Abstract. We propose a novel model of visual contrastmeasurement based on segregated ON and OFF pathways.Two driving forces have shaped our investigation: (1)establishing a mechanism selective for sharp localtransitions in the luminance distribution; (2) generatinga robust scheme of oriented contrast detection. Ourstarting point was the architecture of early stages in themammalian visual system. We show that the circuitbehaves as a soft AND-gate and analyze the scale-spaceselectivity properties of the model in detail. The theo-retical analysis is supplemented by computer simulationsin which we selectively investigate key functionalities ofthe proposed contrast detection scheme. We demon-strate that the model is capable of successfully process-ing synthetic as well as natural images, thus illustratingthe potential of the method for computer vision appli-cations.

1 Introduction

An important goal of early visual information process-ing in man and machine is the robust and reliabledetection of local luminance contrast. A discontinuity ina visual stimulus signals a causally related change in atleast one physical scene parameter (Marr 1982). Thedevelopment of ®lters to detect such discontinuities hasbeen substantially in¯uenced by neurophysiologicalinvestigations concerning the shape of receptive ®eld(RF) pro®les of retinal and cortical neurons. Marr andHildreth (1980) proposed that the RF pro®le ofmammalian retinal ganglion cells can be modeled by aLaplacian-of-Gaussian or di�erence of circular symmet-ric Gaussians, both implementing an isotropic second-order derivative ®ltering stage (Marr 1982). Morerecently, oriented linear ®lters built from Gabor orGaussian derivative functions have been employed

(Daugman 1985; Koenderink and van Doorn 1990).Such families of ®lters have been linked to the RFstructure of cortical simple cells (Pollen and Ronner1983) which sample the image in an optimal fashion andhave been suggested to play a central role in humanfeature detection (Morrone and Burr 1988).

Initial processing stages of the retina and the lateralgeniculate nucleus (LGN) utilize RFs with an isotropiccenter-surround structure. Antagonistic ON-center/OFF-surround and OFF-center/ON-surround cells comprisetwo independent parallel systems capable of signalinglight increments and decrements (Schiller 1992). At thelevel of the striate cortex, ON-contrast and OFF-contrastcells converge onto the same target cells (Fig. 1). Howdo cortical simple cells acquire their polarity-sensitiveelongated RFs? Hubel and Wiesel (1962) hypothesizedthat a simple cell sub-®eld is generated directly by ex-citatory synaptic inputs from a row of LGN neuronswhose RF centers overlap the sub-®eld. The ON sub-®eldwould be generated from ON-center LGN cells, and theOFF sub-®eld by OFF-center LGN cells. The precisemechanisms responsible for orientation selectivity re-main, however, a subject of intense investigation (see arecent review by Vidyasagar et al. 1996). Nevertheless,there is evidence that an initial selectivity is broughtabout by the thalamo-cortical input arrangement (e.g.,Ferster 1988), in line with the original Hubel and Wiesel(1962) proposal.

Local contrast information near localized luminancetransitions (or boundaries) not only signals the presenceof a discontinuity but also carries information about thesurface qualities of neighboring image regions. Inlightness computation of ¯at scenes, such as in the Re-tinex theory (Land and McCann 1971), by measuringthe ratio of luminances at either side of a step transition,information about surface re¯ectance may be recovered.Such a class of lightness algorithms ®rst di�erentiatesthe image and subsequently thresholds the result in or-der to maintain only contrast responses coinciding withsharp-edge-like transitions. Shallow gradients, i.e., thosechanges that are generated by a gradual illuminationgradient, are thus ``discounted.'' The edge information

Biol. Cybern. 81, 515±532 (1999)

Interaction of ON and OFF pathwaysfor visual contrast measurement

Heiko Neumann1, Luiz Pessoa2, Thorsten Hanse1

1 UniversitaÈ t Ulm, FakultaÈ t fuÈ r Informatik, Abteilung Neuroinformatik, Oberer Eselsberg, D-89069 Ulm, Germany2 Universidade Federal do Rio de Janeiro, Programa de Engenharia de Sistemas e ComputacË aÄ o±COPPE Sistemas, Ilha do Fundaà o, Rio deJaneiro, RJ-21945970, Rio de Janeiro, Brazil

Received: 5 June 1998 / Accepted in revised form: 12 May 1999

Correspondence to: H. Neumann(e-mail: hneumann@neuro.informatik.uni-ulm.de)

left after threshold suppression is then integrated in or-der to generate a lightness representation (Horn 1974;Blake 1985; Arend and Goldstein 1987). A ®xedthreshold, however, cannot be determined for generalscenes. If a low threshold is chosen, shallow gradientsmay be registered; if a high threshold is chosen, smallcontrast step-like transitions will be missed. This high-lights a problem for the edge detection mechanismsmentioned above since their response amplitude is scaledwith the local magnitude of the luminance gradient.What is needed is a mechanism that is selective to theshapes of the luminance transitions ± i.e., whether theyare abrupt or shallow.

In this paper, we develop a contrast detection circuit,referred to as a simple cell circuit, that exhibits suchselectivity building upon a previous simpli®ed imple-mentation (Neumann and Pessoa 1994). The model usessegregated streams for ON contrasts and OFF contrasts.The proposed circuit (1) produces strong responseswhenever ON and OFF retinal responses occur next toeach other, such as at a luminance edge, while (2) pro-ducing minor responses when only ON or OFF responsesare present, such as for gradual luminance gradients atramp edges. To achieve this, the circuit behaves as a softAND-gate whose responses have two components: alinear combination of ON and OFF channel contrast re-sponses, and a multiplicative, or gating-like, componentwhenever inputs to the ON and OFF sub-®elds are spa-tially adjacent, or juxtaposed. In essence, the combinedcomputational strategy implements a robust scheme oforiented contrast detection and, in addition, establishesa mechanism that is speci®cally selective for sharp localluminance transitions.

In Sect. 2, we will introduce the proposed circuit forcontrast detection. For this purpose, we ®rst provide aformal description of the functionality of each modelstage, including a center-surround pre-processing stageand the elements of the non-linear circuit itself. We theninvestigate the circuit's capability of contrast detectionproviding an analysis of the steady-state response. Thespatial sampling of the input sub-®elds varies with theparametrization of the spatial input blurring of the ini-tial contrast measures which amounts to a multi-scalescheme of Gaussian derivative operators. Following this

theoretical analysis, we show results of computer simu-lations in which we selectively investigate key function-alities of the proposed processing scheme. Finally, werelate the present model to other proposals that alsohave made use of non-linear or AND-gating processingfor edge detection, most notably those of Marr andHildreth (1980) and Iverson and Zucker (1995).

2 Neural model for oriented contrast measurement

Figure 2A shows the outline of the model. The input tothe system is composed of two streams or channels,namely ON and OFF. The channels carry activations frominitial processing stages having concentric center-sur-round RFs, as found in the retina and LGN. The modelitself consists of a series of processing stages, each ofwhich consists of a two-dimensional (2D) ®eld (or grid)of processing units or cells. The input is also encoded as a2D ®eld of activity. All connections between modelstages are topographically organized such that a spatiallocation �i; j� at a given stage connects to location �i; j� inthe target ®eld. The activation levels at individual modelstages represent the output value of the respective stages.

2.1 Model stages

2.2.1 Center-surround interactionsThe input luminance distribution, L, is processed by cellshaving center-surround, antagonistic, RFs, such asmammalian retinal ganglion cells. The model includesboth ON and OFF pathways that measure the degree of

Fig. 1. Cortical simple cell receptive ®elds (RFs). Hubel and Wiesel(1962) hypothesized that lateral geniculate nucleus (LGN) cells withconcentric, center-surround RFs project to the cortex with the properarrangement so as to generate elongated RFs from unoriented ones.Schematic representation of how ON-center LGN cells connect withON regions of cortical cells, and OFF-center LGN cells connect withOFF regions

Fig. 2. A Proposed circuit of contrast cell sensitive to light-darkcontrast polarity. B Final competition of cells of opposite contrastpolarity at each spatial location. Two types of forward interactionshave been utilized. Those denoted with an arrow at the end in anexcitatory fashion, those with an oval at the ending of a link areinhibitory. LD Light-dark; DL dark-light

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local luminance contrast in input images. Thus, at everyspatial location there are ON and OFF cells. These two®elds of cells implement lateral inhibition that contrast-enhance, or sharpen, the input luminance distribution ±cells respond strongly to (unoriented) luminance discon-tinuities.

A mechanism of local center-surround processing canbe approximated by ®ltering the input distribution withdi�erence-of-Gaussian ®lters such as were originallyproposed to model the receptive ®eld structure of retinalganglion cells (Rodieck 1965; Enroth-Cugell and Rob-son 1966):

net� � L Gr� and netÿ � L Grÿ ; �1�

where net� and netÿ are net center and net surroundinputs, respectively, for ON-channel retinal ganglioncells. The input luminance distribution is denoted by L,Gr� and Grÿ are the respective Gaussian ®lters (para-metrized by di�erently scaled space constants r� and rÿ,respectively, with r� < rÿ), and implements thespatial convolution operator.

In classical proposals, such as, for example, in Rod-ieck (1965) or Marr and Hildreth (1980), the center andsurround contributions combine linearly to determinethe cell's response. Other approaches have speci®ed vi-sual adaptation processes that render responses sensitiveto luminance ratios. Such formalisms adopt multiplica-tive, or shunting, mechanisms (Grossberg 1970; Hodg-kin 1964; Furman 1965). Formally, the di�erentcomputational approaches can be denoted in a gener-alized framework such that an ON-center OFF-surroundinteraction (generating y� activation) results in

d

dty� � ÿasy

� � �bs ÿ csy��net� ÿ �ds � gsy

��netÿ :

�2�

Here as, bs, cs, ds and gs are constants, and net�, netÿ arethe total excitatory/inhibitory inputs to y� (see Eq. 1).The rest of the paper focuses on the description of asimple cell circuit that is sensitive to oriented contrast.Here, the arrangement of ON/OFF -contrast is relevantand, therefore, we can restrict ourselves to the simplestcase having linear ganglion cell responses that generateON and OFF channel inputs.

