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COUPLED SHAPE DISTRIBUTION-BASED

SEGMENTATION OF MULTIPLE OBJECTS

Andrew Litvin and William C. Karl

March 2005

Boston University

Department of Electrical and Computer Engineering

Technical Report No. ECE-2005-01

BOSTON

UNIVERSITY

COUPLED SHAPE DISTRIBUTION-BASED

SEGMENTATION OF MULTIPLE OBJECTS

Andrew Litvin and William C. Karl

fg

Boston UniversityDepartment of Electrical and Computer Engineering

8 Saint Mary’s StreetBoston, MA 02215www.bu.edu/ece

March 2005

Technical Report No. ECE-2005-01

This work was partially supported by AFOSR grant F49620-03-1-0257,National Institutes of Health under Grant NINDS 1 R01 NS34189, Engi-neering research centers program of the NSF under award EEC-9986821.The MR brain data sets and their manual segmentations were providedby the Center for Morphometric Analysis at Massachusetts General Hos-pital and are available at http://www.cma.mgh.harvard.edu/ibsr/.

Summary

In this paper we develop a multi-object prior shape model for use in curve evolution-based image segmentation. Our prior shape model is constructed from a familyof shape distributions (cumulative distribution functions) of features related to theshape. Shape distribution-based object representations possess several desired prop-erties, such as robustness, invariance, and good discriminative and generalizing prop-erties. Further, our prior can capture information about the interaction between mul-tiple objects. We incorporate this prior in a curve evolution formulation for shapeestimation. We apply this methodology to problems in medical image segmentation.

i

Contents

1 Introduction 1

2 Prior work 1

3 General formulation and shape distribution prior 3

3.1 Segmentation by energy minimization based on curve evolution. . . . 33.2 The Shape Distribution Concept . . . . . . . . . . . . . . . . . . . . . 33.3 A prior energy based on shape distributions . . . . . . . . . . . . . . 43.4 Gradient flow computation . . . . . . . . . . . . . . . . . . . . . . . . 6

4 Results 9

5 Conclusions 10

A Appendix. Analytical computation of the minimizing flows for dis-

tribution difference priors 11

A.1 Feature class #1. Collection of inter-point distances . . . . . . . . . . 12A.2 Feature class #2. Multiscale “curvatures” . . . . . . . . . . . . . . . 15A.3 Feature class #3. Relative inter-object distances . . . . . . . . . . . . 21

ii

List of Figures

1 An example of constructing a shape distribution for a curve (left) basedon curvature measured along the boundary. The middle graph showscurvature function κ(s) measured along the curve and the sketch ofthe cumulative distribution function H(κ) of curvature is shown onthe right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Feature value sets used in this work illustrated for a curve C discretizedusing 6 nodes. Feature class #1 (left): interpoint distances d13..d15.Feature class #2 (center): interpoint angles α−1,1,2..α−n,1,n (n = 2)are shown. Feature class #3 (right): feature set values for curve C aredefined as shortest signed distances from the curve Γ to nodes of thecurve C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Synthetic 2 shape example: (a) Noisy image; Solid black line shows thetrue objects boundaries; dashed white lines - initial boundary position;(b) Segmentation with curve length prior; (c) - Segmentation with newmultiobject shape distribution prior including all three feature classes.Solid black line shows the true objects boundaries; solid white lines -final boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

4 Brain MRI segmentation: (a) Multiple structures and interactions usedfor feature class #3; (b) Segmentation with independent object curvelength prior. (c) Segmentation using multiobject PCA technique in[21] (d) Segmentation with new multiobject shape distribution prior.Solid black line shows the true objects boundaries; solid white line -final segmentation boundary. . . . . . . . . . . . . . . . . . . . . . . . 9

5 Illustration of feature value computation for feature class #2 . . . . . 156 4 cases of relative location of points. . . . . . . . . . . . . . . . . . . 167 Sequential computation of the angles for a particular “base” point s1,

starting from r = 1 (assuming inside of the curve is up-wards). . . . . 16

8 Local perturbation of the curve at point ~Γ(s1 + s2). Perturbation

εβ(s1 + s2) is infinitely small comparing to |~Γ(s1, s1 + s2)| . . . . . . . 179 Illustration of 2 cases when the sign of the angle increment dα(1) is

different for the same curve perturbation εβ(s). . . . . . . . . . . . . 18

iii

Segmentation of multiple objects 1

1 Introduction

The use of prior information about shape is essential when image intensity alonedoes not provide enough information to correctly segment objects in a scene. Suchcases arise due to the presence of clutter, occlusion, noise, etc. In this work weconcentrate on curve evolution-based image segmentation approaches [1] and priorshape models matched to this framework. In a typical curve evolution scheme, theregions of interest in the image (i.e. the shapes or objects) are represented by non-self-intersecting closed contours. Curve evolution methods allow convenient handlingof object topology, efficient implementation, and possess variational and associatedprobabilistic interpretations. In a typical curve evolution implementation, the shape-capturing curve is evolved under the combined action of two classes of forces: thosedependent on the observed image data (data-dependent forces) and those reflectingprior knowledge about the segmented shape or boundary (regularizing forces). Thiswork is focused on the development of shape models suitable for such prior shapeforce terms in curve evolution.

Desirable qualities in a shape-based prior include invariance with respect to trans-formations such as scale, translation, and rotation, independence from knowledge ofcorrespondence, robustness of the resulting solution to initialization, the ability togenerate the prior model from training data, and subsequent robustness to smalltraining set size. Shape distributions have been used in the computer graphics com-munity [16] to characterize shapes and, more recently, have been successfully appliedto shape classification problems. They were shown to possess the desired propertiesof robustness, invariance, and flexibility. To date, however, shape distributions havenot been used in an estimation context.

In this work we develop a novel multi-object prior based on such shape distri-butions for use in estimation. We then present a method to use this prior in curveevolution-based segmentation problems. Finally, we suggest the promise of this priorfor challenging medical image segmentation tasks through an example. In our for-mulation, the shape prior is constructed by designing a shape similarity measure pe-nalizing the difference between shape distributions extracted from boundary curvesunder comparison.

In Section 2 we give an overview of existing shape modeling approaches and themotivations behind our technique. In Section 3 we review the curve evolution frame-work and introduce the shape distribution concept. Section 4 presents our experi-mental results and Section 5 concludes this paper.

