi;g1 ...gkp nq = 0,

where n is a normal vector field to boundary S = ∂Udetermined by the gauge invariant metric g, and theinitial and boundary conditions are similar to that in(Law 2).

Note that in physics one normally considers coor-dinate transformations coinciding with some generallinear group action on the local fibers. Such transfor-mations do induce changes in the frame field, co-framefield, metric, connection and consequently curvaturetwo-forms. But the Yang-Mill action [46, 108], i.e. thenorm of the curvature tensor multiplied by a canonicalform and integrated over a compact Riemannian man-ifold, are gauge invariant. In our exposition we ensureright from the start gauge invariance is inherited by allfields whether scalar or not. Thus also the curvaturetwo-forms are right from the start gauge invariant.

In order to find an exact continuous approximativesolution translate the above nonlinear Cauchy prob-lems (Law 3) into an equivalent Cauchy problem fora quasi-linear parabolic system [52]. A semi-discreteapproximative solution can be obtained by solving acoupled system of nonlinear ordinary differential equa-tions equivalent to (Law 3) by means of the theory ofcontrolled stochastic processes [133]. A discrete so-lution to (Law 2) can be found in terms of nonlinearinhomogeneous a-symmetric stochastic matrices [195](see also (Example 3)). Analogously fully finite dif-ference schemes can be formulated through discreteordered calculus [117] in the line of noncommutativegeometry [43].

Example 3. We illustrate our dynamic scale-spaceparadigm for the images considered in (Example 2)

of Section 3.2.1. The aim of such filtering schemes isto retain stable and reproducible input data concerningthe short-wave and long-wave transmission functionof the solar irradiation [202]. The governing integralequations with reflective boundary conditions for ourparadigm is given by:

δτ R =∑

χ

∇�χ R

cosh2

(√g(∇�

χ R, ∇�χ R

))dχ ∧ dR,

with χ a frame vector field in any spatio-temporaldirection.

We have shown in Figs. 6–10 the Meteosat imagein Fig. 1 at one scale and the segmentation in fore-and background dynamics on the basis of the signa-tures mentioned in Section 3.2.1 at the same scale, re-spectively. As to be expected the dynamic scale-spaceparadigm preserves larger scale edges and alike longerover scale. Furthermore, it also leads to perceptualgrouping of the physical field dynamics thus opera-tionalising physical objects such as edges and junc-tions. There’s, of course, a boundary condition inter-fering with the analysis on the spatio-temporal imagedomain that is possibly not supported by the input dataoutside the image domain. It’s, therefore, in case stud-ies concerning the atmospheric dynamics better not toinclude these boundary analyses (unless we have earth

Figure 6. Spatial slice of Meteosat image in the visual spectrum atscale τ = 2 at noon.


Figure 7. Spatial back- and foreground dynamics of Meteosatimage in the visual spectrum at scale τ = 2 at noon.

Figure 8. Temporal back- and foreground dynamics of Meteosatimage in the visual spectrum at scale τ = 2 at noon.

covering input data). Nevertheless, the partitioning byedges and alike produce a natural and direct multi-scalesegmentation in terms of micro-, meso- and synopticscale of physical regimes for the atmospheric dynamics[172].

3.2.3. Dynamic Scale-Space Segmentation. A seg-mentation of the dynamic scale-space of the various

Figure 9. Partitioning of spatial slice of a Meteosat image in thevisual spectrum on the basis of σphase at scale τ = 2 at noon.

Figure 10. Partitioning of spatial slice of a Meteosat image in thevisual spectrum on the basis of σrr at scale τ = 2 at noon.

sequences of input images comes about by using simplythe following signature στ :

σ τedge = sign(dτ

(∇�ετ

ετ

)dτ(ετ )),

with

ετ = ∂R

∂τ

∂

∂τ.


This signature, coinciding with the signature of the sec-ond order derivative of the radiance field R with re-spect to τ , yields a scale-space segmentation in termsof figure–ground dynamics, sources and sinks. Notethat similar remarks can be made as in (Example 2)concerning the influence of the geometry involved inthe data acquisition. Furthermore, that it might be moreappropriate to retrieve first currents or equivalences forthe Volterra processes involved in the atmospheric ra-diation dynamics, and to use them to steer the dynamicfiltering [195, 196, 199]. The images can supply uswith particular mean heat momentum fields by follow-ing the segmented regions over space-time. Last butnot least, that in case stable and reproducible data areneeded during model run it’s more plausible to restrictthe filtering on a cone in space-time pointing in the pastand normalise the filter output in a unique manner (thelatter normalisation procedure is not mandatory andcan even be inconsistent with observed dynamics). Inthis manner the observed history of the radiance fieldsin the different spectra determines an initialisation ofthe transmission functions in the long- and short-waveband [202].

3.2.4. Classification of Dynamic Scale-Space andParadigms. In Section 3.2.1 and Section 3.2.2 wehave given the exchange principles for the variousgauge invariant equivalences in integral and differen-tial form. In (Example 3) and Section 3.2.3 we demon-strated that a segmentation of a dynamic scale-space infore and background induced physical field dynamicsis possible. Of course, this segmentation is dependenton the characteristics of the equivalences presented inSection 3.1 as well as those of the exchange principlespresented in current Section 3.2. The question ariseshow to quantify and qualify these two types of charac-teristics.

3.2.4.1. Dynamic Scale-Spaces. Concerning the de-scription of linear scale-spaces various methods havebeen proposed based on differential and integral geom-etry [195, 196, 199, 206], invariant theory [191, 193,197, 201], singularity theory [45, 105, 106, 126, 138],logical filtering methods [20, 130], fingerprints of zero-crossings [267], topological filtering methods [54, 68,90, 109, 111, 123, 146, 157, 196, 235, 236] and pri-mal sketches [136]. Above some scale these methodsyield (partial) equivalence of the induced physical fielddynamics despite small scale dynamic perturbationsconsistent with a postulated gauge group [196, 201].

The latter gauge group here models a pseudo-groupcapturing in addition the possible non-integrable per-turbations of the induced physical field dynamics. Thispartial equivalence is of major importance in, e.g., pat-tern and character recognition, stereo vision and opticflow measurements [196]. Of course, the same phys-ical framework applied to retain equivalences of themathematical, physical and logical framework carriesover to other possibly nonlinear and nonlocal dynamicscale-space paradigms [195, 196, 199].

As the exchange principles for the equivalences arewritten in divergence form we have simultaneouslyfound local and nonlocal conserved densities with as-sociated fluxes that coincide with our topological cur-rents. Above a particular scale the (partial) equivalenceof an ensemble of similarly prepared physical field dy-namics will be retained from the scaled equivalences.

3.2.4.2. Dynamic Scale-Space Paradigms. There ex-ists extensive mathematical literature on the subject ofconservation laws of systems of evolution equationsand alike [10–12, 22, 166, 250]. Normally one retainssuch laws for systems not in divergence form throughthe use of symbolic packages [19, 23, 74, 77–79, 89,208, 209, 210]. Furthermore, given one exchange prin-ciple for an equivalence one may derive a hierarchyof infinitely many conservation laws [10–12, 22, 166].The latter related conserved densities and fluxes shouldnot be confused with our initially fabricated equiv-alences and topological currents. For each exchangeprinciple for an equivalence this problem of determin-ing conservation laws could arise.

Actually determining whether an exchange princi-ple substantiates a system of conservation laws can beviewed as a way of categorising classes of exchangeprinciples. In mathematics there have been developedseveral classification methods for in particular systemsof partial differential equations based on symmetries[88, 114, 141, 250], and curvatures [175–177]. Formore details on such a categorification the reader is re-ferred to [206]. The logical framework of the dynamicscale-space paradigm is then latent in its topologicalcategorification.

4. Unification

In Section 2 we have presented an extensive overviewof (dynamic) scale-space paradigms. In this section wetry to bring them under the same umbrella through theuse of our paradigm. Instead of spelling out for every


scale-space paradigm how to retain it from ours we justlimit ourselves to very simple but convincing examples.For an extensive treatment the reader is asked to consult[196, 205].

Example 4. In respect to the topological interactionmechanisms formalised in our dynamic scale-spaceparadigm it might be interesting to figure out whethera Lie theoretic approach, as proposed by Bluman andKumei [21] and brought to the attention of the com-puter vision community in Ref. [196, 203], can yieldisomorphic dynamical systems to ours (see also Section2.8). A question that then rises might be the following:Can Bluman’s and Kumei’s work be linked to ours?

Assuming the two-state coupling free energy:

Fr = − log F

the topological current is readily observed to be:

j F = − 1

F2

dF

dxdx ∧ dF.

Thus it appears that Bluman and Kumei’s work canreadily be covered by our physical approach. Inter-esting remains, however, whether a Lie theoretic ap-proach will also be valuable like our mathematical,physical and logical approach in deriving nonlocal dy-namic scale-space paradigms.

Note that the connection on space-time and dynam-ics covered by F are chosen flat such that curvature andtorsion aspects do not appear in this current.

Concerning the morphological and other scale-spaceparadigms we emphasise that the corresponding equiv-alences and conservation laws are of a much more intri-cate form than, e.g., in the case of the linear scale-spaceparadigm [195, 196]. It’s clear that the common ingre-dients of the paradigms are the postulation or statisticalassessment of gauge group, constructing related frameand co-frame fields, metrics, connections, equivalencesand exchange principles.

Example 5. Euclidean shortening flows on spatialgrey-valued images are readily defined on the basisof a suitable metric connection on the net of isophotesand flowlines [196].