The steady-state contrast input to the simple cellcircuit is given by the equilibrium form of Eq. (2) (i.e.,d=dt y� � 0). By setting as � ds and cs � gs � 0, we ob-tain zero DC contrast channel inputs

c� � bsas

�R net� ÿ netÿ� � and

cÿ � bsas

�R netÿ ÿ net�� ��3�

(R�x� denotes a recti®cation operation max�x; 0��:3

2.1.2 ON/OFF input to contrast cell sub-®eldsThe spatial branches of contrast cells receive excitatoryinputs from the ON and OFF contrast channels. Thisinput is sampled separately using elongated weightingfunctions with a smoothly decaying coupling strength.Before integration of contrast activity for excitatory andinhibitory sub-®elds, the activities in both contrastchannels undergo local competition at each spatiallocation. Such competitive interaction is necessary togenerate su�cient initial orientation tuning of the cell.Otherwise, with separate blurring of each input channelactivity, simple cells will be activated even for anorientation orthogonal to the local contrast. In formalterms, the input to either sub-®eld of the circuit ismodeled by

p� �R��c� ÿ cÿ� k�e � and

pÿ �R��cÿ ÿ c�� kÿe � ;�4�

(see Fig. 2), where the kernel k determines an elongatedspatial sampling function of orientation e.

These inputs are used to detect local contrast changesof a given polarity, such as the presence of edges or otherabrupt luminance variations. In our simulations, weused elongated 2D Gaussian weighting functions formodeling the sensitivity pro®les of k�e (see Fig. 3), withspace constants rm and rM to denote the minor axis andthe elongation axis, respectively. In formal terms, theanisotropic weighting kernel can be described by

ke � Nr exp ÿ12~xTRT

e CmMRe~xÿ �� �

; �5�where Nr is a scaling constant, ~xT � �x y� denotes the(transposed) vector of the spatial location, CmM is thediagonal matrix

CmM � 1=r2m 0

0 1=r2M

� �

;

determining the spreading along the principal axes, andRe determines the matrix of rotation in 2D �x; y� space

Re �cos e sin e

ÿsin e cos e

� �

:

Fig. 3. E�ective spatial sensitivity pro®les of oriented contrast cellswith odd-symmetric ON and OFF input. The model uses elongatedGaussian pro®les, k�e and kÿe , both with space constants rm � 3 andrM � 2rm; relative o�set has been set to s � 3, orientation is e � 0�

(see text)

3 In the case of having imbalanced center-surround interaction,an additional stage of cross-inhibition between contrast channelshelps to eliminate any residual DC-level activation (see Neumann1996).

517

The axis of elongation indicated by e determines thepreferred orientation of the cell.

In all cases, we used balanced weighting functions forexcitatory and inhibitory sub-®elds, such that k�e � kÿe .By varying the space constants rm and rM of the elon-gated weighting functions, one obtains kernels of dif-ferent spatial extent in length and width. This de®nes afamily of kernels denoted as kS�e and kSÿe , where S de-notes the spatial scale (see the notion of scale below inSect. 3). The relative spatial o�set s of these sub-®elds(see Fig. 3) varies as a function of the space constant rm,such that we have s�r� � rm.

2.1.3 Sub-®eld opponent interactionsThe second stage of the circuit receives direct linearexcitatory inputs from each ON or OFF sub-®eld, as wellas an opponent divisive interaction between channels.The steady-state activity of the ON sub-®eld is given by

q� � p�

ac � bcpÿ ; �6�

where ac and bc are constants of this shunting inhibitoryinteraction. For the OFF sub-®eld, we assume the samesetting of constants such that at equilibrium the qÿ-response is generated by having the p� and pÿ inputsinterchanged in the right-hand side of Eq. (6).

2.1.4 Direct input and post-opponency signals combinedThe third stage receives channel-speci®c inputs fromboth the ®rst (excitatory) and second (inhibitory) stages.Formally, ON-channel equilibrium activation is

r� � p�

cc � dcq�; �7�

where cc and dc are constants of this second stage of thecircuit. Again, we assume the same setting of constantsin both channels. The OFF-channel activations rÿ arethen obtained by exchanging `�' and `ÿ' indices in theright-hand side of Eq. (7).

The functionality of the sub-circuits realizing eachseparate sub-®eld channel is to generate a self-inhibition,thus normalization, of the input activity distribution.The opponent interaction between these channels andthe corresponding overall net activity will be discussedbelow under functional aspects.

The ®nal response for a light-dark (LD) cell is ob-tained by pooling ON and OFF activities. We formalizethis as a simple linear combination by summing both ractivities and get

zLD � r� � rÿ : �8�The dark-light (DL) response, zDL, is obtained in acorresponding manner, such that the respective ON andOFF input streams are reversed in their spatial arrange-ment.

2.1.5 Mutual inhibition of cellsCells sensitive to opposite contrast polarity at the samespatial position ®nally undergo mutual inhibition (see

Fig. 2B). In functional terms, such a competitive stagehelps to sharpen the activity pro®le generated by eachcell sensitive to opposite contrast polarity alone. The®nal cell responses are computed as

ZLD �R�zLD ÿ zDL� and

ZDL �R�zDL ÿ zLD� :�9�

2.2 Relationship to cortical simple cell physiology

Several design issues and functional properties of themodel have strong relationships to ®ndings about simplecell physiology. For example, we utilize oriented spatialweightings to model elongated sub-®elds for integratingnon-oriented input from cells with a concentric RFoutline (see Fig. 1). This is consistent with evidence thatan initial selectivity is brought about by the thalamo-cortical input arrangement (e.g., Ferster 1988), in linewith the original Hubel and Wiesel proposal (1962).Furthermore, we have utilized a local competition ofcontrast channels before integrating sub-®eld activities.This approach implements the opponent inhibitionmechanism for simple cell sub-®elds as described inFerster (1989; see also Tolhurst and Dean 1990). Also,using partially overlapping spatially o�set sub-®eldweighting functions has been suggested by the ®ndingsof Heggelund (1981, 1986) as well as Ferster (1988).

Our circuit integrates the activity from the corre-sponding sub-®eld (ON or OFF) in an excitatory fashion,while the opponent in¯uences this by inhibitory action.This scheme now realizes a mechanism of opponent in-hibition between sub-®elds, similar to those in othersimple cell models (see Pollen and Ronner 1983). Themutual inhibition of cells with opposite polarity selec-tivity follows the description in Liu et al. (1992), whosuggested the mutual inhibition between members ofpairs of phase-related opposite contrast simple cells.Also, Ferster (1988) suggested that competition betweensimple cells of opposite contrast polarity occurs at eachspatial location. In all, although we do not attempt toexplicitly model the underlying mechanisms of corticalsimple cells (e.g., orientation selectivity and spatial inputsampling), our model incorporates key features of sim-ple cell physiology. We therefore suggest that our modelcircuit also serves as an abstract description of infor-mation processing at early stages in the mammalian vi-sual system.

2.3 Functionality of the circuit

2.3.1 Equilibrium responseIt is possible to gain insight into the precise functionalityof the model, especially with regard to its AND-gatebehavior, by combining the corresponding expressions.The activation r in the ON and OFF branches can thus becomputed analytically. An assumed symmetry relation-ship between both channels can be achieved by settingthe identity dc � bccc. In this case, for a LD cell, we get

518

r� � p�

accc � dc�p� � pÿ� ac � bc pÿ� � and

rÿ � pÿ

accc � dc�p� � pÿ� ac � bc p�� � :

�10�

The ®nal response for a contrast cell selective to LDpolarity is computed as zLD � r� � rÿ. Using Eq. (10)we get

zLD � 1

accc � dc�p� � pÿ� ac p� � pÿ� � � 2bc p�pÿ� � :

�11�The response of a DL selective cell, zDL, is obtained in acorresponding manner. In the ®nal steady-state re-sponse, however, the roles of p� and pÿ are inter-changed.

This demonstrates the non-linear interaction of ac-tivity between the two branches. The basic functionalproperties of the model could be highlighted by thefollowing more compact notation of Eq. (11) that reads

z � L � p� � pÿ� � �N � p� � pÿ� � : �12�

Here, L and N denote transformations of the initialcontrast activities in the ON and OFF channels thatgenerate the inputs to the circuit. In particular, we haveL � ac � C�c�; cÿ� and N � 2bc � C�c�; cÿ� withC�c�; cÿ� � �accc � dc�p� � pÿ��ÿ1

. In C���, the inputactivities p� for each individual lobe are themselvesfunctions of c� and cÿ (see Eq. 4). Input to the modelcircuit is integrated linearly from both channels. Spa-tially adjacent activity (in the ON and OFF pathways) issignaled by an additional correlational (gating-type)component. The relative contribution of additive andgated activities is controlled by the (shunting) parame-ters ac and bc in Eq. (6). Moreover, the activity self-normalizes with respect to the total input activity fromthe ON and OFF channels [function C�c�; cÿ� above]. Themodel thus relates to the scheme proposed by Carandiniand Heeger (1994) in which the activity of corticalneurons is normalized through division of pooledactivity from a large number of cells.