2 Prior work

Different approaches have been proposed for the inclusion of prior information in de-formable curve-based image segmentation. The most common regularization methodfor curve evolution penalizes a quantity such as total curve length or object area[1, 13]. Such “generic” penalties are stationary along the curve, in that every point


on the boundary experiences the same effect. Such priors can remove object pro-trusions and smooth salient object detail when the boundary location is not wellsupported by the observed data, since they seek objects with short boundaries orsmall area.

Deformable template approaches construct prior models based on allowable de-formations of a template shape. Some approaches are based on representing andmodeling shape as a set of landmarks (see [6, 5] and references therein). Otherapproaches use principal component analysis based on parameterized boundary co-ordinates or level-set functions to obtain a set of shape expansion functions [22, 24]that describe the subspace of allowable shapes. Still other approaches construct morecomplex parametric shape representations, such as the MREP approach in [18], ordeformable atlas based approach in [4]. These methods are effective when the spaceof possible curves is well covered by the modeled template deformations as obtainedthrough training data, but may not generalize well to shapes unseen in the trainingset.

Motivated by such limitations in existing approaches and by the representationalrichness of the ideas in [26, 16], we propose construction of a shape prior based onan energy which penalizes the difference between a set of feature distributions of agiven curve and those of a prior reference set. Such prior shape distributions capturethe existence of certain visual features of a shape regardless of the location of thesefeatures. Shape distributions have been successfully applied to shape classificationtasks (e.g. [10, 16] and references therein), and these results indicate that shape distri-butions are robust, invariant, flexible shape representations with good discriminativeproperties. In this work, we apply such shape distribution-based models to problemsof boundary estimation in medical imaging, suggesting their potential.

Simultaneous multiple object segmentation is an important direction of researchin medical imaging. Indeed, in many problems, the positions of segmented parts arehighly correlated and can be used to further constrain the resulting boundary esti-mation. In [25, 20, 8] the authors extend the PCA shape model to constrain multipleobject locations. In [7], a PDM model was extended to model multiple shapes. In[9, 17] the authors mutually constrained shapes by penalizing quantities such as areaof overlap between different objects. These approaches do not include shape specificinformation regarding the different objects. Our approach shares a common idea with[15], where the concept of a force histogram was used to characterize shapes. In [15]histogram-based descriptors were used solely in discrimination while our focus is touse similar descriptors in estimation problems.

In this work we show how the shape distribution-based modeling approach can beextended to encode relationships between different segmented objects. This modelholds the potential for richer descriptions of object shape. In contrast to existingapproaches, this model directly encodes properties of a class of shapes (versus simplypenalizing points of high curvature), yet does not depend on the specific embeddingof a shape, and thus generalizes well to unseen objects. We present a curve evolutionprocess which drives a given set object curves so that their corresponding shape dis-


tributions will match those of a target. This approach seems to provide an interestingalternative shape prior, well matched to curve evolution approaches, with promise forchallenging segmentation problems.

3 General formulation and shape distribution prior

3.1 Segmentation by energy minimization based on curve

evolution.

First, we review the overall approach of segmentation in a curve evolution frameworkbased on energy minimization. We assume that each object (shape) in the image isdefined by a separate closed curve. An energy functional is defined for each object(in contrast to the multi-phase approach in [23]) and each boundary is evolved tominimize the overall energy. For a single object, then, the solution curve C∗ is soughtas:

C∗ = argminC

E(C) (1)

The energy E(C) consists of a data term and a shape prior term:

E(C) = Edata(C) + αEprior(C) (2)

The data term Edata favors the fidelity of the solution to the image data. This termis sensor and application specific. In this paper we use two definitions of Edata:the bimodal image energy in [2] and the information theoretic energy in [12]. Ourcontribution is focused on the shape prior term Eprior, which we base on the conceptof shape-distributions, discussed in Sec. 3.2. The positive scalar α is a regularizationparameter that balances the strength of the prior and data.

The gradient curve flow minimizing (2) can be found using variational approachesor shape-gradient approaches [11]. In this work we utilize variational approaches.The curve is then evolved according to the gradient curve flow:

dC

dt= −∇E(C) (3)

were t is an artificial time parameter. The curve flow can be interpreted as a forceacting on a curve. We implement the curve evolution in (3) via the level-set frame-work [19]. Note that this framework is very general, and our new prior term canbe coupled with many existing data terms (ex. [2, 12]). In the multiobject case thesimultaneous evolution of multiple boundaries effectively minimizes an energy whichis a sum of terms (2) corresponding to the individual objects.

3.2 The Shape Distribution Concept

Distributions of features measured over shapes in a uniform way, are called shapedistributions [16]. As shown by recent shape classification experiments [16], such


kH(k)

s1

k

Shape

Figure 1: An example of constructing a shape distribution for a curve (left) based oncurvature measured along the boundary. The middle graph shows curvature functionκ(s) measured along the curve and the sketch of the cumulative distribution functionH(κ) of curvature is shown on the right.

shape distributions can capture the intuitive similarity of shapes in a flexible waywhile being robust to a small sample size and invariant to object transformation.

In a continuous formulation, shape distributions are defined as sets of cumulativedistribution functions (CDFs) of feature values (one distribution per collection offeature values of the same kind) measured along the shape boundary or across theshape area. Joint CDFs of multiple features can also be considered, although inthis work we only consider one-dimensional distributions. An illustrative example ofthe shape distribution idea is shown in Figure 1, using boundary curvature as thefeature. We define the shape distribution for a class or set of shapes as an averageof the cumulative distribution functions corresponding to the individual shapes inthe group. This approach is equivalent to combining feature value representations(continuous functions or discrete sets of values) measured on individual shapes intoa single set and then defining the overall CDF of the resulting set.

We will call a set of feature values extracted from a shape a “feature class”. In theexample of Figure 1 the feature class is boundary curvature. Separate feature classescapturing different characteristics of shapes can be combined in a single framework,creating a more versatile prior. We distinguish two types of feature classes. For thefirst type of feature, which we term autonomous, the values for a particular curveare computed with reference only to the curve itself. For the second type of feature,which we term directed, the feature values are computed with reference to the curvesof other objects. By incorporating directed feature classes into our shape models weprovide a mechanism for modeling the relationships between different objects in ascene and, thus, create a framework for multi-object segmentation.