Finally, we would like to point out that regularisedanisotropic scale-space paradigms [219] and other

paradigms that not physically justify iterated regular-isation or sub-pixel measurements, can be comple-mented to obtain faithful super-resolution results onlyif a realisation and knowledge about the underlyingpseudo-group action in particular related to the systemnoise and the fluctuations of the external physical fielddynamics is feasible and available.

5. Extensions

We obtain a covariant or topological dynamic scale-space paradigm, that is invariant under a group of(volume preserving) diffeomorphisms of an image, bydefining the topological currents on the basis of thefore- and background dynamics contained by the out-lay and connection of the cells retained by segmentingthe image by means of the signatures σ s

edge and σ tedge

(see also (Example 2); [196, 198, 205]).

Example 6. Using the initial segmentation of thespatio-temporal image in (Example 2) in fore- andbackground dynamics we can define between the ad-jacent and enveloping regions the following two-statecoupling free energy:

Fr = − log(F + |∇F |)

such that the topological current reads:

j F = − 1

(F + |∇F |)2(∇F + ∇(|∇F |)) ∧ dF,

in which xα = xi , t parametrise the cells of the Me-teosat image. These scale-space paradigms can be sub-jected to specific criteria like the number of holes to beretained during the dynamic filtering process.

In the previous example only F and |∇F | are used toconstrain the flow. What about the use of higher orderequivalences to steer the redistribution process.

Example 7. Using the initial segmentation of thespatio-temporal image in (Example 2) in fore- andbackground dynamics we can define between the ad-jacent and enveloping regions the following two-statecoupling free energy depending also on the curvatureaspects of ∇F :

Fr = − log(F + |∇F | + |∇ ∧ ∇F |)


such that the topological current reads:

j F = − 1

(F + |∇F | + |∇ ∧ ∇F |)2

∗ (∇F + ∇(|∇F |) + ∇|∇ ∧ ∇F |) ∧ dF.

Another type of extensions consists of defining dy-namic scale-space paradigms preserving particulargeometric and topological invariants related, for ex-ample, to knots, links and braids. It may happen thatself-passages are prohibited under the flow prevent-ing knots and alike to change type. Nevertheless, sucha flow may relax geometric structures living on suchphysical objects like knots. In this case we are close todefine similar equations like that of Einstein but evenfor nonlocal topological and other geometric aspectsof physical field dynamics [204].

Example 8. The geometric and topological invariantsmay be used as actions to derive equations of mo-tion upon varying canonical coordinates, metric andconnection. After solving these equations structuresliving on those topologically invariant objects can bediffused using the (nonlocal) Beltrami-Laplace oper-ator consistent with the found metric and connection.These topological invariants and energies come intoplay in tracing the transitions in dynamical processes.Furthermore, these functional equivalences may formnew factors in statistical partition functions, informa-tion or topological entropy measures, to be introducedin Section 3.2, that engender a hierarchy of dynamicscale-space paradigms. This type of equivalences arequite familiar in differential topology and quantumfield theory [156].

Note that these paradigms should be distinguishedfrom topological description methods of dynamicscale-spaces [68, 111, 196, 235, 236], that, however,retain their value even for these scale-space paradigms.However, the description methods have to be adaptedand made consistent with the considered dynamicscale-space paradigms.

6. Conclusion

We have presented a physical framework to obtain andgeneralise existing (dynamic) scale-space paradigmsby applying modern geometric, statistical physicaland logical concepts. After identifying the relevant

gauge group in a vision task we were able to formu-late conceptual and data driven dynamic scale-spaceparadigms. The gauge group supports then a framefield and co-frame field consistent with the physicalfield dynamics. Subsequently, the gauge group allowsa construction of a suitable metric and/or connectionsuch that equivalences for the physical field dynamicscan be recovered. We showed how a simple geomet-ric analysis of a two-dimensional grey-valued imagecan supply us with its edges, ridges, courses and in-flection lines; the interfaces between physically essen-tially different physical field dynamics invariant under(volume-preserving) (diffeo)morphisms. The equiva-lences, retrieved by, e.g., performing directed circuitor path integrals over the frame field along physicalobjects allow in turn the construction of a partitionfunction that forms the basis for the derivation of (dy-namic) exchange principles for the equivalences them-selves. These paradigms are essentially developed forany physical observable and not constrained to merelyscalar fields. They encompass aslo density fields andtopological invariants of the physical field dynamics.We illustrated our framework by deriving a figure-ground segmentation of a Meteosat image in the visualspectrum at two scales.

We presented the methods that are around to clas-sify the dynamic scale-spaces themselves as well astheir related paradigms. The standard methods appliedin linear scale-spaces carry over to other paradigms,whereas for the paradigms themselves we mentionbedseveral new schemes to lay bare their logical frame-work.

We concluded our exposition by mentioning variousissues concerning unification and extension of existingdynamic scale-space paradigms.

The different mathematical, physical and logical as-pects of our dynamic scale-space paradigm can bestated as follows:

• Our dynamic scale-space paradigm is gauge invari-ant. The dynamic scale-spaces as well as their de-scriptions in terms of equivalences are invariant un-der the considered gauge group representing a setof passive and active transformations of the physicalfield dynamics.

• Renormalisation group invariance covering gaugegroup invariance ensures unification and extensionof existing scale-space paradigms.

• Conservation laws or superposition principles resid-ing in equivalences remain valid throughout filtering


phase. Filtered images of equivalences may revealtopological changes between consecutive images.But the latter changes do not imply that they resultin a violation of those laws or principles.

• The topological currents for the equivalences aremeasures for the mixing of the complexity of thephysical field dynamics caused by some potentialfor the corresponding equivalences.

• (Non)locality of exchange principles is obtainedthrough those of topological currents and equiva-lences.

• A simplification of the set of equivalences occurson increasing scales or better applying scaled recur-sion operators. These operators can in this respect beseen as forgetful functors, well known from categorytheory, that map a rather complex representation ofinduced physical field dynamics to a relatively lesscomplex one. Recall that conservation laws or su-perposition principles are still being satisfied. Onlythe number of perceivable equivalences in each con-secutive scaled version reduces through the typicalrecombinational procedure. This data compressionand reduction procedure leads also to a decrease inthe number of degrees of freedom in which phys-ical measurements, i.e., equivalences, can be per-formed. Using the conservation laws as constraintsor stopping criteria for the scaling mechanism sen-sibly gives a mean to come to minimal descriptionlength algorithms.

• Inhomogeneity in a spatio-temporal and dynamicalsense. This implies that the exchange of free energydepends on location as well as on the currents activein the vision system. How the external, induced andstored physical field dynamics interact or intertwinewith each other seems to be an open issue. How-ever, the interactions between them can readily bedescribed in terms of the functional equivalences aspresented in Section 3.1 [206].

• Anisotropy in the sense that along any frame vectorfield the exchange of free energy can be differentdepending on the variations over directions in theadjacent factors in the statistical partition function.

• A-symmetry across interfaces between regions withdifferent induced physical field dynamics. Thus forv a change in direction has very large implicationsfor the applied exchange principle as well as thedirectionally dependent contribution to the topolog-ical current. The exchange principle or scaled recur-sion operator in each region might then be differ-ent, because of the fact that the physical dynamics

involved in the physical field dynamics dictates a spe-cific canonical reparametrisation and redistribution.E.g., textures and alike have, therefore, to be sub-jected to their own optimal filtering scheme. Defectsin them can then readily be detected by exploring thehierarchy of subimages in the dynamic scale-space.

• Our dynamic scale-space paradigm allows a restric-tion of a particular exchange principle to a specificregion in the image such that there is no recombina-tion of information with its surrounding. The equiv-alences can yield measures such that at the bordersof physical objects a reflective boundary conditioncan be imposed. Furthermore, other equivalencesmay play the role as some kind of stopping crite-ria for the exchange principle after which a new ex-change principle continues a filtering more adaptedto the physical field dynamics of the initial regionand its neigherest neighbours (that need not neces-sarily have to be literally adjacent). Measures for thetotal energy enclosed and the standard variation ofthe energy within a region might be taken as equiv-alences to perform such constrained dynamic scale-space paradigms.

• Dynamic scale-spaces can be described by our phys-ical framework in terms of geometric and topologi-cal equivalences for regions bounded by discontinu-ity sets and alike. A dynamic scale-space consistentsegmentation and interpretation has been achievedirrespective type of scale-space paradigm. Visiontasks [203] can be captured in terms of symmetries[141, 166, 170], curvatures [175] and (non)local con-served densities [10–12, 22, 77–79, 89] involved inthe exchange principles for the induced physical fielddynamics. Note that through potential symmetriesone may retain also nonlocal equivalences similar toours.

• Our dynamic scale-space paradigm unifies existingscale-space paradigms by stipulating the physics in-volved in the physical field dynamics. The lattermeans that the physics itself induces the frame ofreference in terms of a suitable dynamic scale-spaceparadigm.

• Our dynamic scale-space paradigm extends existingscale-space paradigms in various aspects. First ofall the paradigm restricts itself not merely to scalarentities. Curvature and torsion two-forms and morecomplex gauge invariant topological structures of thedynamics can belong also to the main observablesin a concise representation and interpretation of thephysical field dynamics. The paradigm can adapt or


couple to the particular geometric and topologicalpecularities of the physical field dynamics. We claimthat the proposed fundamental equations in [2, 3, 4,5] are certainly not fundamental enough for solvingcomplex tasks in computer vision or pattern recog-nition. Our paradigm, however, supplies the com-puter vision community with a tool to achieve suchtasks although possibly a renormalisation procedurehas to be accomplished in case one wants to achievesuperresolution.