2.3.2 Linear and non-linear responsesIn all, the model behaves as a soft AND-gate thatcombines linear and non-linear (gating) components ofsub-®eld activity. Adjacent ON and OFF signal con®gu-rations generate an extra boosting activation in additionto responses of reduced magnitude that are generatedfor other signal con®gurations. In terms of the circuitshown in Fig. 2A, it is the opponent inhibition of thesecond stage (signals q� and qÿ) associated with thewithin-channel inhibition (from the second to the thirdstages) that implements the soft AND-gate behavior.These interactions produce a mechanism of disinhibitionthat generates large outputs only when both ON and OFF

channel inputs are large. To understand this, considerthe case where only ON signals are input to the model;the OFF pathways are thus shut down. Responses will besmall since the (within channel) inhibition from the

second to the third stages will attenuate the input signalvia a normalization mechanism. Now consider the casewhen there are potent inputs to both ON and OFF

channels. The cross-inhibition between the ®rst andsecond stages will largely reduce second stage activities(signals q� and qÿ). These, in turn, will not be able toinhibit stage three signals and the original inputs will beable to combine at the last stage (zLD and zDL) since theyreach it via the ``side pathways'' from the ®rst to thirdstages. We see that the presence of inputs in bothchannels leads to a disinhibition in the circuit and,hence, powerful responses. In all, the circuit detectswhen there are adjacent ON and OFF signals, hencegenerating the soft AND-gate behavior.

The relative proportion of linear and non-linear re-sponses is controlled by the parameters ac and bc. Thenet input scaling for the initial ON and OFF channelprocessing also directly contributes to the contrast cellresponse in a systematic fashion. ON and OFF -channelinput to oriented sub-®elds is given by

p� / bsas

�R��net� ÿ net�� k�e � �13�

(compare Eqs. 3 and 4). We observe for a given netinput that the relative proportion of the non-linearcontribution in Eq. (11) is b0 � bsbc=as. For any furtheranalysis, one must therefore take into account the rangeof activities generated in the initial center-surround pre-processing stage. For convenience, an activity range of�0::1� from initial center-surround ®ltering has beenensured in all our simulations.

The key idea behind our model is that contrastchanges are better localized by a circuit that is selectivelysensitive to abrupt luminance transitions. These transi-tions will be invariably associated with adjacent ON andOFF contrast signals (for some spatial scales; see below).A linear combination scheme is, of course, sensitive tosuch transitions, but not sensitive enough such that itshows only di�erential variations in response once pre-viously adjacent activities in the ON and OFF channelshave been shifted farther apart (see also du Buf 1994).

3 Families of RFs and selectivity to spatial scale

3.1 Elementary properties

3.1.1 MotivationThe visual system is faced with the problem of measur-ing the relevant structure of the outside world. Real-world objects only exist as meaningful entities overcertain ranges of measurement scales. An examplestressed by Marr and Hildreth (1980) is the appearanceof the ®ne-textured veins of a leaf on a small scale butthe overall shape and luminance variation appearing ona much coarser scale. Furthermore, more gradualintensity variations often relate to illumination variationor changes in surface orientation, whereas abruptchanges in the luminance distribution are often causedby transitions in surface properties, such as re¯ectance

519

(Horn 1974). A vision system, in general, has no advancedknowledge about the appropriate scales for analysis.Therefore, in order to deal with such a multitude ofproperties of real-world objects, themachinery of a visionsystem has to process the input luminance distribution ondi�erent scales (see, for example, Witkin 1983; Koende-rink 1984; Lindeberg and ter Haar Romeny 1994; and fora short tutorial, Lindeberg 1996).

The raw luminance distribution is processed by aninitial center-surround mechanism. After opponent in-hibition, this activity is integrated by oriented o�setweighting functions which de®ne the excitatory and in-hibitory sub-®elds to an oriented cell. The signals for ON

and OFF contrast responses are recti®ed such that thecontrast channels carry a single positively bounded datarepresentation from the initial ®ltering stage. By utilizingweighting functions of di�erent sizes, kS�e and kSÿe , thesecontrast activities are blurred and sampled at increasedrelative spatial o�sets; thus a family of cells sensitive todi�erent spatial scales is introduced. The spatial scaleattribute S itself is a function of the space constant r,thus S � S�r� (see Eq. 5).

3.1.2 Generation of scale-space derivative kernelsA contrast edge is processed by an initial center-surround ®ltering stage. The results are half-waverecti®ed and segregated into two separate representa-tions for ON and OFF contrast, respectively. The resultsof individual ON and OFF responses are blurred byoriented spatially scaled weighting functions to generatethe input for the opponent contrast cell sub-®elds ofdi�erent sizes. This non-linear processing cascade isshown to be identical to a ®rst-order derivative kernelapplied to the activation generated by the center-surround processing.

To simplify the mathematical analysis we approxi-mate the initial center-surround ®ltering by a Laplacian-of-Gaussian (LoG) operator. Consider a unit stepfunction as input that is modeled as a Heaviside functionH (Bracewell 1978). Input processing results in a pro®leof a ®rst-order derivative of a Gaussian, D1G, which isan odd-symmetric function with positive and negativelobes. For a representation to carry positive signal re-sponses only, the positive and negative lobes are half-wave recti®ed. We get

c��u� �R�LoGr1�u� H�u�� � Rd

duGr1�u�

� �

and

cÿ�u� �R�ÿLoGr1�u� H�u�� � R ÿ d

duGr1�u�

� �

;

with R�x� � max�x; 0�. These responses are subsequently®ltered by a Gaussian blurring function whose resultsare then combined to generate the edge response of thecircuit. We get s�u� � R�d=duGr1�u�� Gr2�u� �R

�ÿd=du Gr1�u�� Gr2�u�. Since R�d=duGr1�u�� � d=duGr1�u� � 1ÿH�u�� � and R�ÿd=duGr1�u�� � ÿd=duGr1�u� �H�u�; we can rewrite the ®nal response to get

s�u� � d

duGr1�u� Gr2�u� �

d

duG ����������

r21�r2

2

p �u� : �14�

This demonstrates the equivalence of our hierarchicalprocessing scheme based on segregated representationsof ON and OFF activations and an approach that utilizes®lters derived from an analytic description of ®rst-orderderivative operations. It also veri®es that the subsequentblurring of separated contrast channel responses of onefrequency band ± generated by the initial center-surroundprocessing stage ± produces results equivalent to thoseachieved with a bank of scaled center-surround ®lters.

3.1.3 Opponent o�set blurring approximates®rst-order derivative ®lteringThe kernels k� were spatially o�set by an amount srelative to the reference location of the target cell. Theycollect responses in the ON and OFF contrast channels tobe further processed in the circuit. The di�erence ofspatial o�set Gaussians resembles the pro®le of a ®rst-order derivative operation. Consider the model of aparametrized ramp edge transition in a local �u; v�-gaugecoordinate system where the u-axis is oriented along thecontrast pro®le. Formally, the di�erence of o�setGaussians can then be written as a convolution of theD1G pro®le with a unit impulse of width D � 2s, thentaking the limit, D ! 0. We get

limD!0

1

DG�u� ÿ G�uÿ D�� � � lim

D!0

d

duGr�u� PD�u�

� �

� d

duGr�u� ;

where PD�u� � H�u� D

2� �H�D

2ÿ u�:

3.1.4 Responses as a function of scaleWe now consider the processing of a luminance transi-tion with a ramp pro®le of a priori unknown width(compare du Buf 1993, 1994). The suggested model issensitive to the spatial adjacency of retinal ON and OFF

signals. Spatial proximity, however, is a relative measurethat will depend on the spatial scale of processing. Whatmay be spatially proximal for a large scale may be distantfor a small one. Sensitivity to spatial scale endows modelcells to selectively process a given visual structure, suchas the ramp transition shown in Fig. 4. For a widetransition ramp, the initial center-surround processinggenerates isolated ON and OFF contrast responses whichare located at the shoulders (``knees'') of the ramptransition. A cell with a RF selective to odd symmetriccontrast variations located at position x � 0 at the centerof the ramp for the smallest scale receives no input fromeither blurred ON or OFF contrast channels (comparescale 1 in Fig. 4 middle, bottom). Cells slightly o�set toeither side of this central location receive input from onlyone sub-®eld (lobe) and thus generate normalized linearresponses. As the blurring proceeds, more di�usedistributions of ON and OFF contrast responses appear,which eventually meet at the center of the ramp such thatthey appear to be juxtaposed for the coarse scale atlocation x � 0 (scales 3 and 4 in Fig. 4, bottom). Nowintegrated inputs from both sub-®elds add and asuperimposed component is generated by the productof the magnitude of these inputs. If the product is

520

su�ciently high in amplitude, the latter contributionfacilitates the target cell activation by an additional``boost.'' The spatial juxtaposition of ON and OFF inputactivation to the circuit signals a transition which issharply localized relative to the current spatial scale.

The circuit is thus selective to inputs with sharpcontrast transitions. The quantitative properties of thecell responses in scale-space can be derived from theactivation given in Eq. (11). Below, we present the re-sults of this investigation.

3.2 Quantitative analysis of scale-space response

In this subsection, we summarize the quantitative®ndings for responses generated by scaled contrast cellswhich are applied to di�erent idealized luminance

transitions. We ®rst show the results derived for stepedge responses. This motivates the subsequent deriva-tion for the results processing the gradual transition of aramp as discussed above. In both studies, we comparethe results for the non-linear circuit with those generatedby a scheme of linear ®ltering. Detailed results ofinvestigations can be found in Appendices 1 and 2.

We utilize the compact notation of a non-linear re-sponse of the circuit that is given by

z � L � p� � pÿ� � �N

L� p� � pÿ� �

� �

�15�

(compare with Eq. 12). For quantitative analysis, weomit the gain L and only consider the expression givenin brackets. For the gain of the non-linear component,we de®ne

N0 � N

Lbs � ds� � � 2

bcac

bs � ds� � �16�

(see Appendix 2). Based on these de®nitions, we relatethe linear and non-linear response components.