3.3 A prior energy based on shape distributions

We now introduce our formulation of the shape prior in the continuous domain. LetΦ ∈ Ω be a continuously defined feature (e.g. curvature along the length of a curve),and let λ be a variable spanning the range of values Λ of the feature. Let H(λ) bethe CDF of Φ:

H(λ) =

∫

Ω h

Φ(Ω) < λ

dω∫

Ω dω(4)


where h(x) is the indicator function, which is 1 when the equality is satisfied and 0otherwise.

We define the prior energy Eprior(C) for the boundary curve C in (2) based onthis shape distribution as:

Eprior(C) =M∑

i=1

wi

∫

Λ

[

H∗i (λ) − Hi(C, λ)

]2dλ (5)

where M is the number of different distributions (i.e. feature classes) being used torepresent the object, Hi(C, λ) is the distribution function of the ith feature class forthe curve C, and the non-negative scalar weights wi balance the relative contributionof the different feature distributions. Prior knowledge of object behavior is capturedin the set of target distributions H∗

i (λ). These target distributions H∗i can correspond

to a single prior shape, an average derived from a group of training shapes, or can bespecified by prior knowledge (e.g. the analytic form for a primitive, such as a square).In practice, we compute the feature values, the corresponding shape distributions, andevolution forces dC/dt by discretizing curves and their properties through uniformsampling.

We use three specific feature classes in our experiments in this work, which wedefine below and illustrate in Figure 2.

• Feature class # 1. Inter-node Distances (Autonomous feature type).The feature value set F consists of the distances between nodes of the discretecurve. The distances are further normalized by the mean distance, making theset scale invariant.

F =dij | (i, j) ∈ S

mean(

dij | (i, j) ∈ S) (6)

The set S defines the subset of internodal distances used in the feature. Weuse S in a multi-scale way to define collections of nodes within a given scaledistance along the object boundary as follows:

(i, j) ∈ S(a,b) if(j − i) ∗ ds

L∈ [a, b] (7)

where a and b are the lower and upper bounds of the interval respectively; ds isthe distance between neighboring boundary nodes and L is the total boundarylength. In the experiments presented in this paper we used 4 different, non-overlapping intervals.

• Feature class # 2. Multiscale curvature (Autonomous feature type).The feature value set F consists of the collection of angles between nodes ofthe discrete curve.

F = 6 i−j,i,i+j (i, j) ∈ S (8)


1

4

32

56 Γ

CD1D2

D3

D4

Figure 2: Feature value sets used in this work illustrated for a curve C discretizedusing 6 nodes. Feature class #1 (left): interpoint distances d13..d15. Feature class#2 (center): interpoint angles α−1,1,2..α−n,1,n (n = 2) are shown. Feature class #3(right): feature set values for curve C are defined as shortest signed distances fromthe curve Γ to nodes of the curve C.

where 6 (ijk) is the angle between nodes i, j, and k. Again, the set S definesthe subset of internodal angles used in the feature and again we used S in amulti-scale way, as described in Feature class #1.

• Feature class # 3. Relative inter-object distances. (Directed feature type)The feature value set F consists of the collection of shortest signed distancesbetween each node of C and the boundary of another object Γ (negative fornodes of C inside Γ). These distances are normalized by the average radiusof Γ with respect to its center of mass. This feature class encodes the relativeposition of C with respect to Γ. Note this is a directed distance involvingdifferent objects and can be obtained from the signed distance function of theobjects.

3.4 Gradient flow computation

In order to use the energy in (5) as a prior in a curve evolution context we must beable to compute the curve flow that minimizes it. For simplicity, we consider herean energy defined on just a single feature class. Since (5) is additive in the differentfeature classes, minimizing flows for single individual feature classes can be addedwith the corresponding weights to obtain the overall minimizing flow. The energy fora single feature class is given by:

E(C) =∫ [

H∗(λ) − H(C, λ)]2

dλ (9)

Because the energy depends on the whole curve in non-additive way, the minimizingflow at any location on the curve also depends on the whole curve, and not just thelocal curve properties (as it does in the case of a curve length minimization prior).

The minimizing flow and its computation will, of course, depend on the specificsof the feature classes chosen. In [14], a numerical scheme was proposed to estimatethe flow. Such a scheme can be employed for any definition of the feature class;although, it is computationally expensive. Here we introduce an efficient approach to


analytically compute the minimizing flows for the feature classes presented previouslyusing a variational framework. The resulting flows guarantee reduction of the energyfunctionals (9). It is important to note that such a variational approach can be usedto obtain the minimizing flow for a variety of feature classes and not just the threeclasses used in this work.

Here we briefly outline the steps required to analytically compute the flow andpresent final results for our specific feature classes.

1. Find the Gateaux semi-derivative of the energy in (9) with respect to a pertur-bation β. Using the definition of the Gateaux semi-derivative, the linearity ofintegration, and the chain rule we obtain

G(E, β) = 2∫ [

H∗(λ) − H(Γ, λ)]

G[

H(γ, λ), β]

dλ (10)

2. If the Gateaux semi-derivative of a linear functional f exists, then according tothe Rietz representation theorem, it can be represented as

G(f, β) =< ∇f, β > (11)

were ∇f is the gradient flow minimizing the functional f . Therefore, we mustfind the boundary functional representation ∇H(Γ, λ) for the feature such that

G[

H(Γ, λ), β]

=< ∇H(Γ, λ), β >.

3. The overall flow minimizing (9) is then given by

∇E = 2∫ [

H∗(λ) − H(Γ, λ)]

∇(

H(Γ, λ))

dλ (12)

A detailed derivation of the gradient flows for the three features used in this workcan be found in appendix A. We simply summarize those results here. For featureclass #1 the minimizing flow is given by:

∇E(Γ)(s) = 2∫

t∈S

~n(s) ·~Γ(s, t)

|~Γ(s, t)|[

H∗(|~Γ(s, t)|) − H(Γ, |~Γ(s, t)|)]

dt (13)

where Γ is the parameterized curve as a function of arc length X(s), Y (s) with

s ∈ 0, 1, ~Γ(s, t) is a vector with coordinates X(t) − X(s), Y (t) − Y (s), and ~n(s)is the outward normal at X(s), Y (s). The flow at each s is an integral over thecurve, indicating the non-local dependence of the flow on the curve properties. Theexpression under the integral can be interpreted as a force acting on a particular pairof locations on the curve, projected on the normal direction at s.