Our paradigm is not set up for merely spatio-temporal scalar images. It is equally well applicable to atime sequence of multi-spectral stereo images. Colour,optic flow and stereo images can be analysed and pro-cessed in a dynamics consistent manner. Of course,there are other disciplines than computer vision andpattern recognition in which our approach may be valu-able as a data analysis and processing tool:

• In robotics it can be used to analyse and process anintegrated and coupled vision, audio, motor, haptic,tactile and vestibular system to perform coordinatedand controlled autonomous actions.

• In audio sciences it can serve as a mean to analyseand process speech and music, to elucidate the tone,chromaticity, discourse and prosody, and to dwell onstyle-figures.

• In linguistic sciences it can serve also as a mean toanalyse and process text, to elucidate discourse andprosody and to dwell on style-figures.

• In multimedia analysis and processing it can sup-port various tasks concerning integrated multi-mediamanagement. Not only content with respect to audio,video, text, haptic, tactile, motor and vestibular, butalso content with respect to user and network pro-files, software and hardware languages, natural lan-guages, queries (searches and retrieval) and varioustelematics systems, e.g. B2B-processes, can be ef-fectively and efficiently dynamically analysed, pro-cessed and steered [207].

• In political, economic and ecological sciences it canpermit a sustainable development of eco-systems andalike [206].

• Learning theories based on parameter estimationand Hidden–Markov models can be replaced by acoupling and linking these theories and models tothe physical field dynamics underlying learning pro-cesses.

Notes

1. A recursion operator [162] is formally an operator that gener-ates a hierarchy of generalised local or nonlocal symmetries ofPDES with IBC upon iteratively applying it to the correspondingfundamental scale operator.

2. Chirality of a medium concerns the left- or right-handednessof its microstructure. This material property manifests itself asa nonlocal phenomenon via the dependence of the polarisa-tion and magnetisation of the medium on the circulation of theelectric and magnetic field, respectively [87, 134]. This hand-edness plays also a crucial role in detecting ridges, coursesand inflection lines in (Example 2). Moreover, chirality or par-ity happening upon mutation of a room is one of the mostintriguing mysteries in knot theory because some detectionmethods based on Wilson link operators or vertex models doonly partly reveal it, whereas others such as those based onChern–Simons theories remain totally indifferent to chirality[118].

References

1. R.L. Adler, A.G. Konheim, and M.H. McAndrew, “Topologicalentropy,” Amer. Math. Soc., Vol. 114, pp. 309–319, 1965.

2. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Ax-iomes et equations fondamentales du traitement d’images,” C.R. Acad. Sci. Paris, Vol. 315, pp. 135–138, 1992.

3. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Axioma-tisation et nouveaux operateurs de la morphologie mathema-tique,” C. R. Acad. Sci. Paris, Vol. 315, pp. 265–268, 1992.

4. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Equationsfondamentales de l’analyse de films,” C. R. Acad. Sci. Paris,1992.

5. L. Alvarez, F. Guichard, P.L. Lions, and J.M. Morel, “Axiomsand fundamental equations of image processing,” Arch. for Ra-tional Mechanics, Vol. 123, pp. 199–257, 1993.

6. L. Alvarez and J.M. Morel, “Morphological approach to multi-scale analysis,” in Geometry-Driven Diffusion in Computer Vi-sion, Kluwer Academic Publishers: Dordrecht, 1994, pp. 229–254.

7. A. Alvarez, J. Weickert, and J. Sanchez, “A scale-space ap-proach to nonlocal optical flow calculations,” in Scale–SpaceTheories in Computer Vision, Lecture Notes in Computer Sci-ence, Springer: Berlin, 1999, Vol. 1682, pp. 235–246.

8. L. Ambrosio and H.M. Soner, “Level set approach to mean cur-vature flow in arbitrary co-dimension,” Journal of DifferentialGeometry, Vol. 43, pp. 837–865, 1996.

9. P.J. Amit, Field Theory, the Renormalisation and Critical Phe-nomena, McGraw-Hill: New York, 1978.

10. S.C. Anco and G. Bluman, “Direct construction of conservationlaws from field equations,” Physical Review Letters, Vol. 78,pp. 2869–2873, 1997.

11. S.C. Anco and G. Bluman, “Direct construction method forconservation laws of partial differential equations,” 1998.

12. S.C. Anco and G. Bluman, “Integrating factors and first inte-grals for ordinary differential equations,” European Journal ofApplied Mathematics, Vol. 9, pp. 245–259, 1998.

13. J. Babaud, A.P. Witkin, M. Baudin, and R.O. Duda, “Unique-ness of the Gaussian kernel for scale–space filtering,” IEEE


Trans. Pattern Analysis and Machine Intelligence, Vol. 8, pp.26–33, 1986.

14. D. Bar-Natan, “Perturbative aspects of the topological quantumfield theory,” Ph.D. thesis, Princeton University, 1991.

15. D. Bar-Natan, “On the Vassiliev knot invariants,” Topology,Vol. 34, pp. 423–472, 1995.

16. E. Bertin, S. Marchand-Maillet, and J.-M. Chassery, “Opti-misation in Voronoi diagrams,” in Mathematical Morphologyand Its Applications to Image Processing. Kluwer AcademicPublishers: London, 1994, pp. 209–216.

17. S. Beucher, “Segmentation d’images et morphologiemathematique,” Ph.D. thesis, Ecole des Mines, France, 1990.

18. S. Beucher, “Watershed, hierarchical segmentation and water-fall algorithm,” in Mathematical Morphology and Its Appli-cations to Image Processing, Kluwer Academic Publishers:Dordrecht, 1994, pp. 69–76.

19. F. Beukers, J. Sanders, and J. Wang, “One symmetry does notimply integrability,” Journal of Differential Equations, Vol.146, pp. 251–260, 1998.

20. J. Blom, “Topological and geometrical aspects of image struc-ture,” Ph.D. thesis, Utrecht University, The Netherlands, 1992.

21. G. Bluman and S. Kumei, “A remarkable nonlinear diffusionequation,” Journal of Mathematical Physics, Vol. 21, pp. 1019–1023, 1979.

22. G. Bluman and S. Kumei, Symmetries and Differential Equa-tions, Springer-Verlag: New York, 1989.

23. A.V. Bocharov, “DELiA: A system for exact analysis of dif-ferential equations using S. lie approach,” Technical ReportDELiA 1.5.1, Beaver Soft Programming Team, New York,1991.

24. R. Bott and C. Taubes, “On the self-linking of knots,” Journalof Mathematical Physics, Vol. 35, pp. 5247–5287, 1994.

25. A.M. Bruckstein and A.N. Netravali, “On differential invari-ants of planar curves and recognising partially occluded planarshapes,” in Proc. of Visual Form Workshop, Capri, Italy, 1990.

26. A.M. Bruckstein, G. Sapiro, and D. Shaked, “Evolutions ofplanar polygons,” International Journal of Pattern Recognitionand Artificial Intelligence, Vol. 9, pp. 991–1014, 1995.

27. A.M. Bruckstein and D. Shaked, “On projective invariantsmoothing and evolutions of planar curves and polygons,”Journal of Mathematical Imaging and Vision, Vol. 7, pp. 225–240, 1997.

28. R.L. Bryant, S.S. Chern, R.B. Gardner, H.L. Goldschmidt, andP.A. Griffiths, Exterior Differential Systems, Springer-Verlag:New York, 1991.

29. P.J. Burt and E.H. Adelson, “The Laplacian pyramid as a com-pact image code,” IEEE Trans. Communications, Vol. 9, pp.532–540, 1983.

30. E. Calabi, P.J. Olver, C. Shakiban, A. Tannenbaum, and S.Haker, “Differential and numerically invariant signature curvesapplied to object recognition,” International Journal of Com-puter Vision, Vol. 26, pp. 107–135, 1998.

31. E. Calabi, P.J. Olver, and A. Tannenbaum, “Affine geometry,curve flows, and invariant numerical approximations,” Adv. inMath., Vol. 124, pp. 154–196, 1996.

32. G. Calugareanu, “Sur les classes d’isotopie des noeuds tridi-mensionels et leurs invariants,” Czechoslovak Math. J., Vol. 11,pp. 588–625, 1961.

33. E. Cartan, “Les espaces a connexion conforme,” Ann. Soc. Plon.Math., Vol. 2, pp. 171–221, 1923.

34. E. Cartan, Geometrie des Espaces de Riemann, Gauthiers-Villars: Paris, 1928.

35. E. Cartan, Lecons sur la Theorie des Espaces a ConnexionProjective, Gauthiers-Villars: Paris, 1937.

36. E. Cartan, La Theorie des Groupes Finis et Continus et laGeometrie Differentielle traitees par la Methode du RepereMobile, Gauthiers-Villars: Paris, 1937.

37. E. Cartan, “Les problemes d’equivalence,” in OeuvresCompletes, Gauthiers-Villars: Paris, 1952, Vol. 2, pp. 1311–1334.

38. E. Cartan, Sur les varietes a connexion affine et la theorie dela relativite generalisee, Gauthiers-Villars: Paris, 1955.

39. E. Cartan, Lecons sur la Geometrie des Espaces de Riemann,Gauthier-Villars: Paris, 1963.

40. V. Caselles, F. Catte, T. Coll, and F. Dibos, “A geometric modelfor active contours,” Numerische Mathematik, Vol. 66, pp. 1–31, 1993.

41. F. Catte, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selectivesmoothing and edge detection by nonlinear diffusion,” SIAM J.Num. Anal., Vol. 29, pp. 182–193, 1992.