3.2.1 Step edge responseA step edge pro®le of height h is de®ned by a Heavisidefunction,

fstep�u� � h �H�u� ; �17�

in which the u-axis is taken in the direction orthogonalto the local orientation tangent to the luminancecontrast. Processing this input ®rst by a center-surround®lter (approximated by a Laplacian-of-Gaussian) sub-sequently followed by a ®rst-order (Gaussian) derivative®ltering generates a response pro®le with a uniquemaximum at the step location u � 0. For the non-linearcircuit, the segregated contrast responses always appearto be juxtaposed at either side of the contrast step. Therelevant component of response (contribution given inbrackets in Eq. 15) of the non-linear circuit at thelocation of the luminance step is given by

snlstep�u � 0� � h������

2pp

r31�N

0 1������

2pp

er

� �

: �18�

This shows that for a linear as well as a non-linear cell,the response is maximal for the smallest scale, i.e., itsmoothly drops as blurring increases. Furthermore, itshows that the non-linear contribution only becomessigni®cant for a gain N

0 �������

2pp

er � 7r. Consideringthe de®nition of the gain in Eq. 16, it is reasonable totreat the parameter bc as a non-zero function of r.

3.2.2 Ramp edge responseA ramp edge pro®le of height h and width R is de®ned by

framp�u� � h �H�u� PR�u� : �19�

where PR�u� denotes a ®nite pulse of width R. In thelimit as R ! 0, the ramp converges to a step function asde®ned above. Again, we utilize the sequence ofprocessing steps of center-surround ®ltering and the

Fig. 4. Responses of a scaled family of contrast cells sensitive to anodd-symmetric dark-light (DL) luminance transition (ramp contrast).Top Input stimulus is shown together with schematic ON and OFF

responses generated by the initial stage of center-surround processing.Responses of either contrast channel appear at the ``knee points'' ofthe plateaus. Depending on the width of the ramp relative to the sizeof the initial center-surround ®lter, they appear in spatial isolation (asshown in the sketch) or juxtaposed. Middle Increased blurring ofinitially isolated contrast responses for a wide transition ramp. Fourdi�erent scale selectivities are shown with reference to the location atthe center of the ramp (x � 0). These blurred activations provide theinputs to cells sensitive to DL transitions. Bottom Responses ofcontrast cells for the four scales that integrate their input to sub-®eldsfrom increasingly distant o�sets [the display of responses for light-dark cells has been omitted in order to keep the sketch simple]. For acell at a more coarser scale, ON and OFF inputs appear to bejuxtaposed such that the non-linear component can contribute via itsignition. As a consequence, a unique maximum response is generatedas a function of the scale

521

®rst-order Gaussian derivative operation. The latter isapproximated by the individual blurring of the contrastresponses followed by a numerical di�erentiation for®nal sampling of the inputs from the sub-®elds of thecontrast cell. Now, depending on the ratio between theoperator scale and the width of the ramp, r=R, the ON

and OFF contrast responses appear in isolation (r � R)or juxtaposed similar to the case of a step transition(r � R). For the position u � 0 at the center of the ramp,we can now treat the response as a function of scale. Forthe linear model, a relative maximum in response occursat

rmax �R

2���

3p : �20�

For the analysis of scale-space response for the non-linear circuit, we investigate the additional contributionfrom (blurred) ON and OFF contrast responses sampledfrom o�sets relative to the ramp center. The relevantcomponent (bracket part in Eq. 15) of the correspond-ing response for the circuit is

snlramp�u � 0�jr�rmax� 48

���

3p

h������

2pp exp ÿ 3

2

� �

� 1R3

� 1�N0

���

2p

8������

3pp h

Rexp

3

2

� � !

:

�21�This shows that there exists an optimum scale forprocessing a ramp edge of a priori unknown width andheight. The non-linear contribution is itself dependenton the slope of the ramp and becomes signi®cant for again N

0 � 4������

6pp

exp ÿ 32

� �mÿ1 � 4mÿ1, where m � h=R.

The non-linear circuit shows a multitude of desiredprinciples. For one, the additional ``boosting'' of activityhelps to drive the cell output to generate a response thatis maximal in absolute terms. It is shown in Appendix 2that the non-linear cell tuned to an optimum scale gen-erates absolute maximum amplitude responses. In ad-dition, the contribution generated by the non-linearitydepends on the scale r (step) or on the slope m (ramp) ofthe luminance transition. The parameter to control thee�cacy of the non-linear contribution in N

0 is bc, thegain of opponent shunting inhibition in the circuit (seeEq. 6). The gating of contrast activations ± adding to the(linear) cell response ± enables the circuit to preferen-tially respond to sharp transitions, irrespective of theirheight. This selective functional property was one of theprincipal motivations for the development of the circuitin the context of the modeling of functional mechanismsfor neural contrast and brightness processing (Neumannand Pessoa 1994; Pessoa et al. 1995).

4 Simulations

In this section, we evaluate the model on the basis of aseries of computer simulations that demonstrate thefunctionality. The section is organized so as to start withthe demonstration of the usefulness of the approach for

image processing purposes. We tested its robustness tonoise in comparison with a corresponding pure linearmodel and demonstrated its selectivity in automaticscale selection, all based on synthetic test images. Thecomputational relevance is also demonstrated by meansof natural images from a test data set. Since we have alsoshown that the model accounts for some relevantproperties of cortical simple cells that are usuallybelieved to behave almost linearly, we conducted anexperiment showing that our non-linear circuit alsoshows substantial linear behavior. We then ®nally showresults that demonstrate the usefulness of the model as a®rst step in a processing hierarchy in which initialmeasurements are grouped together in order to generatemeaningful pieces for shape recognition. In particular,we show the model's strength in the initial measurementfor subsequent integration by long-range groupingprocesses.

In all simulations, responses are shown after mutualinhibition (ZLD and ZDL). Luminance values of inputimages have been normalized to the range �0::1�. Theinitial center-surround processing stage involves shunt-ing interaction and subsequent half-wave recti®cation togenerate segregated ON and OFF channels (see Sect. 2.1).Model parameters for this initial stage were set toas � 0:5, bs � 1:0, and ds � 0:1 (as already pointed outabove, we set cs � gs � 1:0 in order to yield symmetricnormalization in both channels). Parameters of isotropicGaussians were set to r� � 1:0 and rÿ � 3:0, thus ob-taining a 1 : 3 center-surround ratio. The model pa-rameters of the non-linear circuit were set to ac � 1:0,bc � 10000:0, cc � 0:01, and dc � 100:0. Their speci®cchoice is non-critical as long as the linear componentsscaled by ac and cc are small compared to the cross-channel inhibition e�ect. The Gaussian weighting func-tions were elongated by a rM : rm � 2 : 1 ratio; thevariance rm is measured in pixels along the short axis.The separation s grows linearly with the variance (scaleS). Eight discrete, equally spaced orientations wereprocessed. The corresponding linear circuit was ap-proximated simply by eliminating the opponent inter-action between the sub-®elds in the non-linear contrastcell circuit (stages q� and r�, see Fig. 2). Thus, thescheme used for comparison directly integrates the ac-tivity from both sub-®elds, namely z � p� � pÿ.

4.1 Image processing

As indicated, we demonstrate the capacity of the circuiton the basis of synthetic as well as real camera imagesfrom a test set. Test simulations run on synthetic datawere to justify the robustness of the model against noiseand the selectivity of the mechanisms to scale. Resultsgenerated for natural images are intended to serve forcomparison of performance with other known methods.

4.1.1 Synthetic imagesA ®rst strict test of an image processing algorithmconsists of probing it with noisy images. Figure 5 showsa synthetic image containing an elliptic region embedded

522

in a lighter background (left) and a cross-section pro®le.The entire image was corrupted by Gaussian noise (half-width 50% amplitude). Figure 5 (middle) shows theoutput of the new model revealing that it is capable ofaccurate contour localization, even in the presence ofnon-trivial noise levels. It is also instructive to comparethe performance of the circuit with the linear scheme. Asshown in Fig. 5 (right), the linear scheme is less robustto noise as it also shows less selectivity in terms of edgelocalization. Note that no post-processing, such as ®nalthresholding operations, was performed ± these could beused to remove the low-intensity spurious signals due tonoise.

In order to evaluate the theoretical derivations on themechanism's selectivity to spatial scale we have con-ducted a series of experiments based on the processing ofscaled gradual luminance transitions, namely ramps.Figure 6 illustrates the spatial frequency selectivity ofthe non-linear model by comparing its behavior with theanalogous linear scheme. For the non-linear contrastcell, as the spatial scale of operation is increased, cellswhose centers are closer to the middle of the rampeventually become ignited by the non-linear proportionof the response. That is, by using larger elongatedGaussian operators that sample the blurred ON and OFF

sub-®eld inputs target cells conjointly receive input fromo�set positions as they appear to be juxtaposed in acoarser scale ± a maximum response is produced at themiddle of the ramp. This behavior should be contrastedto that of the linear scheme. It can be recognized thatalthough maxima evolve at the middle of the ramp, they

are only local such that the absolute maximum responseappears at a ®ne spatial scale, localized to the left andright of the ``knee'' points.

4.1.2 Natural imagesNatural images provide a good test of the imageprocessing capabilities of our model of contrast detec-tion. In particular, we can assess the contrast localiza-tion properties of the model by again comparing itsoutput with that produced by an analogous linearscheme. Figure 7 illustrates the better contrast localiza-tion properties of our circuit when compared to thelinear scheme. We see that much sharper ``edge signals''are generated by the circuit e�ectively registering thecontour outlines present in the image.