For feature class #2, the minimizing flow is given by


(a) (b) (c)

Figure 3: Synthetic 2 shape example: (a) Noisy image; Solid black line shows the trueobjects boundaries; dashed white lines - initial boundary position; (b) Segmentationwith curve length prior; (c) - Segmentation with new multiobject shape distribu-tion prior including all three feature classes. Solid black line shows the true objectsboundaries; solid white lines - final boundary.

∇E(Γ)(s) =

−∫

t∈S

cos(β) cos(n(s),Γ+)+cos(γ) cos(n(s),Γ−)

sin αif α 6= π

sin(~n(s), ~Γ−)) otherwise

× a · r(s,t)

bc−

f (s−t)

√

1 −(

~n(s − t) · ~Γ−

|~Γ−|

)2

|~Γ−|−f (s+t)

√

1 −(

~n(s + t) · ~Γ+

|~Γ+|

)2

|~Γ+|

×

[

H∗(α(s, t)) − H(Γ, α(s, t))]

dt (14)

where r(s,t) and f (s+t) take values -1 and 1 and indicate the sign of change of the angleα(s, t) = 6 (~Γ−, ~Γ+) with respect to along-the-normal perturbation of the point Γ(s)

and Γ(s + t) respectively, ~Γ+ = ~Γ(s, s + t); ~Γ− = ~Γ(s, s − t); a = |~Γ+ − ~Γ−|; b =

|Γ−|; c = |Γ+|; β = 6 (−~Γ+, ~Γ− − ~Γ+); γ = 6 (−~Γ−, ~Γ+ − ~Γ−).Finally, for feature class #3, relating the curve Γ to another object Ω, the gradient

flow is given by:

∇E(Γ)(s) = ~n(s) · ~∇DΩ(s)

[

H∗

(

DΩ(s)

R(Ω)

)

− H

(

Γ,DΩ(s)

R(Ω)

)]

(15)

where DΩ(s) is value of signed distance function generated by curve Ω at the pointon the curve Γ given by X(s), Y (s), and R(Ω) is the mean radius of the shape Ωrelative to its center of mass.


23

41

5

6

7

8

lenticularnucleus

caudatenucleus

lenticularnucleus

caudatenucleus

(a) (b)

(c) (d)

Figure 4: Brain MRI segmentation: (a) Multiple structures and interactions usedfor feature class #3; (b) Segmentation with independent object curve length prior.(c) Segmentation using multiobject PCA technique in [21] (d) Segmentation withnew multiobject shape distribution prior. Solid black line shows the true objectsboundaries; solid white line - final segmentation boundary.

4 Results

In this section we apply our shape distribution based prior to both a synthetic anda real example. The real data example arises in segmentation of brain MRI. Wecompare both single object and multi-object priors. The benefit of using a multi-object prior is expected to be greater when the object boundary is not well supportedby the observed image intensity gradient or when initialization is far from the trueboundary.

In the first experiment we apply our prior to a synthetic 2-object segmentationproblem with very low SNR, simulating two closely positioned organs, shown in Fig-ure 3, panel (a). Both objects in the ground truth image have the same knownintensity. The background intensity is also known as in the model [2]. Gaussian IID


noise (SNR= -18dB) was added to this bimodal image to form the noisy observedimage. The data term Edata in (2) and the corresponding data-term gradient curveflow are formed according the data model in [2].

In Figure 3 we show the results obtained by segmenting this image using energyminimizing curve evolution based on two different shape priors: (b) shows the resultswith a curve length penalty; (c) shows the results with our multiobject shape dis-tribution prior including all 3 feature classes. The prior target distributions for case(c) were constructed using the true objects in (a). The regularization parameter waschosen in each case to yield the subjectively best result. The curve length prior resultin (b) yields an incorrect segmentation for one of the objects or leads to a collapseof one of the contours. With the directed feature class included in the segmenta-tion functional (c), both objects can be correctly segmented since the energy termcorresponding to feature class #3 effectively prevents intersection of boundaries.

In our second example we apply our techniques to 2D MRI brain data segmen-tation. A data set consisting of 12 normal adult subjects was used. Manual expertsegmentations of the subjects was provided and those of 11 of these subjects was usedas training data to construct our shape prior. The prior was then applied to segmentthe data of the omitted subject. The eight numbered structures shown in Figure 4,panel (a) were simultaneously segmented. For the data dependent energy term Edata,we used the information theoretic approach of [12] by maximizing the mutual infor-mation between image pixel intensities and region labels (inside or outside), thereforefavoring segmented regions with intensity distributions that were different from thebackground intensity distribution.

In Figure 4 we present our results. Panel (b) gives the segmentation with astandard curve length prior applied independently to each object. One can see thatStructures 1 and 4 are poorly segmented, due to their weak image boundaries. Inpanel (c) we present the result given by the multi-shape PCA technique in [21] using5 principle components defining the subspace of allowable shapes. The segmentationis sought as the shape in this subspace, optimizing the same information-theoreticcriteria [12] as used with our shape prior. The usage of the same data term simplifiesthe comparison with our approach since only the shape model components of themethod are different. One can see that structures 2,5,6, and 7 are not segmentedproperly due to the poor generalization by the PCA prior. Expanding the subspaceby choosing 10 PCA components did not improve the result given by this method.Finally, our result is shown in panel (d). We obtain satisfactory segmentation for thestructures for which PCA method failed (2,5,6,7), while performing equally well forstructures 1,3,4 and 8. The choice of initialization did not significantly influence ourresults.

5 Conclusions

In this paper we present a shape distribution-based object prior for use in curve evo-lution image segmentation. This prior allows encoding of multi-object information.


We apply a variational approach to analytically compute the energy minimizing curveflows for three feature classes. We investigate the application of our shape distribu-tion prior to medical image segmentation involving multiple object boundaries. Inour experiments we achieved the performance superior to that obtained using thetraditional curve length minimization methods and a multi-shape PCA shape priorreported in the literature.