42. R. Claypoole, G. Davis, W. Sweldens, and R. Baraniuk, “Non-linear wavelet transform for image coding,” in Proceedings ofthe 31-st Asilomar Conference on Singals, Systems, and Com-puters, Vol. 1, 1997, pp. 662–667.

43. A. Connes, Geometrie non commutative, InterEditions: Paris,1990.

44. J.L. Crowley, “A representation for visual information,” Ph.D.thesis, Carnegie-Mellon University, Robotics Institute, Pitts-burgh, Pennsylvania, 1981.

45. J. Damon, “Local Morse theory for solutions to the heat equa-tion and Gaussian blurring,” Jour. Diff. Eqtns., 1993.

46. R.W.R. Darling, Differential Forms and Connections,Cambridge University Press: New York, 1994.

47. D.L. Donoho, “Anisotropy scaling and discontinuities alongcurves,” in Scale–Space Theories in Computer Vision, LectureNotes in Computer Science, Springer: Berlin, Vol. 1682, 1999.

48. D.M. Dubois (Ed.), First International Conference on Com-puting Anticipatory Systems: CASYS’97, AIP Conference Pro-ceedings Vol. 437, American Institute of Physics, College Park,MD, 1998.

49. D.M. Dubois (Ed.), Second International Conference on Com-puting Anticipatory Systems: CASYS’98, AIP Conference Pro-ceedings Vol. 465, American Institute of Physics, College Park,MD, 1999.

50. C.N. de Graaf, S.M. Pizer, A. Toet, J.J. Koenderink, P. Zuidema,and P.P. van Rijk, “Pyramid segmentation of medical 3D im-ages,” in Medical Computer Graphics and Image Communica-tions, 1984, pp. 71–77.

51. D.H. Eberly, “Geometric methods for analysis of ridges in n-dimensional images,” Ph.D. Thesis, Department of ComputerVision, The University of North Carolina, Chapel Hill, NorthCarolina, 1994.

52. S.D. Eidel’Man, Parabolic Systems, North-Holland Publish-ing Company and Wolters-Noordhoff Publishing: Amsterdam,1962.

53. EUMETSAT, User Handbook, meteorological archive and re-trieval facilities edition, issue 2, 1998.

54. A.V. Evako, R. Kopperman, and Y.V. Mukhin, “Dimensionalproperties of graphs and digital spaces,” Journal of Mathemat-ical Imaging and Vision, Vol. 6, pp. 109–119, 1996.


55. O. Faugeras, “Cartan’s moving frame method and its applica-tions to the geometry and evolution of curves in the Euclidean,affine and projective planes,” in Applications of Invariancein Computer Vision, Lecture Notes in Computer Science,Springer-Verlag: Berlin, 1994, Vol. 825, pp. 11–46, 1994.

56. O. Faugeras and R. Keriven, “Scale–spaces and affine curva-ture,” in Proceedings Europe–China Workshop on GeometricModelling and Invariants for Computer Vision, 1995, pp. 17–24.

57. O. Faugeras and R. Keriven, “Some recent results on the pro-jective evolution of 2-D curves,” in Proceedings of the Inter-national Conference on Image Processing, Lausanne, Switzer-land, 1995, Vol. III, pp. 13–16.

58. J. Favard, Cours de Geometrie Differentielle Locale, Gauthier-Villars: Paris, 1957.

59. M. Fels and P. Olver, “Moving co-frames, I: A practical algo-rithm,” Acta Appl. Math., Vol. 51, pp. 161–213, 1998.

60. M. Fels and P. Olver, “Moving co-frames, II: Regularisationand theoretical foundations,” Acta Appl. Math., Vol. 55, pp.127–208, 1999.

61. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, andM.A. Viergever, “Families of tuned scale–space kernels,” inProceedings of the European Conference on Computer Vision,1992, pp. 19–23.

62. L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koenderink, andM.A. Viergever, “Cartesian differential invariants in scale–space,” Journal of Mathematical Imaging and Vision, Vol. 3,pp. 327–348, 1993.

63. L.M.J. Florack, “The syntactical structure of scalar images,”Ph.D. thesis, Utrecht University, The Netherlands, 1993.

64. L.M.J. Florack, A.H. Salden, B.M. ter Haar Romeny, J.J.Koenderink, and M.A. Viergever, “Nonlinear scale–space,” inGeometry-Driven Diffusion in Computer Vision, Kluwer Aca-demic Publishers: Dordrecht, 1994, pp. 339–370.

65. L.M.J. Florack, A.H. Salden, B.M. ter Haar Romeny, J.J.Koenderink, and M.A. Viergever, “Nonlinear scale–space,” Im-age and Vision Computing, Vol. 13, pp. 279–294, 1995.

66. L.M.J. Florack, Image Structure, Vol. 10 of ComputationalImaging and Vision Series, Kluwer Academic Publishers:Dordrecht, 1996.

67. L.M.J. Florack and M. Nielsen, “The intrinsic structure of theoptic flow field,” International Journal of Computer Vision,Vol. 27, pp. 263–286, 1998.

68. L.M.J. Florack and A. Kuijper”, “The topological structure ofscale–space images,” Journal of Mathematical Imaging andVision, Vol. 12, pp. 65–79, 2000.

69. M. Freedman, Z.-X. He, and Z. Wang, “Mobius energies ofknots and unknots,” Ann. Math., Vol. 139, pp. 1–50, 1994.

70. J. Garding, “Shape from texture for smooth curved surfaces inperspective projection,” Journal of Mathematical Imaging andVision, Vol. 2, pp. 329–352, 1992.

71. J. Garding and T. Lindeberg, “Direct estimation of local surfaceshape in a fixating binocular vision system,” in ProceedingsEuropean Conference on Computer Vision, Lecture Notes inComputer Science, Springer-Verlag: Berlin, 1994, Vol. 800,pp. 365–376.

72. R.B. Gardner, The Method of Equivalence and Its Applica-tions, CBMS-NSF Regional Conference Series in AppliedMathematics, Society for Industrial and Applied Mathematics,Philadelphia, Pennsylvania, 1989.

73. S. Geman and D. Geman, “Stochastic relaxation, Gibbs distri-butions, and the Bayesian restoration of images,” IEEE Trans.Pattern Analysis and Machine Intelligence, Vol. 6, pp. 721–741, 1984.

74. V.P. Gerdt and A.Y. Zharkov, in Proc. ISSAC’90, 1990, pp.250–254.

75. G. Gerig, O. Kubler, R. Kikinis, and F.A. Jolesz. “Nonlinearanisotropic filtering of MRI data,” IEEE Transactions on Med-ical Imaging, Vol. 11, pp. 221–232, 1992.

76. C.R. Giardina and E.R. Dougherty (Eds.), MorphologicalMethods in Image and Signal Processing, Prentice-Hall:Englewood Cliffs, 1988.

77. U. Goktas, H.W. Hereman, and G. Erdmann, “Computationof conserved densities for systems of nonlinear differential-difference equations,” Physics Letters A, Vol. 236, pp. 30–38,1997.

78. U, Goktas and W. Hereman, “Computation of conserved densi-ties for systems of nonlinear differential-difference equations,”Journal of Symbolic Computation, Vol. 24, pp. 591–621, 1997.

79. U, Goktas and W. Hereman, “Computation of conserved den-sities for nonlinear lattices,” Physica D, Vol. 123, pp. 425–436,1998.

80. M.L. Green, “The moving frame, differential invariants andrigidity theorems for curves in homogeneous spaces,” DukeMathematical Journal, Vol. 45, pp. 735–779, 1978.

81. U. Grenander, Lectures in Pattern Theory I, II and III: PatternAnalysis, Pattern Synthesis and Regular Structures, Springer-Verlag: Berlin, 1976-81.

82. G.E. Healey and S.A. Shafer (Eds.), Physics-based Vision:Principles and Practice; Colour, Jones and Bartlett Publish-ers: Boston, London, 1992.

83. F.W. Hehl, J. Dermott McCrea, and E.W. Mielke, Weyl Space-times, The Dilation Current, And Creation Of Gravitating MassBy Symmetry Breaking, Verlag Peter Lang: Frankfurt, 1988, pp.241–311.

84. H.J.A.M. Heijmans, “Mathematical morphology: A geomet-rical approach in image processing,” Nieuw Archief voorWiskunde, Vol. 10, pp. 237–276, 1992.

85. H.J.A.M. Heijmans, “Mathematical morphology as a tool forshape description,” in Proc. of the NATO Advanced ResearchWorkshop Shape in Picture—Mathematical Description ofShape in Grey-Level Images, Vol. 126 of NATO ASI SeriesF, pp. 147–176, 1994.

86. H.J.A.M. Heijmans (Ed.), Morphological Image Operators,Academic Press: San Diego, 1994.

87. Y. Hel-Or, S. Peleg, and D. Avnir, “Charaterization of right-handed and left-handed shapes,” in Proceedings CVGIP:IU,1991, Vol. 53, pp. 297–302.

88. Y. Hel-Or and P. Teo, “Canonical decomposition of steerablefilters,” Journal of Mathematical Imaging and Vision, Vol. 9,pp. 83–95, 1998.

89. W. Hereman, U. Goktas, and A. Miller, “Algorithmic inte-grability tests for nonlinear differential and lattice equations,”Computer Physics Communications, Vol. 115, pp. 428–446,1998.

90. G.T. Herman and Z. Enping, “Jordan surfaces in simply con-nected digital spaces,” Journal of Mathematical Imaging andVision, Vol. 6, pp. 121–138, 1996.