4.2 Linear versus non-linear processing behavior

We have pointed out above (see Sect. 2.2) that ourmodel inherits a number of simple cell properties. Wedeveloped a non-linear circuit that pronounces any inputcon®guration of juxtaposed ON and OFF activation, thusbeing speci®cally selective to sharp luminance transi-tions. In order to justify the relevance of the model, ithas to be demonstrated that the overall behavior doesnot, in general, contradict previous ®ndings in physiol-ogy that have been taken as support for linear mecha-nisms driving simple cells. Candidate tests include thelinear summing principle given increasing stimuluslength and using eigenfunctions of linear systems such

Fig. 5. Synthetic image with noise. Input luminance distribution corrupted by additive Gaussian noise with 50% amplitude of contrast height(top left); pooled activity for all orientation ®elds generated by the non-linear circuit (top middle) and the linear circuit (top right). Correspondingone-dimensional cross-sections of the above shown 2D distributions (bottom row)

523

as spatial sine-wave luminance modulations. Sine-wavestimuli have been used to characterize linear systemssuch as a linear ®ltering operation. In order to demon-strate the validity of our model in this respect, weprocess a spatial cosine luminance waveform (Fig. 8).The result generates a spatial distribution of activity thatcorresponds to the initial cosine modulation and isconsistent with that predicted for a simple cell.

This demonstration shows that our circuit, althoughintrinsically non-linear, shows signi®cant linear behaviorespecially demonstrated for test cases similar to experi-mental studies with sinusoidal luminance modulations.

4.3 Initial measurement and grouping

Glass patterns (Glass 1969) illustrate how the visualsystem is capable of employing local information to

generate global structure. The perception of structuregenerated by a Glass pattern (or a variation of it) is basedon two types of interaction: a local operation of featuremeasurement or token extraction and a long-rangeintegration of local items (Sagi et al. 1993). A test ofthe initial mechanisms underlying Glass pattern percep-tion was provided by a recent investigation by Brookesand Stevens (1991). In their study, they generated Glasspatterns using oriented white dot pair items which wereeither radially or circularly oriented. Due to the contrastsensitivity of oriented cells, the authors hypothesizedthat the inclusion of an opposite polarity dark dotbetween the two white dots of each local pair (``dot-between'' arrangement) would disrupt the perception ofthe structure seen without the distractor. In particular,the experiment was designed such that the disruption of aradial organization caused the pattern to be more likelyperceived as circular. In contrast, adding a black dot to

Fig. 6. Scale-space selectivity. A ramp luminance distribution is processed at small, medium and large spatial scales. For better visualization ofthe responses, we plot one-dimensional cross-sections of the ramp as well as of responses generated with the non-linear circuit (top) and linearscheme (bottom). Responses are shown for light-dark cells (solid line) and dark-light cells (dotted line). The ramp widths were parametrized byR � 32 units

Fig. 7. Processing a real camera image (left). Pooled activity of all orientation ®elds generated by the non-linear circuit (center) versus the linearmodel (right). Note that no further post-processing, such as local non-maximum suppression, has been applied to the data

524

disrupt a circular organization made by white dot pairitems increases the tendency to perceive a radial orga-nization. As a control, the ``dot-between'' case wasmodi®ed so that the disrupting black dot was placedbeside one of the white dots (``dot-adjacent'' case).Figure 9 shows the Glass patterns that are composed bythe local arrangements of Fig. 10.

Brookes and Stevens (1991) reasoned that if the per-ception of structure intimately depends on localizedmechanisms sensitive to local contrast direction, thenthe perceived structure should be lost since the localoriented ``dot-between'' arrangement largely disruptsresponses for the orientation of the white dot pair item.Even more, in the ``dot-between'' case, a strong responsein the orientation orthogonal to the item is predictedwhile the ``dot-adjacent'' case should have no disturbinge�ect (it may be even more supportive). The perceptualresults con®rmed this prediction. In an alternative-forced-choice judgement, there was a signi®cant increasein the subjects' responses to orthogonal apparent orga-nization (reversing concentric and radial) when the blackdot was placed in the middle between the pairs of whitedots. The placement of the black dot adjacent to a white

dot had no in¯uence on the correct judgement of per-ceptual organization.

The experimental setting described above demon-strates that small di�erences in the local arrangement ofstructure can cause categorical changes in global struc-ture. Such behavior is suggestive of mechanisms that aresensitive to precise contrast arrangements. Motivated bythis observation, we stimulated our circuit with the localdot arrangements shown in Fig. 10 and compared theresults with those generated by a linear model. Eachpattern was pre-processed by the initial stage of center-surround interaction, segregating representations of ON

and OFF contrast. Figure 11 displays the results. In orderto gain more insight into the orientation speci®city of thegenerated activations, we produced ``needle'' diagramsto encode individual response magnitudes for the dif-ferent orientations. The certainty of orientation selec-tivity serves as an indicator for the clarity of subsequentlong-range integration mechanisms for grouping [Sagiet al. (1993); see computational mechanisms by, for ex-ample, Grossberg and Mingolla (1985)]. For the whitedot pair item (Fig. 11, left), the non-linear responseshows a clear dominance along the orientation of the dot

Fig. 8. Processing results of the non-linear circuit generated for a one-dimensional spatial luminance sine wave. The input signal is shown on theleft, the responses for two simple cells sensitive to opposite contrast (dark-light, light-dark) are shown in the middle and in the right panel,respectively. As predicted for a pure linear cell, symmetric responses occur for opposite contrast sensitivities at the positive and negative slopes ofthe sine wave

Fig. 9. Displays of the di�erent variants of Glass patterns from the study of Brookes and Stevens (1991). Patterns with white dot-pair items,``dot-between'' and ``dot-adjacent'' are shown from left to right. Each item in the radially oriented patterns is composed of local pairs of equalpolarity and local triples of mixed polarity dots (see ®gure below). Due to reduced resolution and additional discretization, the perceptual e�ectmay be signi®cantly reduced. However, in our original displays and the plates shown by Brookes and Stevens (1991), the variations producesigni®cant e�ects

525

pair. The linear model also shows a certain preference inthat direction but the response is much more blurred andthe orientations are much more uncertain. For the ``dot-between'' stimulus (Fig. 11, middle), the non-linear cir-cuit generates a categorically di�erent pattern with astrong orthogonal contrast orientation. This coincideswith the observation in psychophysical experiments thatthe correct organization is disrupted and a perception ofthe orthogonal orientation appears. In the ``dot-adja-cent'' case (Fig. 11), the correct organization is not

largely disturbed and the orientation is mostly along thesame polarity dot pair; between the pair of white andblack dots, contrast responses of high amplitude aregenerated which support the overall orientation prefer-ence. In comparison to the result of the white dot pairalone, the spatial distribution of responses appears moreblurred. We suggest that the increased blurring weakensthe response in the subsequent stage of integration forgrouping and perceptual organization. The responsesgenerated by the linear model also show preferences

Fig. 10A±C. Local dot arrangements used in the Brookes and Stevens (1991) experiments to demonstrate categorical changes in the appearanceof Glass patterns. AA local pair of white dots is shown as it appears in the original version of a Glass pattern. Stimulus variants were obtained byincluding a third dot of opposite polarity (black). The dot was either placed B in between the two white dots (``dot-between''), or C beside one ofthe white dots (``dot-adjacent'')

Fig. 11. Results of processing the localitems used by Brookes and Stevens (1991)to generate the Glass pattern variants.Results of processing for the three items(white dot pair, ``dot-between'', ``dot-adjacent'') are shown for the non-linearcircuit (top row) and for the linear model(bottom row). For a better visualization ofindividual contributions for di�erent ori-entations, we show ``needle'' diagrams inwhich the activities for the spatial orien-tations are plotted at each location. Thelength of each oriented needle encodes therelative magnitude of response (the mag-nitudes are normalized with respect to themaximum response that appears in thedisplay). The results indicate the individ-ual dominances in orientation responsethat correspond to that of the perceptualappearance of the variants of Glassstimuli (see text)

526

corresponding to those of the non-linear circuit, al-though weaker. In addition, as in the dot pair item, thedistribution appears much more blurred and more un-certain in orientation space.

5 Discussion

We have introduced a circuit for non-linear contrastdetection. The development was motivated by ananalysis of the process of local structure measurementwithin luminance distributions. The proposed mecha-nism incorporates the local detection of contrast bymeasuring the amplitude of luminance changes, while atthe same time accounting for the sharpness of any suchchange. The importance of the latter functionality wasmotivated by our earlier investigations on brightnessperception (Pessoa et al. 1995). However, as discussed inSect. 1, such functionality is also important for robustcontrast estimation in Retinex-like algorithms, whereshallow gradients need to be discounted. In fact, weclaim that processes of contrast measurement need to beconsidered from a broader perspective, one in whichissues such as the determination of surface qualities,ratio processing, to name a couple, are adequately takeninto account. Mechanisms only based on contrastamplitude measurements may be too limited to accountfor general processing needs.

The circuit for contrast detection produces non-zeroresponses whenever ON or OFF responses from initialcenter-surround processing are present. This allows thecircuit to signal shallow luminance gradients as in thecase of ramps or harmonic modulations. The sharpnessof a luminance transition, for example, at a luminanceedge, is encoded by the spatial juxtaposition of such ON

and OFF responses. Relative to the sampling scale of theproposed mechanism, the circuit registers any juxta-posed input con®guration producing an extra strongresponse component with respect to the individual linearcontributions.

The response properties of the circuit were embeddedwithin the framework of scale-space processing, in par-ticular the mechanism of scale selection. With referenceto linear models of contrast processing based, for in-stance, on Gaussian derivatives, we have speci®ed con-ditions for the generation of absolute maximum scale-space responses and veri®ed these results throughspeci®c computer experiments. We suggest that therepresentation generated by the circuit is uniquelyselective to intrinsic parameters of the luminance tran-sition of a priori unknown width and height.