A Appendix. Analytical computation of the min-

imizing flows for distribution difference priors

Definitions:s, s1, s2, p ∈ [0, 1] - curve arc length parameterization variablesλ - variable spanning the range of the feature values for a given feature classΓ - curve~Γ(s) = x(s), y(s) - curve coordinates~Γ(s1, s2) = ~Γ(s1)−~Γ(s2) - point-to-point vector with coordinates x(s1)−x(s2), y(s1)−y(s2)β(s) ∈ R1 - continuously differentiable function defining the magnitude of the per-turbation at point s along the local normal~n(s) - unit length normal vector as point s~β(s) = β(s)~n(s) - deformation vector at point s

H(Γ, λ) - cumulative distribution function defined on curve ΓH∗(λ) - prior cumulative distribution functionh(x) = h(x > 0) - indicator functionG(f) - Gateaux derivative

The following derivations are similar to those made in [3]. We refer the reader to[3] and references therein for more details and functional analysis background. Recallthat our shape distribution difference prior term for one feature class is

E =∫ [

H∗(λ) − H(Γ, λ)]2

dλ (16)

Gradient curve flow minimizing the energy in eq. 16 can be computed usingvariational approach as follows

1. Computing Gateaux semi derivative of the energy in eq. 16 with respect toperturbation β. Using the definition of the Gateaux semi derivative, linearityof the integration and the chain rule we obtain

G(E, β) =∫

G[

(H∗(λ) − H(Γ, λ))2]

dλ =

2∫ [

H∗(λ) − H(Γ, λ)]

G[

H(Γ, λ), β]

dλ (17)


2. If Gateaux semi derivative of a linear functional f exists, due to Rietz repre-sentation theorem, it can be represented as

G(f, β) =< ∇f, β > (18)

were ∇f is the gradient (flow) minimizing the functional f .

3. The flow minimizing the energy in eq. 16 is then given by

∇E = 2∫ [

H∗(λ) − H(Γ, λ)]

∇(H(Γ, λ))dλ (19)

Below we perform the above three steps for feature claases defined in this paper.For each of the feature classes we obtain the expression for the corresponding gradientcurve flow.

A.1 Feature class #1. Collection of inter-point distances

More definitions:d(~Γ(s1), ~Γ(s2)) - Euclidean distance between points s1 and s2 on the curve

For this feature class, the cumulative distribution function for a curve Γ is definedas:

H(Γ, λ) =

1∫

0

1∫

0

h(

d(~Γ(s1), ~Γ(s2)) < λ)

ds1ds2 (20)

Now we apply the perturbation εβ(s) to the curve Γ, yielding deformed curve

~Γ′(s) = ~Γ(s) + εβ(s)~n(s) (21)

It is easy to see that for small ε the distance between 2 points on the perturbedcurve is

d(~Γ′(s1), ~Γ′(s2)) = d(~Γ(s1), ~Γ(s2)) + ε [β(s1)~n(s1) − β(s2)~n(s2)] ·~Γ(s1, s2)

|~Γ(s1, s2)|(22)

It means that distance between 2 points on the curve is augmented by a projectionof difference between deformation vectors on the vector ~Γ(s1, s2).

We can now write the Gateaux derivative G [H(Γ, λ), β] according to its definition

G [H(Γ, λ), β] = limε→0

H(Γ′, λ) − H(Γ, λ)

ε=

limε→0

1∫

0

1∫

0

[

h(

d(~Γ′(s1), ~Γ′(s2)) < λ)

− h(

d(~Γ(s1), ~Γ(s2)) < λ)]

ds1ds2

ε=

limε→0

1∫

0

1∫

0

[

h(

d(~Γ(s1), ~Γ(s2)) + ε [β(s1)~n(s1) − β(s2)~n(s2)] ·~Γ(s1,s2)

|~Γ(s1,s2)|< λ

)

−

ε

− h(

d(~Γ(s1), ~Γ(s2)) < λ)]

ds1ds2

1(23)


We introduce the simplified notation:d = d(~Γ(s1), ~Γ(s2))dβ = [β(s1)~n(s1) − β(s2)~n(s2)]

dΓ =~Γ(s1,s2)

|~Γ(s1,s2)|

nx(s), ny(s) = ~n(s) - components of the normal vector at s

Gateaux derivative can now be written as

G [H(Γ, λ), β] = limε→0

−1∫

0

1∫

0[h (d − λ + εdβ · dΓ) − h (d − λ)] ds1ds2

ε(24)

In order to find the limit, we use smooth approximation of the indicator function:

h(x) −→ φα(x) =atan

(xα

)

+ 1

2(25)

where the parameter α defines the degree of approximation. In the limiting case ofα = 0 the approximation becomes an equality. We perform the derivation using theapproximation and at the last step consider the limiting case of α = 0.

Gateaux derivative becomes:

G [H(Γ, λ), β] = limε→0

−12

1∫

0

1∫

0atan

(λ−dα

)

+ atan(

εdβ·dΓ+d−λα

)

ε(26)

For small ε, using the Taylor expansion we obtain

atan

(

εdβ · dΓ + d − λ

α

)

= atan

(

d − λ

α

)

+ atan′

(

d − λ

α

)(

εdβ · dΓ

α

)

+ O(ε2) (27)

Now we can find the Gateaux derivative as follows

G [H(Γ, λ), β] = limε→0

−12

1∫

0

1∫

0atan′

(d−λα

) (εdβ·dΓ

α

)

ε=

− 1

2α

1∫

0

1∫

0

atan′

(

d − λ

α

)

(dβ · dΓ) ds1ds2 = − 1

2α

1∫

0

1∫

0

ds1ds2

1 +(

d−λα

)2 (dβ · dΓ) =

− 1

2α

1∫

0

1∫

0

ds1ds2

1 +(

d−λα

)2 [(nx(s1)β(s1) − nx(s2)β(s2))(x(s1) − x(s2))+

+(ny(s1)β(s1) − ny(s2)β(s2))(y(s1) − y(s2))] =

− 1

2α

1∫

0

1∫

0

nx(s1)(x(s1) − x(s2)) + ny(s1)(y(s1) − y(s2))

1 +(

d−λα

)2 ds2

β(s1)ds1 +

1

2α

1∫

0

1∫

0

nx(s2)(x(s1) − x(s2)) + ny(s2)(y(s1) − y(s2))

1 +(

d−λα

)2 ds1

β(s2)ds2 (28)