91. D. Hilbert, Theory of Algebraic Invariants, Cambridge Univer-sity Press: New York, 1993.


92. D.H. Hubel and T.N. Wiesel, “Receptive fields, binocular in-teraction, and functional architecture in the cat’s visual cortex,”Journal of Physiology, Vol. 160, pp. 106–154, 1962.

93. R.A. Hummel, “The scale–space formulation of pyramid datastructures,” in Parallel Computer Vision, Academic Press: NewYork, 1987, pp. 187–123.

94. T. Iijima, “Basic theory on the normalisation of a pattern,”Bulletin of Electrical Laboratory, Vol. 26, pp. 368–388, 1962.

95. T. Iijima, “Basic equation of figure and observational trans-formation,” Transactions of the Institute of Electronics andCommunication Engineers of Japan, Vol. 54-C(11), pp. 37–38, 1971.

96. T. Iijima, “Basic theory on the construction of figure space,”Transactions of the Institute of Electronics and CommunicationEngineers of Japan, Vol. 54-C(8), pp. 35–36, 1971.

97. T. Iijima, “A suppression kernel of a figure and its mathematicalcharacteristics,” Transactions of the Institute of Electronics andCommunication Engineers of Japan, Vol. 54-C(9), pp. 30–31,1971.

98. T. Iijima, “Basic theory on normalisation of figure,” Transac-tions of the Institute of Electronics and Communication Engi-neers of Japan, Vol. 54-C(11), pp. 24–25, 1971.

99. T. Iijima, “A system of fundamental functions in an abstractfigure space,” Transactions of the Institute of Electronics andCommunication Engineers of Japan, Vol. 54-C(11), pp. 26–27,1971.

100. T. Iijima, “Basic theory on feature extraction of figures,” Trans-actions of the Institute of Electronics and Communication En-gineers of Japan, Vol. 54-C(12), pp. 35–36, 1971.

101. T. Iijima, “Basic theory on the structural recognition of a fig-ure,” Transactions of the Institute of Electronics and Commu-nication Engineers of Japan, Vol. 55-D(8), pp. 28–29, 1972.

102. T. Iijima, “Theoretical studies on the figure identification bypattern matching,” Transactions of the Institute of Electronicsand Communication Engineers of Japan, Vol. 54-D(8), pp. 29–30, 1972.

103. B. Jawerth and W. Sweldens, “An overview of the theory andapplications of wavelets,” in Proc. of the NATO Advanced Re-search Workshop Shape in Picture—Mathematical Descriptionof Shape in Grey-Level Images, Vol. 126 of NATO ASI SeriesF, 1994, pp. 249–274.

104. G.R. Jensen, Higher Order Contact of Submanifolds of Homo-geneous Spaces, Springer-Verlag: Berlin, 1977.

105. P. Johansen, S. Skelboe, K. Grue, and J.D. Andersen, “Repre-senting signals by their top-points in scale-space,” in Proceed-ings of the 8-th International Conference on Pattern Recogni-tion, 1986, pp. 215–217.

106. P. Johansen, “On the classification of top-points in scale–space,” Journal of Mathematical Imaging and Vision, Vol. 4,pp. 57–68, 1994.

107. J.-J. Jolion and A. Rozenfeld, A Pyramid Framework for EarlyVision, Kluwer Academic Publishers: Dordrecht, The Nether-lands, 1994.

108. A. Kadi and D.G.B. Edelen, A Gauge Theory of Dislocationsand Disclinations, Springer-Verlag: Berlin, 1983.

109. S.N. Kalitzin, B.M. ter Haar Romeny, A.H. Salden, P.F.M.Nacken and M.A. Viergever, “Topological numbers and sin-gularities in scalar images; scale–space evolution properties,”Journal of Mathematical Imaging and Vision, Vol. 9, pp. 253–269, 1998.

110. S.N. Kalitzin, B.M. ter Haar Romeny, and M.A. Viergever,“Invertible orientation bundles on 2D scalar images,” in Scale–Space Theories in Computer Vision, Lecture Notes in ComputerScience, Springer-Verlag: Berlin, 1997, Vol. 1252, pp. 77–88.

111. S.N. Kalitzin, B.M. ter Haar Romeny, and M.A. Viergever,“On topological deep-structure segmentation,” in ProceedingsInternational Conference on Image Processing, pp. 863–866,1997.

112. S.N. Kalitzin, B.M. ter Haar Romeny, and M.A. Viergever,“Invertible apertured orientation filters in image image anal-ysis,” International Journal of Computer Vision, Vol. 31, pp.145–158, 1999.

113. S.N. Kalitzin, J. Staal, B.M. ter Haar Romeny, and M.A.Viergever, “Frame-relative critical point-sets in image analy-sis,” in Proc. Conf. on Computer Analysis of Images and Pat-terns, CAIP99, Lecture Notes in Computer Science, Vol. 1354,Springer-Verlag, Lijubljana, Slovenia, in press.

114. N. Kamran, R. Milson, and P.J. Olver, “Invariant modules andthe reduction of nonlinear partial differential equations to dy-namical systems,” Adv. in Math., to appear.

115. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes: Active con-tour models,” in Proc. IEEE First Int. Comp. Vision Conf., 1987,pp. 259–268.

116. A. Katok and B. Hasselblatt, Introduction to the Modern Theoryof Dynamical Systems, Cambridge University Press: New York,1995.

117. L.H. Kauffman, “Noncommutativity and discrete physics,”Physica D, pp. 125–138, 1998.

118. R.K. Kaul, “Chern–Simons theory, knot invariants, vertex mod-els and three-manifold invariants,” in Frontiers of Field The-ory, Quantum Gravity and Strings, Puri, India, 1996. ReportIMSc-98/04/15, Institute of Mathematical Sciences, Taramani,Madras, India.

119. A. Khinchine, Mathematical Foundations of Information The-ory, Dover Publications: New York, 1957.

120. S. Kichenassamy, A. Kumar, P.J. Olver, A. Tannenbaum, andA. Yezzi, “Conformal curvature flows: From phase transitionsto active vision,” Arch. Rat. Mech. Anal., Vol. 134, pp. 275–301,1996.

121. B.B. Kimia, “Towards a computational theory of Shape,” Ph.D.thesis, Department of Electrical Enginering, McGill University,Montreal, Canada, 1990.

122. B. Kimia, A. Tannenbaum, and S. Zucker, “On optimal con-trol methods in computer vision and image processing,” inGeometry-Driven Diffusion in Computer Vision, 1994, pp. 307–338.

123. B.B. Kimia, A. Tannenbaum, and S.W. Zucker, “Shapes,shocks, and deformations, I,” International Journal of Com-puter Vision, Vol. 15, pp. 189–224, 1994.

124. R. Kimmel, K. Siddiqi, B.B. Kimia, and A. Bruckstein, “Shapefrom shading: Level set propagation and viscosity solutions,”International Journal of Computer Vision, Vol. 16, pp. 107–133, 1995.

125. H. Kleinert, Gauge Fields in Condensed Matter, Vol. 1–2.World Scientific Publishing: Singapore, 1989.

126. Jan J. Koenderink, “A hitherto unnoticed singularity of scale–space,” IEEE Trans. Pattern Analysis and Machine Intelli-gence, Vol. 11, pp. 1222–1224, 1989.

127. J.J. Koenderink, “The structure of images,” Biol. Cybern., Vol.50, pp. 363–370, 1984.


128. J.J. Koenderink, “Scale-time,” Biol. Cybern., Vol. 58, pp. 159–162, 1988.

129. J.J. Koenderink and W. Richards, “Two-dimensional curvatureoperators,” Journal of the Optical Society of America-A, Vol.5, pp. 1136–1141, 1988.

130. J.J. Koenderink and A.J. van Doorn, “A description of thestructure of visual images in terms of an ordered hierarchyof light and dark blobs,” in Second Int. Visual Psychophysicsand Medical Imaging Conf., IEEE Cat. No. 81 CH 1676-6,1981.

131. J.J. Koenderink and A.J. van Doorn, “Dynamic shape,” Biol.Cybern., Vol. 53, pp. 383–396, 1986.

132. J.J. Koenderink and A.J. van Doorn, “Local features of smoothshapes: Ridges and courses,” in Proc. SPIE Geometric Methodsin Computer Vision II, Vol. 2031, pp. 2–13, 1993.

133. H.J. Kushner, Probability Methods for Approximations InStochastic Control and For Elliptic Equations, Vol. 129 ofMathematics In Science And Engineering, Academic Press:New York, 1977.

134. A. Lakhtakia, V.K. Varadan, and V.V. Varadan, Time-HarmonicElectromagnetic Fields in Chiral Media, Lecture Notes inPhysics, Springer-Verlag: Berlin, 1989, Vol. 335.

135. C. Landi and P. Frosini, “Size functions and morphologicaltransformations,” Acta Applicandae Mathematicae, Vol. 49, pp.85–104, 1997.

136. T. Lindeberg, “Scale–space for discrete signals,” IEEE Trans.Pattern Analysis and Machine Intelligence, Vol. 12, pp. 234–245, 1990.

137. T. Lindeberg and J. Garding, “Shape from texture from a multi-scale perspective,” in Proceedings 4th International Confer-ence on Computer Vision, 1993, pp. 683–691.

138. T. Lindeberg, Scale–Space Theory in Computer Vision, TheKluwer International Series in Engineering and Computer Sci-ence., Kluwer Academic Publishers: Dordrecht, The Nether-lands, 1994.

139. T. Lindeberg and J. Garding, “Shape-adapted smoothing in es-timation of 3-D depth cues from affine distortions of local 2-Dstructure,” in Proceedings European Conference on ComputerVision, Lecture Notes in Computer Science, Springer-Verlag:Berlin, 1994, Vol. 800, pp. 389–400.