5.1 Relation to other proposals

Our scheme shares important features with othercontrast measurement proposals including the well-known Marr and Hildreth (1980) edge detection pro-posal. Marr and Hildreth's main idea is that it is possibleto detect edges by linking the outputs of ON and OFF

center-surround cells (such as LGN cells) through a

logical AND gate. Such a scheme is based on the ideathat spatially adjacent ON and OFF responses representthe in¯ection point of a sharp transition ± thus the zero-crossings of its second-order derivative indicate the edgetransition (Poggio 1983). A closer inspection of theprocess of zero-crossing detection indicates that itutilizes an oriented multiplicative combination of initialON/OFF responses based on a simple ®rst-order di�erencemechanism. The zero-crossing detection can be formal-ized using our previous notation used in Sect. 2. A LD(DL) transition is detected via

zLD �max�c�i;j � cÿi�1;j; c

�i;j � cÿi;j�1

�and

zDL �max cÿi;j � c�i�1;j; cÿi;j � c�i;j�1

h i �22�

(compare with Eqs. 3, 4 and 11).While our model provides a circuit capable of real-

izing an AND-gate-like behavior, it does not compute alogical AND gate, but instead a soft gate. This propertyis one of the key di�erences between the present pro-posal and the Marr and Hildreth method. Althoughimportant, edge detection should not be the sole aim ofcontrast measurement. As stated above, what is neededis a general, robust method of contrast measurementthat incorporates not only physical contrast assessment,but also other factors, such as transition form. In ad-dition, image contours are present not only at edges butat other luminance distributions [see, for example, Fig. 8of Pessoa (1996)]. Moreover, utilizing a numerical dif-ference scheme, simple Fourier interpretation renderssuch a mechanism highly sensitive to noise. This is notthe case in our model as we use Gaussian low-pass ®ltersbefore sampling ON and OFF inputs.

The analysis of the di�erential structure of imagecurves and contrast outlines led Zucker and colleagues(Dobbins et al. 1990; Iverson and Zucker 1995) to pro-pose a syntactic scheme de®ning a language of logical/linear operators. In this formalization, operators com-posed of tangentially separated sub-®elds were designedto selectively respond to contrast (``edge'') and linefeatures while enabling the operators to automaticallysuppress ``false'' responses. The characterizing feature ofZucker's logical/linear scheme is that it selectively sup-presses any response that is generated by a luminancestructure to which a contrast or line operator does not®t. Yet, it generates the responses in a linear fashion ±but only at locations that match the ®lter's structure. Inour case, the circuit signals the presence of a relevant``sub-optimal'' structure, i.e., a structure that does notmatch the spatial arrangement the circuit is most sensi-tive to. In the case of optimal structures (e.g., steptransitions), however, the presence of such a sharpcontrast is signaled by an extra amount of high ampli-tude response that raises the activation over that of thelinear response alone. This allows the decision as to whatconstitutes signi®cant structure to be postponed, allow-ing more global stages of visual integration to selectivelyfocus and enhance the visual structure that initially maynot appear as dominant (in terms of amplitude re-sponse). In all, the non-linearities in Zucker's and in our

527

scheme di�er: in Zucker's approach, the non-linearityfunctions as an early local decision mechanism, while inour model it serves as an emphasizing, or ``boosting,''mechanism to generate a high amplitude or energy re-sponse.

While other computational models have not explicitlydiscussed any scale-space-related processing properties,our scheme nicely generalizes to the multi-scale frame-work of edge detection. We have demonstrated how thenon-linearity in the response to juxtaposed ON and OFF

input can be utilized to signal the optimal operator scalefor a gradual luminance transition of a certain spatialextent. The multi-scale scheme approximates a Gaussian®rst-order derivative scheme that makes use of the non-linearity in a scale-dependent input combination of ad-jacent contrast arrangement.

5.2 Simple cell physiology

Originating from Hubel and Wiesel's proposal (1962), along-standing view of simple cell response is that itdepends on the linear sum of ON and OFF LGN signals(for a review see von der Heydt 1987). Simple cellslinearly sum (or pool) all of their inputs ± weighted bytheir e�ective strength which is de®ned by the RFresponse pro®le. Data supporting the view that simplecells behave as linear devices come from a study bySchumer and Movshon (1984) showing that simple cellsintegrate their inputs in a sum-to-threshold linearmanner. In addition, stimulus-response measurementsof cells using spatial frequency modulated luminancegratings reveal a basically linear relationship whosesensitivity pro®le can be closely approximated by,among others, Gabor/Wavelet pro®les generated by aGaussian weighted sinusoidal spatial modulation (Pollenand Ronner 1983; see Daugman 1985, for a mathemat-ical description), derivative of Gaussian pro®les, oro�set Gaussian pro®les (Heggelund et al. 1983).

Non-linearities utilized in several computationalschemes, like ours, have also been observed in corticalsimple cell behavior, thus indicating that the view ofsuch cells as linear devices is, in general, untenable (seevon der Heydt 1987). For example, Hammond andMacKay (1983) investigated the length-summationcharacteristics and probed cells with bars composed ofopposite polarity segments (i.e., light and dark seg-ments). An OFF sub-®eld was probed by a dark bar thatincluded two light ends, and the same was done for theON sub-®eld using light bars with dark segments at theirends. The inclusion of inverse-polarity segments largelysuppressed cell responses instead of only causing a re-duction of response that is proportional to the length ofthe opposite polarity segments used ± as predicted bylinearity. A more recent investigation by von der Heydtet al. (1991) has demonstrated the existence in areas V1and V2 of cells selectively responsive to oriented highspatial frequency gratings. Cells of this type vividly re-spond to gratings composed of alternating light anddark bars while remaining silent when stimulated byisolated bars of either contrast polarity. This has been

interpreted as evidence for non-linear sub-®eld integra-tion, since the responses could not be reconciled with alinear ®lter model responsive to gratings of the givenspatial frequency. The responses were critically depen-dent on the precise stimulus periodicity, suggesting thatthe cells integrate spatially aligned arrangements of al-ternating ON and OFF LGN input in a non-linear way[see von der Heydt (1987) for an overview and discussionof previous ®ndings in that direction, and Petkov andKruizinga (1997) for an approach to model the key be-havior].

These and several other observations suggest that theview of cortical contrast selective cells as being basicallylinear devices is too limited. In addition, there is growingevidence that they are sensitive to more than just localcontrast magnitude. We suggest that an analysis basedon a broader view of the functionality of early stages ofvisual processing may guide the development of modelsof visual contrast measurement.

Acknowledgements. We are grateful to the anonymous reviewersfor their critical comments that helped to improve the manuscript.The collaborative research is supported by the German-BrazilianAcademic Research Collaboration Program (DAAD-CAPES,Probral 1997, Project No. 70712). The research work by H.N. andT.H. is performed in the Collaborative Research Center on the``Integration of symbolic and subsymbolic information processingin adaptive sensory-motor systems'' (SFB-527), which is located atthe University of Ulm and is funded by the German ScienceFoundation (DFG). L.P. is also supported in part by a CNPq/Brazil grant (520419/96-0).

Appendix 1: A linear Gaussian edge detectorin scale-space

This section draws upon previous work by Neumann and Otten-berg (1992a,b) who investigated the optimum scale-space responsefor one-dimensional odd-symmetric intensity transitions based on alinear pre-processing stage. In particular, we utilize here the cas-cade processing of the initial center-surround processing stagefollowed by an operation that realizes a low-pass ®ltered ®rst-orderderivative. We use an oriented gauge coordinate system �u; v� asintroduced above in order to express the derivative attributes interms of matched directional derivative operations. We utilize thisframework to relate the ®ndings on the evolution of scale-spaceresponse to the concept of automatic scale selection, which has onlyrecently been suggested by Lindeberg (1996).4 It should be notedthat for the analysis of the scale-space response, we only analyzehere the numerator term of the normalized response. Since wefurther investigate the relation between linear and non-linear re-sponses in relation to their scale-selection properties (see Appen-dix 2), the denominator component for normalization was omittedhere. We start with the analysis of the response to a step edge, thenproceed to the more general case of a ramp transition.

Response for a step edge luminance transition

A step edge pro®le of height h is de®ned by an amplitude-scaledHeaviside function H�:�, thus having

4 Thanks to Maria Quiteria (Inst. of Oceanographic Studies, Riode Janeiro) for providing some of the insights and a fresh view onrecent modeling approaches.

528

fstep�u� � h �H�u� :

The model consists of a sequence of principal processing stages.The ®rst step consists of a center-surround mechanism de®ning anisotropic band-pass ®ltering operation. We approximate the (nor-malized) di�erence-of-Gaussian by a Laplacian-of-Gaussian (LoG)operation. The response is generated by the spatial convolution ofthe step pro®le with the impulse response of the LoG ®lter,fstep�u� LoGr�u�. We get5

LoGstep�u� � hd

duGr�u� :

The LoG-®ltered pro®le is processed subsequently by a ®rst-order(Gaussian) derivative (D1G) ®lter. At its limit, for r ! 0, theGaussian derivative will converge to d=du d�u�. The ®nal spatialpro®le of contrast cell responses for its minimum spatial scale limitsis given by

sstep�u� � h � d2

du2Gr�u� �

h

r21ÿ u2

r2

� �

Gr�u� :

The response amplitude at the step edge location, u � 0, for anormalized Gaussian weighting function is then

sstep�u � 0� � h

r2Gr�u�

����

���� �

h������

2pp

� r3

����

���� :

The result shows that the response taken as a function of scale r isstrong monotonically decreasing such that the maximum responseappears for the smallest scale.