Changing the order of integration in the second integral we obtain

G [H(Γ, λ), β] =

− 1

α

1∫

0

1∫

0

nx(s1)(x(s1) − x(s2)) + ny(s1)(y(s1) − y(s2))

1 +(

d−λα

)2 ds2

β(s1)ds1 (29)

According to eq. (18), the expression in square brackets is the gradient

∇H(Γ)(s) = − 1

α

1∫

0

nx(s)(x(s) − x(t)) + ny(s)(y(s) − y(t))

1 +(

d−λα

)2 dt =

− 1

α

1∫

0

n(s) · dΓ(s, t)

1 +(

d−λα

)2 dt (30)

Gradient flow minimizing energy in eq. (16) is therefore

∇E(Γ)(s) = −2∫

λdλ [H∗(λ) − H(Γ, λ)]

1

α

1∫

0

n(s) · dΓ(s, t)

1 +(

d−λα

)2 dt

=

−2∫

λdλ [H∗(λ) − H(Γ, λ)]

1∫

0

αn(s) · dΓ(s, t)

α2 + (d − λ)2 dt

(31)

For α ≈ 0, the expression inside the integral is non zero only when d = λ.Changing the order of integration we obtain:

∇E(Γ)(s) = −2

1∫

0

~n(s) ·~Γ(s, t)

|~Γ(s, t)|dt∫

λ[H∗(λ) − H(Γ, λ)]

α

α2 + (d − λ)2dλ =

−2

1∫

0

~n(s) ·~Γ(s, t)

|~Γ(s, t)|[

H∗(|~Γ(s, t)|) − H(Γ, |~Γ(s, t)|)]

dt (32)

The obtained expression has an intuitive interpretation. The flow at each points on the curve is computed as the integral along the curve. For each point t on thecurve the expression under the integral is the projection of the normal vector at sonto the vector pointing from s to t, modulated by the difference between currentand model distributions evaluated at the distance between s and t. This projectionis intuitively the projection of the “force” acting on the feature (link between s andt). The computational complexity of the flow computation is O(N 2) were N is thenumber of nodes of the discretized curve.


A.2 Feature class #2. Multiscale “curvatures”

A.2.1 Computation of feature values.

We recall that the feature values are the “support” angles α, see figure 5. Thecumulative distribution function is computed on the set of angles measured for allpossible combinations of “base” point s1 and symmetrical “side” points s1 − s2 ands1 + s2 as

H(Γ, λ) =

1∫

0

1∫

0

h(α(s1 − s2, s1, s1 + s2) < λ)ds1ds2 (33)

We choose to define the “inner” angle α as shown on figure 5 as the angle be-tween vectors ~dΓ(s1, s1 − s2) and ~dΓ(s1, s1 + s2) measured always in the same half-

space, separated by the difference vector between ray directions ~dΓ(s1− s2, s1 + s2) =~Γ(s1,s1−s2)

|~Γ(s1,s1−s2)|− ~Γ(s1,s1+s2)

|~Γ(s1,s1+s2)|. This half-space is fixed to be the inside of the curve for s2 = 0.

By other words, the angle α must be a continuous function of s2, and α(s2=0) = π.

s_1

s_1−s_2

s_1+s_2

n(s_1)

α

G(s_1+s_2)

G(s_1−s_2)

n(s_1−s_2)

n(s_1+s_2)

b

c

a

γ

β

Figure 5: Illustration of feature value computation for feature class #2

Figure 6 illustrates 4 cases of relative locations of points on the curve. It canbe seen that unambiguous determination of the angle from rays ~Γ(s1, s1 − s2) and~Γ(s1, s1 + s2) is impossible. We therefore determine the angle α sequentially for s2

increasing from 0 to 1.For each “base” point s1 the angle α is computed sequentially for s2 increasing

from 0 to 1. This process is illustrated in Figure 7. For each base point s1, we startfrom s2 = 0; r(s1, s2) = sign(κ(s1)); α(s1, s2) = π. Flag function r(s1, s2) for s2 > 0 isdefined as

r(s1, s2) =

1 if α(s1, s2) <= π−1 otherwise

(34)


Figure 6: 4 cases of relative location of points.

1

2

3

4

1

2

3

4

r=1r=1

r=1

r=−1

Figure 7: Sequential computation of the angles for a particular “base” point s1,starting from r = 1 (assuming inside of the curve is up-wards).

We define a mean direction vector as

~Γm(s1 − s2, s1 + s2) =~Γ(s1, s1 − s2)

|~Γ(s1, s1 − s2)|+

~Γ(s1, s1 + s2)

|~Γ(s1, s1 + s2)|(35)

In the process of computing α(s1, s2), as s2 increases by ds, we capture the change oforientation of the mean direction vector by looking at the scalar product

C = ~Γm(s1 − s2, s1 + s2) · ~Γm(s1 − s2 − ds, s1 + s2 + ds) (36)

If C becomes less then zero for some s2 = l we change the sign of r(s1, l) for consecutive

values of s2 > l. Angle between rays 6 (~Γ(s1, s1−s2), ~Γ(s1, s1+s2)) ∈ [0, π] is measuredas inverse cosine. After sequential computation is performed for all s1 we end upwith 2D functions 6 (s1, s2) and r(s1, s2). The latest is used to correct the values of6 (~Γ(s1, s1 − s2), ~Γ(s1, s1 + s2)) that must be greater then π. Using these definitions,angles α(s1, s2) are computed as

α(s1, s2) = (2π)1 − r(s1, s2)

2+ r(s1, s2)acos

~dΓ(s1, s1 − s2)

| ~dΓ(s1, s1 − s2)|·

~dΓ(s1, s1 + s2)

| ~dΓ(s1, s1 + s2)|

(37)


A.2.2 Flow computation

In order to find the energy (eq. (16)) minimizing flow we perform essentially the sameprocedure as for feature class #1. We refer the reader to the previous section foromitted details.