140. T. Lindeberg, “Direct estimation of affine deformations ofbrightness patterns using visual front-end operators with au-tomatic scale selection,” in Proc. 5th International Conferenceon Computer Vision, 1995, pp. 134–141. I

141. I. Lisle, “Equivalence transformations for classes of differen-tial equations,” Ph.D. thesis, University of British Columbia,Vancouver, Canada, 1992.

142. M. Nielsen, R. Maas, W.J. Niessen, L.M.J. Florack, and B.M.ter Haar Romeny, “Binocular stereo from grey-scale image,”Journal of Mathematical Imaging and Vision, Vol. 10, pp. 103–122, 1999.

143. R. Malladi, J.A. Sethian, and B.C. Vemuri, “A fast level setbased algorithm for topology-independent shape modelling,”Journal of Mathematical Imaging and Vision, Vol. 6, pp. 269–289, 1996.

144. S.G. Mallat, “A theory for multiresolution signal decomposi-tion: The wavelet representation,” IEEE Trans. Pattern Analysisand Machine Intelligence, Vol. 11, pp. 674–694, 1989.

145. G. Matheron, Random Sets and Integral Geometry, Wiley: NewYork, 1975.

146. M.H. McAndrew and C.A. Osborne, “A survey of algebraicmethods in digital topology,” Journal of Mathematical Imagingand Vision, Vol. 6, pp. 139–159, 1996.

147. N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H.Teller, and E. Teller, “Equation of the state calculations byfast computing machine,” Journal of Chemical Physics, Vol.21, pp. 1087–1091, 1953.

148. F. Meyer, “Minimum spanning forests for morphological seg-mentation,” Mathematical Morphology and Its Applications toImage Processing, Kluwer Academic Publishers: Dordrecht,The Netherlands, 1994, pp. 77–84.

149. F. Mokhtarian and A. Mackworth, “Scale-based descriptionof planar curves and two-dimensional shapes,” IEEE Trans.Pattern Analysis and Machine Intelligence, Vol. 8, pp. 34–43,1986.

150. T. Moons, E. Pauwels, L. Van Gool, and A. Oosterlinck, “Foun-dations of semi-differential invariants,” Journal of Mathemat-ical Imaging and Vision, Vol. 14, pp. 25–47, 1995.

151. D. Mumford and J. Shah, “Boundary detection by minimisingfunctionals,” in Proceedings IEEE Conference on ComputerVision and Pattern Recognition, 1985, pp. 22–26.

152. D. Mumford and J. Shah, “Optimal approximations by piece-wise smooth functions and associated variational problems,”Communications on Pure and Applied Mathematics, Vol. XLII,pp. 577–685, 1989.

153. D. Mumford, “Bayesian rationale for the variational formula-tion,” in Geometry-Driven Diffusion in Computer Vision, Com-putational Imaging and Vision, Kluwer Academic Publishers:Dordrecht, The Netherlands, 1994, pp. 135–146.

154. D. Mumford, “The statistical description of visual signals,” inICIAM 95, Akademie Verlag: Berlin, 1996.

155. D. Mumford, “Pattern Theory: A unifying perspective,” in Per-ception as Bayesian Inference, Cambridge University Press:New York, 1996, pp. 25–62.

156. C. Nash, Differential Topology and Quantum Field Theory,Academic Press: London, 1990.

157. M. Newman, “A fundamental group for grey-scale digital im-ages,” Journal of Mathematical Imaging and Vision, Vol. 6, pp.139–159, 1996.

158. W.J. Niessen, J.S. Duncan, L.M.J. Florack, B.M. ter HaarRomeny, and M.A. Viergever, “Spatiotemporal operators andoptic flow,” in Physics-Based Modeling in Computer Vision,pp. 78–84, 1995.

159. M. Nitzberg and T. Shiota, “Nonlinear image filtering withedge and corner enhancement,” IEEE Trans. Pattern Analysisand Machine Intelligence, Vol. 14, pp. 826–833, 1992.

160. N. Nordstrom, “Biased anisotropic diffusion—a unified regu-larisation and diffusion approach to edge detection,” Image andVision Computing, Vol. 8, pp. 318–327, 1990.

161. N. Nordstrom, Variational Edge Detection, Ph.D. thesis, Uni-versity of California at Berkeley, May 1990.

162. P.J. Olver, Applications of Lie Groups to Differential Equations,Vol. 107 of Graduate Texts in Mathematics, Springer-Verlag:Berlin, 1986.

163. P.J. Olver, “Invariant theory, equivalence problems and the cal-culus of variations,” in Invariant Theory and Tableaux, IMAVolumes in Mathematics and Its Applications, Springer-Verlag:Berlin, 1990, Vol. 19, pp. 59–81.

164. P.J. Olver, G. Sapiro, and A. Tannenbaum, “Classifica-tion and uniqueness of invariant geometric flows,” Comptes


Rendus Acad. Sci. (Paris), Serie I, Vol. 319, pp. 339–344,1994.

165. P. Olver, G. Sapiro, and A. Tannenbaum, “Differential invari-ant signatures and flows in computer vision: A symmetry groupapproach,” in Geometry-Driven Diffusion in Computer Vision,Computational Imaging and Vision, Kluwer Academic Pub-lishers: Dordrecht, The Netherlands, 1994, pp. 255–306.

166. P. Olver, Equivalence, Invariants, and Symmetry, CambridgeUniversity Press: New York, 1995.

167. P.J. Olver, G. Sapiro, and A. Tannenbaum, “Invariant geomet-ric evolutions of surfaces and volumetric smoothing,” SIAM J.Appl. Math., Vol. 57, pp. 176–194, 1997.

168. P.J. Olver, G. Sapiro, and A. Tannenbaum, “Affine invariantedge maps and active contours,” Acta Appli. Math., to appear.

169. S. Osher and J.A. Sethian, “Fronts propagating with curvaturedependent speed: Algorithms based on the Hamilton-Jacobiformalism,” J. Computational Physics, Vol. 79, pp. 12–49,1988.

170. L.V. Ovsiannikov, Group Analysis of Differential Equations,Academic Press: New York, 1982.

171. P. Perona and J. Malik, “Scale–space and edge detection us-ing anisotropic diffusion,” IEEE Trans. Pattern Analysis andMachine Intelligence, Vol. 12, pp. 629–639, 1990.

172. R.A. Pielke. Meso-scale Meteorological Modeling, AcademicPress: New York, 1984.

173. S. Piunikhin, “Weights of Feynman diagrams, link polynomialsand Vassiliev knot invariants,” Journal of Knot Theory and itsRamifications, Vol. 4, pp. 163–188, 1995.

174. W.F. Pohl. “The self-linking number of a closed space curve,”Journal of Mathematics and Mechanics, Vol. 17, pp. 975–985,1968.

175. J.F. Pommaret, Systems of Partial Differential Equations andLie Pseudogroups, Gordon and Breach: New York, 1978.

176. J.F. Pommaret, Differential Galois Theory, Gordon and Breach:New York, 1983.

177. J.F. Pommaret, Lie Pseudogroups and Mechanics, Gordon andBreach: New York, 1988.

178. L. Pontryagin, Topological Groups, Princeton University Press:Princeton, N.J., 1946.

179. F. Preteux, “Watershed and skeleton by influence zones: Adistance-based approach,” Journal of Mathematical Imagingand Vision, Vol. 1, pp. 239–256, 1992.

180. M. Proesmans, E. Pauwels, and L. Van Gool, “Coupledgeometry-driven diffusion equations for low level vision,”in Geometry-Driven Diffusion in Computer Vision, Compu-tational Imaging and Vision, Kluwer Academic Publishers:Dordrecht, The Netherlands, 1994, pp. 191–228.

181. T. Richardson and S. Mitter, “Approximation, computation,and distortion in the variational formulation,” in Geometry-Driven Diffusion in Computer Vision, Computational Imag-ing and Vision, Kluwer Academic Publishers: Dordrecht, TheNetherlands, 1994, pp. 169–190.

182. J.H. Rieger, “Generic evolutions of edges on families of dif-fused grey-value surfaces,” Journal of Mathematical Imagingand Vision, Vol. 5, pp. 207–217, 1995.

183. R. Rosen, Anticipatory Systems, Pergamon, New York, 1985.184. R. Rothe, “Zum problem des talwegs,” Sitz. ber. d. Berliner

Math. Gesellschaft, Vol. 14, pp. 51–69, 1915.185. N. Rougon and F. Preteux, “Deformable markers: Mathe-

matical morphology for active contour models control,” in

Proceedings SPIE’s International Symposium on Optical Ap-plied Science and Engineering—Image Algebra and Morpho-logical Image Processing II, Vol. 1568, pp. 78–89, 1991.

186. N. Rougon, “Elements pour la reconnaissance des formestridimensionnelles deformables. Applications a l’imageriebiomedicale,” Ph.D. thesis, Telecom Paris—Department Im-ages, Paris: France, 1993.

187. N. Rougon, “On mathematical foundations of local deforma-tion analysis,” in Proceedings SPIE’s International Symposiumon Optical Applied Science and Engineering—MathematicalMethods in Medical Imaging II, Vol. 2035, pp. 2–13, 1993.

188. N. Rougon, “Geometric Maxwell equations and the struc-ture of diffusive scale-spaces,” in Proceedings SPIE Confer-ence on Morphological Image Processing and Random ImageModeling—SPIE’s 1995 International Symposium on OpticalScience, Engineering and Instrumentation, Vol. 2568, pp. 151–165, 1995.

189. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total vari-ation based noise removal algorithms,” in ModelisationsMatematiques pour le traitement d’images, 149–179. INRIA,1992.

190. L.I. Rudin and S. Osher, “Total variation based image restora-tion with free local constraints,” in Proc. First IEEE Internat.Conf. on Image Processing, 1994, pp. 13–16.

191. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever,“Local and multilocal scale-space description,” in Proc. ofthe NATO Advanced Research Workshop Shape in Picture—Mathematical Description of Shape in Grey-Level Iimages, Vol.126 of NATO ASI Series F, Berlin, 1994, pp. 661–670.

192. A.H. Salden, L.M.J. Florack, B.M. ter Haar Romeny, J.J. Koen-derink, and M.A. Viergever, “Multi-scale analysis and descrip-tion of image structure,” Nieuw Archief voor Wiskunde, Vol.10, pp. 309–326, 1992.

193. A.H. Salden, B.M. ter Haar Romeny, L.M.J. Florack, J.J. Koen-derink, and M.A. Viergever, “A complete and irreducible setof local orthogonally invariant features of 2-dimensional im-ages,” in Proceedings 11th IAPR Internat. Conf. on PatternRecognition, The Hague, The Netherlands, 1992, pp. 180–184.

194. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Im-age structure generating normalised geometric scale spaces,”in Volume Image Processing ’93, pp. 141–143, Utrecht, TheNetherlands, 1993.

195. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever,“Modern geometry and dynamic scale–space theories,” in Proc.Conf. on Differential Geometry and Computer Vision: FromPure over Applicable to Applied Differential Geometry, Nord-fjordeid, Norway, 1995, in press.

196. A.H. Salden, “Dynamic Scale–Space Paradigms,” Ph.D. thesis,Utrecht University: The Netherlands, 1996.

197. A.H. Salden, “Invariant theory,” in Gaussian Scale–SpaceTheory, Kluwer Academic Publishers: Dordrecht, The Nether-lands, 1996.

198. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Dy-namic scale–space theories,” in Scale–Space 97, First Interna-tional Conference on Scale–Space Theory in Computer Vision,Utrecht, the Netherlands, 1997, pp. 248–259.

199. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Dif-ferential and integral geometry of linear scale spaces,” Jour-nal of Mathematical Imaging and Vision, Vol. 9, pp. 5–27,1996.


200. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Lin-ear scale–space theory from physical principles,” Journal ofMathematical Imaging and Vision, Vol. 9, pp. 103–139, 1998.

201. A.H. Salden, “Algebraic invariants of linear scale-spaces,” inAsian Conference on Computer Vision, Lecture Notes in Com-puter Science, Springer-Verlag: Berlin, 1998, Vol. 1352, pp.65–72.

202. A.H. Salden, “Satellite image analysis and processing toolsto be used in air-pollution forecast and simulation systems,”System Analysis Modelling Simulation, 4-th ERCIM Environ-mental Modelling Group Workshop, Vol. 39, pp. 131–168,2000.

203. A.H. Salden, J. Weickert, and B.M. ter Haar Romeny, “Blumanand Kumei’s nonlinear scale–space theory,” Technical ReportERCIM-02/99-R057, February 1999.

204. A.H. Salden, B.M. ter Haar Romeny, and M.A. Viergever, “Lin-earised Euclidean shortening flow of curve geometry,” Interna-tional Journal of Computer Vision, Vol. 34, pp. 29–67, 1999.

205. A.H. Salden, “Dynamic scale–space paradigm versus mathe-matical morphology?,” in AeroSense’99, SPIE’s 13-th AnnualSymposium on Aerospace/Defence Sensing, Simulation, andControl, Orlando, Florida, USA, pp. 155–166, 1999.

206. A.H. Salden, “Modeling and analysis of sustainable systems,”ERCIM, 1999, in press.

207. A.H. Salden and S. Iacob, “Dynamic multi-scale image clas-sification,” in SPIE’s International Symposium on ElectronicImaging 2001, Internet Imaging EI24, 2001, in press.

208. J. Sanders and J. Wang, “Combining maple and form to de-cide on integrability questions,” Computer Physics Communi-cations, Vol. 115, pp. 447–459, 1998.

209. J. Sanders and J. Wang, “On the integrability of homogeneousscalar evolution equations,” Journal of Differential Equations,Vol. 147, pp. 410–434, 1998.

210. J.A. Sanders and J. Wang, “Classification of conservation lawsfor KdV-like equations,” Math. Comput. Simulation, Vol. 44,pp. 471–481, 1997.

211. L.A. Santalo, “Integral geometry in general spaces,” in Pro-ceedings International Congress of Mathematics, Vol. 1, Cam-bridge, 1950, pp. 483–489.

212. G. Sapiro and A. Tannenbaum, “Affine invariant scale-space,”International Journal of Computer Vision, Vol. 11, pp. 25–44,1993.

213. G. Sapiro, R. Kimmel, D. Shaked, B.B. Kimia, and A.M. Bruck-stein, “Implementing continuous-scale morphology via curveevolution,” Pattern Recognition, Vol. 26, pp. 1363–1372, 1993.

214. G. Sapiro and A. Tannenbaum, “Image smoothing based on anaffine invariant flow,” in Conference on Information Sciencesand Systems, Johns Hopkins University, 1993, pp. 196–201.

215. G. Sapiro and A. Tannenbaum, “Formulating invariant heat-type curve flows,” in SPIE Conference on Geometric Methodsin Computer Vision II, Vol. 2031, pp. 234–245, 1993.

216. G. Sapiro and A. Tannenbaum, “On invariant curve evolutionand image analysis,” Indiana Journal of Mathematics, Vol. 42,pp. 985–1009, 1993.

217. G. Sapiro and A. Tannenbaum, “Area and length preserving ge-ometric invariant scale–spaces,” IEEE Trans. Pattern Analysisand Machine Intelligence, Vol. 17, pp. 67–72, 1995.

218. G. Sapiro and A. Tannenbaum, “On affine plane curve evolu-tion,” Journal of Functional Analysis, Vol. 119(1), pp. 79–120,1994.

219. O. Scherzer and J. Weickert, “Relations between regularizationand diffusion filtering,” Technical Report DIKU-98/23, Dept.of Computer Science, University of Copenhagen, Denmark,1998.

220. C. Schmidt and R. Mohr, “Combining grey-value invariantswith local constraints for object recognition,” in Proc. Intern.Conf. on Computer Vision and Pattern Recognition, San Fran-cisco, USA, 1996.

221. J. Segman and W. Schempp, “Two ways to incorporate scalein the Heisenberg group with an intertwining operator,” Jour-nal of Mathematical Imaging and Vision, Vol. 3, pp. 79–94,1993.

222. J. Segman and Y.Y. Zeevi, “Image analysis by wavelet-typetransforms: Group theoretic approach,” Journal of Mathemat-ical Imaging and Vision, Vol. 3, pp. 51–77, 1993.

223. J. Serra, Image Analysis and Mathematical Morphology. Aca-demic Press: New York, 1982.

224. J. Shah, “Segmentation by nonlinear diffusion,” in Conferenceon Computer Vision and Pattern Recognition, pp. 202–207,1991.

225. J. Shah, “Segmentation by minimising functionals: Smoothingproperties,” SIAM J. Control and Optimisation, Vol. 30, pp.99–111, 1992.

226. J. Shah, “A nonlinear model for discontinuous disparity andhalf-occlusions in stereo,” in Proc. Computer Vision and Pat-tern Recognition, pp. 34–40, 1993.

227. K. Sivakumar and J. Goutsias (Eds.), Mathematical Morphol-ogy and Its Applications to Image Processing, Kluwer Aca-demic Publishers: London, 1994.

228. K. Sivakumar and J. Goutsias, “Morphological size den-sities: Applications to optimal texture classification andnoise filtering,” in 1997 IEEE/EURASIP Workshop on Non-linear Signal and Image Processing, Causal Productions,CD-ROM, paper 113, Mackinac Island, Michigan, USA,1997.

229. E.H. Spanier, Algebraic Topology, McGraw-Hill: New York,1966.

230. J. Sporring, “The entropy of scale–space,” in Proceedings of 13-th International Conference on Pattern Recognition, Vienna,Austria, 1996.

231. J. Sporring, M. Nielsen, J. Weickert, and O.F. Olsen, “A noteon differential corner measures,” in Proceedings InternationalConference on Pattern Recognition, Brisbane, Australia, 1996,pp. 652–654.

232. J. Sporring and J. Weickert, “Information measures in scale–spaces,” IEEE Trans. Information Theory, Vol. 45, pp. 1051–1058, 1999.

233. J. Staal, S.N. Kalitzin, B.M. ter Haar Romeny, and M.A.Viergever, “Detection of critical structures in scale–space,” inScale-space theories in computer vision, Lecture Notes in Com-puter Science, Vol. 1682, Springer, Berlin, pp. 105–116, 1999.

234. S. Tanimoto and A. Klinger (Eds.), Structured Computer Vi-sion, Academic Press: New York, 1980.

235. S. Tari, J. Shah, and H. Pien, “Extraction of shape skeletonsfrom grey-scale images,” Computer Vision and Image Under-standing, Vol. 66, pp. 133–146, 1997.

236. S. Tari, “Colouring 3D symmetry set: Perceptual meaning andsignificance in 3D,” in AeroSense’99, SPIE’s 13-th Annual In-ternational Symposium on Aerospace/Defence Sensing, Simu-lation, and Control, Vol. 3716, pp. 105–111, 1999.


237. B.M. ter Haar Romeny (Ed.), Geometry-Driven Diffusion inComputer Vision, Kluwer Academic Publishers: Dordrecht,1994.