Response for a ramp edge luminance transition

A ramp edge pro®le of width R and height h can be generated bythe convolution of an amplitude-scaled Heaviside function (seeabove) with a unit-pulse function P�:� of the corresponding width.The pro®le is thus described analytically as

framp�u� � h �H�u� PR�u� :

The response of the model is generated by the spatial convolutionof the ramp pro®le with the impulse response of the LoG-®lter,framp�u� LoGr�u�. We get

LoGramp�u� �h

RGr u� R

2

� �

ÿGr uÿ R

2

� �� �

:

Corresponding to the procedure we have adopted for the step case,further processing of the LoG-®ltered pro®le by a ®rst order(Gaussian) derivative (D1G) ®lter yields

sramp�u� �h

R

d

duGr u� R

2

� �

ÿ d

duGr uÿ R

2

� �� �

:

The response consists of a sum of two Gaussian derivative pro®lesof opposite sign shifted to either side of the central location of theramp transition. For completeness, we can con®rm the result toinclude the special case of a step transition. By taking the limitR ! 0, the response sramp�u� converges to that derived for a step,limR!0 sramp�u� � sstep�u� (see above).

Scale-space processing and responsesat ramp edge location

Depending on the width of the ramp and the relative scaling of thecascaded pre-processing stage, the response pro®le appears in dif-ferent spatial arrangement. We consider a case that has beensketched in Fig. 4. For a scale that is small relative to the transitionwidth (r � R), Gaussian derivatives appear as isolated ON and OFF

channel responses at the ``knees'' where the plateaus meet the ramptransition. For a coarser scale after blurring, the central lobes ofeither ON or OFF responses (depending on a light-dark or a dark-light transition, respectively) meet and overlap at the center of theramp and appear as juxtaposed input contrast responses. Analyti-cally, for a condition R � r (corresponding to R ! 0), we will getthe second-order derivative of a Gaussian pro®le that peaks at thecenter of the ramp (compare with the results of mathematical in-vestigation above and see Fig. 6).

In order to evaluate the behavior of a scale-space edge detectorfor a gradual luminance transition, we evaluate its response at thecenter of the ramp (u � 0). We make use of the even-symmetry ofthe Gaussian, such that we ®nally get the amplitude

sramp�u � 0� � 2h

R�G0

r

R

2

� �

:

We now treat sramp as a function of scale r such that we derive themagnitude function

sramp�r�ju�0 �h������

2pp

r3exp ÿ R2

8r2

� �

:

Figure 12 shows pro®les of scale-space responses of a linear cell forvariable parameter settings of ramp transition widths R and con-stant height h = 1. The responses for a ramp transition shown inSect. 4 (Fig. 6, bottom) represent a coarse sample of the analyticpro®le presented in Fig. 12 (right). For a mechanism capable ofselecting an appropriate scale it needs a proper selection and de-cision criterion. A useful candidate for this selection appears to bethe maximum scale response along u � 0. The unique peak re-sponse along the scale is determined by the zero-crossing of theslope function for sramp�r�ju�0. We get

d

drsramp�r� � ÿ h

������

2pp

r4

R2

4r2ÿ 3

� �

exp ÿ R2

8r2

� �

� 0 :

The solution for the scale-space location of the peak response isgiven by

rmax �R

2���

3p ;

which indicates an operator scaling slightly less than half the ramptransition width. A mechanism reading out such maximum re-sponses must rely on a robust selection criterion that guarantees thecorrespondence with a representation of variable-width contrastedges. We investigate this topic below in terms of absolute maxi-mum contrast responses suggesting a winner-take-all type mecha-nism.

Scale selection for a ramp contrast

In order to investigate the relevant conditions for selecting maximalresponses corresponding to a ramp of certain width, we comparethe activations generated for sramp�rmax� u�0 � sramp�u � 0�

��

��r�rmax

with the absolute height of one lobe in the pro®le of D1G. Inparticular, we investigate the response for sramp�u � ÿR=2� r0�.The latter corresponds to the case of isolated contrast responsesat the ``knees'' of the ramp that are processed by the D1G ®lter.For the condition r0 � R, the response pro®le in the u directionshows

5 For simplicity, we are neglecting the fact that, for this 2D iso-tropic pre-®ltering stage, the e�ective space constant is re �

���

2p

r.Since this refers only to a rescaling of coordinate axes, we furtherdeal with the original scale constant r only.

529

sramp�u � ÿR=2� r0�jr0�R

� h

R

d

duGr�r0� ÿ

d

duGr�ÿR� r0�

|������������{z������������}

�0

2

664

3

775

(see Fig. 6 for comparison). Since the second derivative term al-most vanishes, we can simplify the equations by only consideringthe dominant component for further analysis. This condition cor-responds to a situation where r0 < rmax holds. If we compare theresponses of the two selected cases we get

sramp�u � 0�jr�rmax� 48

���

3p

h������

2pp exp ÿ 3

2

� �

� 1

R3

sramp�u � ÿR=2� r0�jr0�R � h������

2pp exp ÿ 1

2

� �

� 1

Rr2

With the ratio of these two expressions, we get a condition forabsolute maximum response. For this purpose we introduce thenormalized scale-space variable r̂ � r=R. The condition then ®nallyreads

48���

3p

er̂2 > 1 or; equivalently

r̂ >

������������e

48���

3p

r

� 0:18; with e � exp�1� :

We left the ®rst version of the inequality for direct comparison withthe result derived for the non-linear circuit (Appendix 2).

The result for the linear model demonstrates that the identi®edoptimum scale is not guaranteed to yield maximum responses inabsolute terms as related to the overall ®eld of activities. In par-ticular, if r̂ becomes less than approximately 1=5, the responsesramp�u � 0�jr�rmax

declines to a value lower than the maximumresponse for the smallest scale, as demonstrated in Fig. 6 (bottom).If, however, the necessary condition is met, the maximum scaleresponse determines the overall maximum amplitude in the array ofscale-space ®lter responses.

Appendix 2: Scale-space responsefor the non-linear circuit

It has been shown in Appendix 1 that the linear model alreadyshows a selectivity for an optimal scale. The response taken as afunction of scale S�r� yields a unique maximum value, pre-req-uisite for automatic scale selection. However, it was shown thatthis value cannot be guaranteed to be the maximum response inabsolute terms ± as also demonstrated by the simulations shownin Fig. 6.

A compact notation of the response properties of the non-linearcircuit is given by

z � L � p� � pÿ� � �N

L� p� � pÿ� �

� �

(compare with Eq. 12). This demonstrates that the non-linear cir-cuit in its linear processing behavior inherits all the qualitativeproperties shown in Appendix 1. Moreover, for con®gurationsapproaching the optimum scale conditions, i.e., con®gurations withjuxtaposed ON and OFF activations, the results of the linear pro-cessing component are superimposed by the correlation (gating)between ON and OFF responses.

The function L only de®nes a proportionality for bothlinear and non-linear components. For any further analysis ofthe response z, it is therefore su�cient to consider the terms inlarge brackets (see equation above). The non-linear contribu-tion is scaled by the factor N=L. Since we want to relate thecell response to the activations generated by the initial center-surround processing stage, we have to scale this ratio ac-cordingly. Based on the dependency described in Eq. (3), wearrive at

N0 � N

L

bsas

� 2bcac

bsas

;

in which bs=as is the gain of the initial center-surround processingstage, denoted by net� ÿ net�.

Fig. 12. Linear contrast cell response at ramp location u � 0 treated as a function of scale (r). Processing results of a family of ramp pro®les ofunit height (h � 1� is shown as a surface for di�erent widths R (left); the shape of the response pro®le for one ®xed setting of R shows a uniquemaximum in response as a function of scale (right). As it is predicted by the equation for sramp�r�ju�0, a unique shape of scale-space pro®les isshifted in peak location for increasing values of R accompanied by a drop in overall response amplitude

530

Non-linear response for step edges

In Appendix 1, we have analyzed the responses of a linear mech-anism processing a step edge pro®le. The non-linear part of thetotal response is contributed by p� � pÿ� �. We determine the mag-nitudes from the maxima in the D1G pro®le. These correspond tothe o�set points for sub-®eld integration given by s � �r. For thepro®le of a ®rst-derivative of a Gaussian we get

hd

duGr�u � r� � ÿ h

������

2pp

r2exp ÿ 1

2

� �

:

We introduce the functions slinstep � sstep (see Appendix 1) and snlstepand show the non-linear response in terms of the linear responsederived in Appendix 1. The signi®cant part of a step edge response(proportion of response z given in brackets, see above) generated bythe non-linear circuit is then given by

snlstep�u � 0� � h������

2pp

r3�N

0 1������

2pp

er;

� slinstep�u � 0� � 1�N0 1������

2pp

er

� �

:

Two observations can be made from this result: (1) as in the linearcase, the response for a step edge is a monotonically decreasingfunction of scale in which the maximum response is generated atthe smallest scale, and (2) the contribution of the non-linearitydominates for a gain factor

N0 �

������

2pp

er :

Optimum input conditions for non-linear processing

Non-zero correlation between ON and OFF channel activation isavailable when both sub-®elds of a cell integrate positive inputactivation. The initial center-surround responses at the ``knees'' ofthe ramp (see Appendix 1) therefore have to be progressivelyblurred to meet at the center of the ramp. For each Gaussian bumpin the initial LoGramp�u� pro®le we get

h

R�Gr0 u� R

2

� �

Gr�u�� �

� � h

RG ����������

r20�r2

p u� R

2

� �

:

Proper scaling of the blurred pro®le approaches the value deter-

mined for the optimum scale, such that����������������

r20 � r2q

� rmax. The

pro®le showing the di�erence of two broadly tuned o�set Gaussi-ans assembles a D1G ®lter.

Input integration and maximum scale responsefor ramp edges

Since o�sets were symmetric around the center of the ramp, wesample each shifted Gaussian Grmax

�u� R=2� at u � �s � �R=2.For the gating-type input to the non-linear response we thereforeget

p� � pÿ� �jr�rmax� h2

R2� 1

2p � r2max

� 6h2

p � R4;

with rmax � R=�2���

3p

�. We can now write the response amplitude ofthe non-linear circuit at the optimum scale condition, rmax, as acombination of the linear (additive) and non-linear (correlational)input contribution. We introduce the functions slinramp � sramp (seeAppendix 1) and snlramp and present the non-linear response in termsof the linear response derived in Appendix 1. We get

snlramp�u � 0�jr�rmax� 48

���

3p

h������

2pp exp ÿ 3

2

� �

� 1

R3�N

0 6h2

p � R4;

� slinramp�u � 0�jr�rmax

� 1�N0

���

2p

8������

3pp h

Rexp

3

2

� � !