First, we perturb the curve in the normal direction by εβ(s). It is important thatall 3 points (“base” point and 2 “side” points) on the curve change their positions.Let α′ be the angle after the perturbation. Using the continuous approximation ofthe indicator function, we obtain

G(H(λ), β) = limε→0

∫ ∫

ds1ds2 [h((α′) < λ) − h((α) < λ)]

ε=

limε→0

− ∫ ∫ ds1ds2 [φγ(α′ − λ) − φγ(α − λ)]

ε=

limε→0

−12

∫ ∫

ds1ds2

[

atan(α′−λγ

) − atan(α−λγ

)]

ε=

limε→0

−12

∫ ∫

ds1ds2atan′(α−λ

γ)(α′ − α)

ε= lim

ε→0

−12

∫ ∫

ds1ds2γ

γ2+(α−λ)2(α′ − α)

ε(38)

We must therefore compute the angle increment α′ − α resulting from the per-turbation. This increment consists of 3 terms (additive due to the assumption of thesmall perturbation):

α′ − α = dα(1) + dα(2) + dα(3) (39)

where the first 2 terms result from the displacement of “side” points and the thirdterm results from the displacement of the “base” point Γ(s1).

We first consider dα(1) (dα(2) is determined similarly). The local geometry of thecurve perturbation at point s1 +s2 is shown in detail in figure 8. It is easy to see that

n

dG

dG’

p

d_alpha

eps*beta(s)

θ

Figure 8: Local perturbation of the curve at point ~Γ(s1+s2). Perturbation εβ(s1+s2)

is infinitely small comparing to |~Γ(s1, s1 + s2)|

the increment dα(1) can be found as followsp

εβ(s)= sin(Θ)

p = εβ(s)√

1 − cos2 Θ

dα(1) =p

|dΓ| =εβ(s)

√

1 −(

~n(s) · dΓ|dΓ|

)2

|dΓ| (40)


G_m

1

2

s_1

L

P

P

Figure 9: Illustration of 2 cases when the sign of the angle increment dα(1) is differentfor the same curve perturbation εβ(s).

It is important to recognize that the sign of the above angle increment dependson the relative direction of the normal ~n(s1 + s2) and ~Γm(s1, s1 + s2). Two possiblecases are illustrated in figure 9. In case 1, for positive β(s1 + s2), the increment ispositive and for case 2, it is negative. We define a flag function f(s1 + s2) as follows

f(s1 + s2) =

−1 if points L and P are on the same side of Γ(s1, s1 + s2)1 otherwise

(41)Under this definition,

dα(1) =εβ(s1 + s2)

√

1 −(

~n(s1 + s2) ·~Γ(s1+s2)

|~Γ(s1+s2)|

)2

|~Γ(s1 + s2)|f(s1 + s2) (42)

dα(2) =εβ(s1 − s2)

√

1 −(

~n(s1 − s2) ·~Γ(s1−s2)

|~Γ(s1−s2)|

)2

|~Γ(s1 − s2)|f(s1 − s2) (43)

We now proceed to calculate dα(3). Let us assume that angles α′ and α arecomputed as inverse cosines. In such case, for α > π, the sign of the angle incrementmust be changed as follows

dα(3) = (α′ − α)r(s1, s2) (44)

were α′ is the angle resulting after displacing point s1. Using cosine theorem andabbreviations from figure 5, we can write the angle increment due to the displacementof point s1 as

α′ − α = acos

(

a2 − b′2 − c′2

−2b′c′

)

− acos

(

a2 − b2 − c2

−2bc

)

=


acos

a2 −(

b − n(s1) · Γ−|Γ−|

β(s1)ε)2 −

(

c − n(s1) · Γ+|Γ+|

β(s1)ε)2

−2(

b − n(s1) · Γ−|Γ−|

β(s1)ε) (

c − n(s1) · Γ+|Γ+|

β(s1)ε)

−

−acos

(

a2 − b2 − c2

−2bc

)

=

acos

a2 − b2 − c2 + 2β(s1)ε

(

n(s1) · Γ−|Γ−|

b + n(s1) · Γ+|Γ+|

c)

−2bc + 2β(s1)ε(

n(s1) · Γ−|Γ−|

c + n(s1) · Γ+|Γ+|

b)

−

−acos

(

a2 − b2 − c2

−2bc

)

(45)

were ()′ indicates the quantity after point s1 displacement; Γ+ = ~Γ(s1, s1 + s2) and

Γ− = ~Γ(s1, s1 − s2). Using Tailor expansion

m + ε1

n + ε2

=m

n+

ε1

n− ε2m

n2(46)

we obtain

α′ − α = acos

a2 − b2 − c2

−2bc+

β(s1)ε~n(s1) ·(

Γ−|Γ−|

b + Γ+|Γ+|

c)

−bc−

−β(s1)ε~n(s1) ·

(Γ−|Γ−|

c + Γ+|Γ+|

b)

(a2 − b2 − c2)

2b2c2

− acos

(

a2 − b2 − c2

−2bc

)

(47)

Using Tailor expansion of the cosine function we obtain

α′ − α =

acos′(

a2 − b2 − c2

−2bc

)

εβ(s1) ×

~n(s1) ·(

− Γ−c|Γ − | −

Γ+

b|Γ + | −a2 − b2 − c2

2b2c2

(

Γ−|Γ − |c +

Γ+

|Γ + |b))

=

− 1√

cos(α)

εβ(s1)~n(s1)

2bc·(

Γ+

|Γ + |a2 + c2 − b2

c+

Γ−|Γ − |

a2 + b2 − c2

b

)

=

− 1

sin(α)

εβ(s1)a

bc[cos(β) cos(n(s1), Γ+) + cos(γ) cos(n(s1), Γ−)]︸︷︷︸

K

(48)

We finally obtained the expression for dα(3). We only need to resolve the uncer-tainty arising when α ≈ π. In such case, we assume α = π − δ, β = O(δ), γ = O(δ).Using Tailor expansion we can write K as

K =

(

1 − 12β2)

cos(α − 6 (n(s1), Γ−)) +(

1 − 12γ2)

cos( 6 (n(s1), Γ−))

δ=


− cos(δ + 6 (n(s1), Γ−)) + cos( 6 (n(s1), Γ−))

δ=

−cos( 6 (n(s1), Γ−)) − sin( 6 (n(s1), Γ−))δ − cos( 6 (n(s1), Γ−))

δ=

sin(~n(s1), ~Γ(s1, s1 − s2)) (49)

Finally,

dα(3) =

−εβ(s1)a

bc

cos(β) cos(n(s1),Γ+)+cos(γ) cos(n(s1),Γ−)

sin αwhen α 6= π

sin(~n(s1), ~Γ(s1, s1 − s2)) otherwise

r(s1, s2) (50)