238. T.Y. Thomas, The Differential Invariants of GeneralisedSpaces, Cambridge University Press: New York, 1934.

239. D. Thurston, Integral Expressions for the Vassiliev Knot Invari-ants, Harvard University, U.S.A. Senior Thesis, 1995.

240. C.-H. Tze, “Manifold-splitting regularization, self-linking,twisting, writhing numbers of space-time ribbins andpolyakov’s proof of Fermi–Bose transmutations,” Interna-tional Journal of Modern Physics A, Vol. 3, pp. 1959–1979,1988.

241. R. van den Boomgaard, “Mathematical morphology: exten-sions towards computer Vision,” Ph.D. thesis, University ofAmsterdam, Amsterdam, The Netherlands, 1992.

242. R. van den Boomgaard, “The morphological equivalent of theGauss convolution,” Nieuw Archief voor Wiskunde, Vol. 10, pp.219–236, 1992.

243. R. van den Boomgaard and A.W.M. Smeulders, “Morphologi-cal multi-scale image analysis,” in Mathematical Morphologyand its Applications to Signal Processing, Barcelona, Spain,May 1993, pp. 180–185.

244. R. van den Boomgaard and A.W.M. Smeulders, “The mor-phological structure of images, the differential equations ofmorphological scale–space,” IEEE Transactions on PatternAnalysis and Machine Intelligence, Vol. 16, pp. 1101–1113,1994.

245. R. van den Boomgaard and J. Schavemaker, “Image segmenta-tion in morphological scale–space,” in ISMM’94: Mathemati-cal Morphology and Its Application to Signal Processing 2—Poster Contribution—, pp. 15–16, 1994.

246. R. van den Boomgaard and L. Dorst, “The morphologicalequivalent of Gaussian scale–space,” in Gaussian Scale-SpaceTheory, Kluwer Academic Publishers: Dordrecht, 1996, pp.203–220.

247. V.A. Vassiliev, “Cohomology of knot spaces,” Advances in So-viet Mathematics, Vol. 1, pp. 23–69, 1990.

248. V.A. Vassiliev, “Topology of complements to discriminants andloop spaces,” Advances in Soviet Mathematics, Vol. 1, pp. 9–21,1990.

249. L. Vincent and P. Soille, “Watersheds in digital spaces: An effi-cient algorithm based on immersion simulations,” IEEE Trans.Pattern Analysis and Machine Intelligence, Vol. 13, pp. 583–598, 1991.

250. J.P. Wang, Symmetries and Conservation Laws of EvolutionEquations, Ph.D. thesis, Thomas Stieltjes Institute for Mathe-matics, Vrije Universiteit, Amsterdam, The Netherlands, 1998.

251. J. Weickert, Anisotropic Diffusion in Image Processing, Ph.D.thesis, Dept. of Mathematics, University of Kaiserslautern,Germany, 1996.

252. J. Weickert, “Foundations and applications of nonlinearanisotropic diffusion filtering,” in Numerical Analysis, Scien-tific Computing, Computer Science, Akademie Verlag, Berlin,1996, pp. 283–286.

253. J. Weickert, “Nonlinear diffusion scale–spaces: From the con-tinuous to the discrete setting,” in ICAOS ’96: Images, Waveletsand PDEs, Lecture Notes in Control and Information Sciences,Springer-Verlag: London, 1996, Vol. 219, pp. 111–118.

254. J. Weickert, Anisotropic Diffusion in Image Processing,Teubner-Verlag: Stuttgart, Germany, 1998.

255. J. Weickert, “On discontinuity-preserving optic flow,” in Proc.Computer Vision and Mobile Robotics Workshop, Santorini,Greece, 1998, pp. 115–122.

256. J. Weickert, “Fast segmentation methods based on partial dif-ferential equations and the watershed transformation,” in Mus-tererkennung 1998, Springer-Verlag, Berlin, Germany, 1998,pp. 93–100.

257. J. Weickert, S. Ishikawa, and A. Imiya, “On the history ofgaussian scale–space axiomatics,” in Gaussian Scale–SpaceTheory, Kluwer: Dordrecht, 1996, pp. 45–59.

258. J. Weickert and S. Ishikawa, “Linear scale–space has first beenproposed in Japan,” Journal of Mathematical Imaging and Vi-sion, Vol. 10, pp. 237–252, 1999.

259. A. Weil, L’integration dans les groupes topologiques et sesapplications, Gauthiers Villars: Paris, 1938.

260. R.T. Whitaker and S.M. Pizer, “A multi-scale approach tononuniform diffusion,” Computer Vision, Graphics, and Im-age Processing: Image Understanding, Vol. 57, pp. 99–110,1993.

261. R.T. Whitaker and G. Gerig, “Vector-valued diffusion,” inGeometry-Driven Diffusion in Computer Vision, Computa-tional Imaging and Vision, Kluwer Academic Publishers: Dor-drecht, The Netherlands, 1994, pp. 93–134.

262. J.H. White, “Self-linking and the Gauss integral in higher di-mensions,” American Journal of Mathematics, Vol. 91, pp.693–728, 1969.

263. A.P. Witkin, “Scale–space filtering,” in Proc. InternationalJoint Conference on Artificial Intelligence, Karlsruhe, Ger-many, 1983, pp. 1019–1023.

264. L.B. Wolff, S.A. Shafer, and G.E. Healey (Eds.), Physics-basedVision: Principles and Practice; Radiometry, Jones and BartlettPublishers: London, 1992.

265. A. Yezzi, S. Kichenassamy, A. Kumar, P.J. Olver, and A.Tannenbaum, “A geometric snake model for segmentation ofmedical imagery,” Medical Imaging, Vol. 16, pp. 199–209,1997.

266. A. Yuille, L. Vincent, and D. Geiger, “Statistical morphologyand Bayesian reconstruction,” Journal of Mathematical Imag-ing and Vision, Special Issue: Image Algebra and Morpholog-ical Image Processing, Vol. 1, pp. 223–238, 1992.

267. A.L. Yuille and T.A. Poggio, “Scaling theorems for zero-crossings,” IEEE Trans. Pattern Analysis and Machine Intelli-gence, Vol. 8, pp. 15–25, 1986.

268. O.I. Zavialov, Renormalised Quantum Field Theory, KluwerAcademic Publishers: Dordrecht, 1990.

Alfons H. Salden received a M.Sc. degree in Experimental Physicsin 1992 and a Ph.D. on dynamic scale-space paradigms in 1996


both from Utrecht University, The Netherlands. He developedthese paradigms within the Helmholtz Institute and the Image Sci-ences Institute, Utrecht, The Netherlands. From October 1996 tillFebruary 1998 he held a post-doc position at INRIA Sophia-Antipolis(ROBOTVIS), France developing nonlinear image processing andanalysis tools for medical and satellite images. From June 1998 tillDecember 1999 he was an ERCIM-fellow at INRIA Rocquencourt(AIR), France, and GMD-FIRST (DYMOS), Germany working onremote sensing problems related to air-pollution simulation and fore-cast systems, and on modeling and analysis of sustainable systemscomprising ecology, economy and society, respectively. From May2000 he’s a member of the scientific staff of the Telematics Insti-tute, Enschede, The Netherlands concerned with development ofmulti-media management, mobile and e-business systems for theNext Generation Internets.

His main mathematical research interests range from invarianttheory, differential and integral geometry, theory of partial differen-tial and integral equations, knot theory, topology to category theory.Besides these mathematical fields he’s interested in various fieldsin physics, e.g., general relativity, string theory, gauge field theories,statistical physics, meteorology and climatology. In telematics his in-terest has been lately attracted to discrete event systems and formallanguages in relation to content engineering and e-business. Withincomputer vision and science he’s momentarily active in studyingvarious scale-space paradigms through the use of problem solvingenvironments.

Bart M. ter Haar Romeny received a M.Sc. degree in AppliedPhysics from Delft University of Technology in 1978, and a Ph.D.on neuromuscular nonlinearities from Utrecht University in 1983.

After being the principal physicist of the Utrecht University HospitalRadiology Department he joined in 1989 the department of MedicalImaging at Utrecht University as associate professor. He is a perma-nent staff-member of the Image Sciences Institute of Utrecht Univer-sity and the University Hospital Utrecht, and chairman of the DutchBiophysical Society. His interests are mathematical aspects of visualperception, in particular linear and non-linear scale-space theory,computer vision applications, and all aspects of medical imaging.He is author of numerous papers and book chapters on these issues,edited books on non-linear diffusion theory and scale-space theoryin Center Vision and initiated a number of international collabora-tions on those subjects. He is an active teacher in national coursesand international summer-schools.

Max A. Viergever received the M.Sc. degree in Applied Mathemat-ics in 1972 and the D.Sc. degree with a thesis on cochlear mechanicsin 1980, both from Delft University of Technology. From 1972 to1988 he was assistant/associate professor of applied mathematics atthis University. Since 1988 he is professor and head of the departmentof Medical Imaging at Utrecht University, and as of 1996 scientificdirector of the newly established Image Sciences Institute of UtrechtUniversity and the University Hospital Utrecht. He is (co)author ofover 200 refereed scientific papers on biophysics and medical imageprocessing, and (co)author/editor of 10 books. His research interestscomprise all aspects of computer vision and medical imaging. MaxViergever is a board member of IPMI and IAPR, is editor of the bookseries Computational Imaging and Vision of Kluwer Academic Pub-lishers, associate editor-in-chief of IEEE Transactions on MedicalImaging, editor of the Journal of Mathematical Imaging and Vision,and Participates on the editorial boards of several journals.

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