:

Several observations can be made from this result: (1) the responseof the circuit for a ramp edge transition can be predicted for anoptimum scale, (2) the contribution of the non-linearity depends onthe slope of the ramp transition denoted as m � h=R, (3) the plainnon-linearity of the circuit (without the gain N

0) contributes with a26% increase ± times the slope ± in output magnitude. In the caseof discrete signals and images, any transitions are of unit width atminimum, i.e., RP1. For a ramp of unit slope, and the non-linearcomponent to dominate the response, the gain factor N0 has to behigh enough to ``boost'' the output activation. Moreover, with sucha gain factor it can be guaranteed that the magnitude of response ismaximal in absolute terms (compare the analysis for a linear ®lterin Appendix 1). A dominance of the response caused by the non-linear, i.e., multiplicative, contribution from both ON and OFF sub-®eld is achieved for

N0 � 4

������

6pp R

hexp ÿ 3

2

� �

:

Since the signal parameters h and R are not known in advance, thegainN

0 may be chosen high enough to guarantee the dominance ofany non-linear contribution.

References

Arend L, Goldstein R (1987) Lightness models, gradient illusions,and curl. Percept Psychophys 42:65±80

Blake A (1985) Boundary conditions for lightness computation inMondrian world. Comput Vis Graph Image Process 32:314±327

Bracewell RN (1978) The Fourier transform and its applications2nd edn. McGraw-Hill, New York

Brookes RA, Stevens KA (1991) Symbolic grouping versus simplecell models. Biol Cybern 65:375±380

Buf J du (1993) Responses of simple cells: events, interferences, andambiguities. Biol Cybern 68:321±333

Buf J du (1994) Ramp edges, Mach bands, and the functionalsigni®cance of the simple cell assembly. Biol Cybern 69:449±461

Carandini M, Heeger DJ (1994) Summation and division by neu-rons in primate visual cortex. Science 264:1333±1336

Daugman J (1985) Uncertainty relation for resolution in space,spatial frequency, and orientation optimized by two-dimen-sional visual cortical ®lters. J Opt Soc Am A 2:1160±1169

Dobbins AL, Iverson L, Zucker SW (1990) A logical/linear modelof cortical subunit interaction. Proc ARVO'90 (Invest, Opt-halmol Vis Sci 31: 397, Sarasota, Florida, USA, April 29 ±May 4

Enroth-Cugell C, Robson JG (1966) The contrast sensitivity ofretinal ganglion cells of the cat. J Physiol (Lond) 187:517±552

Ferster D (1988) Spatially opponent excitation and inhibition insimple cells of the cat visual cortex. J Neurosci 8:1172±1180

Ferster D (1989) The synaptic inputs to simple cells in the cat visualcortex. In: Lam D, Gilbert C (eds) Neural mechanisms of visualperception. Portfolio Publishing, The Woodlands, Tex, pp63±85

Furman GG (1965) Comparison of for subtractive and shuntinglateral-inhibition in receptor-neuron ®elds. Kybernetik 2:257±274

Glass L (1969) Moire e�ect from random dots. Nature 223:578±580Grossberg S (1970) Neural pattern discrimination. J Theor Biol

27:291±337

531

Grossberg S, Mingolla E (1985) Neural dynamics of form percep-tion: boundary completion, illusory contours, and neon colorspreading. Psychol Rev 92:173±211

Hammond P, MacKay D (1983) In¯uence of luminance gradientreversal on simple cells in feline striate cortex. J Physiol (Lond)337:69±87

Heggelund P (1981) Receptive ®eld organization of simple cells incat striate cortex. Exp Brain Res 42:89±98

Heggelund P (1986) Quantitative studies of enhancement andsuppression zones in the receptive ®eld of simple cells in catstriate cortex. J Physiol (Lond) 373:293±310

Heggelund P, Krekling S, Skottun BC (1983) Spatial summation inthe receptive ®eld of simple cells in the cat striate cortex. ExpBrain Res 52:87±98

Heydt R von der (1987) Approaches to visual cortical function.Rev Physiol Biochem Pharmacol 108:69±150

Heydt von der R, Peterhans E, DuÈ rsteler MR (1991) Grating cellsin monkey visual cortex: Coding texture? In: Blum B (ed)Channels in the visual nervous system: neurophysiology, psy-chophysics and models. Freund, London, pp 53±73

Hodgkin AL (1964) The conduction of the nervous impulse. Liv-erpool University Press, Liverpool

Horn BKP (1974) Computing lightness from an image. ComputGraph Image Process 3:277±299

Hubel DH, Wiesel TN (1962) Receptive ®elds, binocular interac-tion and functional architecture in the cat's visual cortex.J Physiol 160:106±154

Iverson LA, Zucker SW (1995) Logical/linear operators for imagecurves. IEEE (Inst Electr Electron Eng) Trans Pattern Analysisand Machine Intelligence 17:982±996

Koenderink JJ (1984) The structure of images. Biol Cybern 50:363±370

Koenderink JJ, van Doorn A (1990) Receptive ®eld families. BiolCybern 63:291±297

Land EH, McCann JJ (1971) Lightness and retinex theory. J OptSoc Am 61:1±11

Lindeberg T (1996) Scale-space: a framework for handling imagestructures at multiple scales. Lecture notes from the CERNschool of computing 1996. Egmond aan Zee, The Netherlands,8±21 September

Lindeberg T, ter Haar Romeny B (1994) Linear scale-space I: basictheory. In: ter Haar Romeny B (ed) Geometry-driven di�usionin computer vision, Freund, London, pp 1±38

Liu Z, Gaska JP, Jacobson LD, Pollen DA (1992) Interneuronalinteraction between members of quadrature phase and anti-phase pairs in the cat's visual cortex. Vision Res 32:1193±1198

Marr D (1982) Vision. Freeman, San FranciscoMarr D, Hildreth E (1980) Theory of edge detection. Proc R Soc

Lond Ser B 207:187±217Morrone MC, Burr DC (1988) Feature detection in human vision:

a phase-dependent energy model. Proc Royal Soc Lond Ser B235:221±245

Morrone MC, Burr DC, Ma�ei L (1982) Functional implica-tions of cross-orientation inhibition of cortical visual cells. I.

Neurophysiological evidence. Proc R Soc Lond Ser B216:335±354

Neumann H (1996) Mechanisms of neural architecture for visualcontrast and brightness perception. Neural Netw 9:921±936

Neumann H, Ottenberg K (1992a) Estimating ramp-edge attributesfrom scale-space. In: Vandevalle J, Boite R, Moonen M,Oosterlinck A (eds) Signal Processing VI: theories and appli-cations vol I±III (6th Eur Signal Proc Conf ICPR-92), Brussels,Belgium. Elsevier, Amsterdam, pp 603±607

Neumann H, Ottenberg K (1992b) Estimating attributes of smoothsignal transitions from scale-space. Proc 11th Int Conf PatternRecog vol I±IV (ICPR-92), The Hague IEEE Press, Los Ala-mitos, California, pp 754±758

Neumann H, Pessoa L (1994) A simple cell model with multiplespatial frequency selectivity and linear/non-linear responseproperties. Proc of the World Congr on Neural Networks(WCNN-94), vol I±IV, San Diego, USA. LEA, Hillsdale, NewJersey, pp 290±298

Pessoa L (1996) Mach bands: how many models are possible?Recent experimental ®ndings and modeling attempts. VisionRes 36:3205±3227

Pessoa L, Mingolla E, Neumann H (1995) A contrast- and lumi-nance-driven multiscale network model of brightness percep-tion. Vision Res 35:2201±2223

Petkov N, Kruizinga P (1997) Computational models of visualneurons specialized in the detection of periodic and aperiodicoriented visual stimuli: bar and grating cells. Biol Cybern76:83±96

Poggio T (1983) Visual algorithms. In: Braddick OJ, Sleigh AC(eds) Physical and biological processing of images. (Springerseries in information sciences, vol 11) Springer, Berlin Heidel-berg Newy York, pp 128±153

Pollen DA, Ronner SF (1983) Visual cortical neurons as localizedspatial frequency ®lters. IEEE (Inst Electr Electron Eng) TransSystems, Man, and Cybernetics 13:907±916

Rodieck RW (1965) Quantitative analysis of cat retinal ganglioncell responses to visual stimuli. Vision Res 5:583±601

Sagi D, Kova cs L, EoÈ tvoÈ s L (1993) Long range processes involvedin the perception of Glass patterns. Proc. ARVO'93 (InvestOphthalmol Vision Sci 34. Sarasota, Florida, USA, May 2±7, p1130

Schiller P (1992) The ON and OFF channels of the visual system.Trends Neurosci 15:86±91

Schumer RA, Movshon JA (1984) Length summation in simplecells in cat striate cortex. Vision Res 24:565±571

Tolhurst DJ, Dean AF (1990) The e�ects of contrast on the lin-earity of spatial summation of simple cells in the cat's striatecortex. Exp Brain Res 79:582±588

Vidyasagar TR, Pei X, Volgushev M (1996) Multiple mechanismsunderlying the orientation selectivity of visual cortical neu-rones. Trends Neurosci 19:272±277

Witkin AP (1983) Scale-space ®ltering. Proc 8th Int Joint ConfArtif Intell, IJCAI-83. vol I±II, Karlsruhe, Morgan KaufmanPubl., Palo Alto, California pp 1019±1022

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