Combining eq. (38, 39), the Gateaux derivative can be written as

G(H(λ), β) =

−1∫

0

1∫

0

ds1ds2γ

γ2 + (α − λ)2

β(s1 + s2)

√

1 −(

~n(s1 + s2) ·~Γ(s1+s2)

|~Γ(s1+s2)|

)2

|~Γ(s1 + s2)|f(s1 + s2)+

β(s1 − s2)

√

1 −(

~n(s1 − s2) ·~Γ(s1−s2)

|~Γ(s1−s2)|

)2

|~Γ(s1 − s2)|f(s1 − s2) −

−β(s1)a

bc


sin αwhen α 6= π


r(s1, s2)

]

(51)

Using Ritz representation theorem and changing variables of integration we obtainthe gradient flow minimizing H(Γ, λ)

∇H(Γ, λ)(s) =1∫

0

γdt

γ2 + (α(s) − λ)2

a

bc×


sin αwhen α 6= π


r(s1, s2) +

1∫

0

−γdt

γ2 + (α(s − t) − λ)2

√

1 −(

~n(s − t) · ~Γ(s−t)

|~Γ(s−t)|

)2

|~Γ(s − t)|f(s − t) +

1∫

0

−γdt

γ2 + (α(s + t) − λ)2

√

1 −(

~n(s + t) · ~Γ(s+t)

|~Γ(s+t)|

)2

|~Γ(s + t)|f(s + t) (52)

Recall that the gradient of the energy is

∇E(Γ)(s) =∫

dλ[H∗(λ) − H(Γ, λ)]∇H(Γ, λ)(s) (53)


Using eq. (52), changing the order of integration and considering the limiting case ofsmall γ, we obtain

∇E(Γ)(s) =

∫

t∈S

cos(β) cos(n(s),Γ+)+cos(γ) cos(n(s),Γ−)

sin αif α 6= π

sin(~n(s), ~Γ−)) otherwise

× a · r(s,t)

bc−

f (s−t)

√

1 −(

~n(s − t) · ~Γ−

|~Γ−|

)2

|~Γ−|−f (s+t)

√

1 −(

~n(s + t) · ~Γ+

|~Γ+|

)2

|~Γ+|

×

[

H∗(α(s, t)) − H(Γ, α(s, t))]

dt (54)

where r(s,t) and f (s+t) take values -1 and 1 and indicate the sign of change of the angleα(s, t) = 6 (~Γ−, ~Γ+) with respect to along-the-normal perturbation of the point Γ(s)

and Γ(s + t) respectively, ~Γ+ = ~Γ(s, s + t); ~Γ− = ~Γ(s, s − t); a = |~Γ+ − ~Γ−|; b =

|Γ−|; c = |Γ+|; β = 6 (−~Γ+, ~Γ− − ~Γ+); γ = 6 (−~Γ−, ~Γ+ − ~Γ−).We finally compute the closed form solution for the flow minimizing the energy

corresponding to feature class #2. The computational complexity for the discretecurve with N nodes is O(N 2).

A.3 Feature class #3. Relative inter-object distances

This feature class relates 2 shapes. Here we will derive the curve flow for the curve Γ,induced by distribution of distances between curves Γ and Ω. We call Ω a generatingcurve and Γ a target curve. We assume that the signed distance to curve Ω is knownat every point in the image plane. That is done by precomputing the signed distancefunction values DΩ(x, y) on the grid and using linear interpolation to obtain the values

at non integer locations. We compute the gradient of ∇ ~DΩ(x, y) from grid values

using central rule of differentiation and linear interpolation to compute ∇ ~DΩ(x, y)at arbitrary locations. We further normalize DΩ(x, y) by R(Ω) - the mean radius ofshape Ω with respect to its center of mass, so that

DΩ(x, y)′ =DΩ(x, y)

R(Ω)(55)

For this feature class, the set of feature values consists of values of DΩ(x, y)′

sampled along the curve Γ. The distribution function is given by

H(Γ, λ) =

1∫

0

h(

DΩ(~Γ(s))′ < λ)

ds (56)


We can now write the Gateaux derivative G [H(Γ, λ), β]

G [H(Γ, λ), β] = limε→0

H(Γ′, λ) − H(Γ, λ)

ε=

limε→0

−1∫

0h(

DΩ(~Γ(s) + ε~n(s)β(s))′ − λ)

ds +1∫

0h(

DΩ(~Γ(s))′ − λ)

ds

ε=

limε→0

−1∫

0

[

h(

DΩ(~Γ(s)) + εβ(s)~n(s) · ~∇DΩ(~Γ(s))′ − λ)

− h(

DΩ(~Γ(s))′ − λ)]

ds

ε(57)

Using the smooth approximation of the indicator function

G [H(Γ, λ), β] =

limε→0

−1∫

0ds[

φγ(DΩ(~Γ(s)) + εβ(s)~n(s) · ~∇DΩ(~Γ(s))′ − λ) − φγ(DΩ(~Γ(s))′ − λ)]

ε=

limε→0

−12

1∫

0ds[

atan′

(

DΩ(~Γ(s))′−λ

γ

)

εβ(s)~n(s) · ~∇DΩ(~Γ(s))′]

ε=

−1

2

1∫

0

ds

[

γ

γ2 + (DΩ(~Γ(s))′ − λ)2β(s)~n(s) · ~∇DΩ(~Γ(s))′

]

(58)

Using Ritz representation we obtain

∇H(Γ, λ)(s) = −1

2

γ

γ2 + (DΩ(~Γ(s))′ − λ)2~n(s) · ~∇DΩ(~Γ(s))′ (59)

Finally the gradient flow minimizing the energy is

∇E(Γ)(s) =

− limγ−→0

∫

λdλ [H∗(λ) − H(Γ, λ)]

γ

γ2 + (DΩ(~Γ(s))′ − λ)2~n(s) · ~∇DΩ(~Γ(s))′ =

−~n(s) · ~∇DΩ(s)

R(Ω)

[

H∗

(

DΩ(s)

R(Ω)

)

− H

(

Γ,DΩ(s)

R(Ω)

)]

(60)

where DΩ(s) is value of signed distance function generated by curve Ω at the pointon the curve Γ given by X(s), Y (s), and R(Ω) is the mean radius of the shape Ωrelative to its center of mass.

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