Genetic Algorithm Driven Statistically Deformed Modelsfor Medical Image Segmentation

Chris McIntoshcmcintos@sfu.ca

Ghassan Hamarnehhamarneh@sfu.ca

Medical Image Analysis LabSchool of Computing Science

Simon Fraser University, Burnaby, BC

ABSTRACTWe present a novel evolutionary computing based approachto medical image segmentation. Our method complementsthe image-pixel integration power of deformable shape mod-els with the high-level control mechanisms of genetic al-gorithms (GA). Specifically, the GAs alleviate typical de-formable model weaknesses pertaining to model initializa-tion, deformation parameter selection, and energy functionallocal minima through the simultaneous evolution of a largenumber of models. Furthermore, we constrain the evolu-tion, and thus reduce the size of the search-space, by usingstatistically-based deformable models whose deformationsare intuitive (stretch, bulge, bend) and driven in-terms ofprincipal modes of variation of a learned mean shape. Wedemonstrate our work through the application to segmen-tation of the corpus callosum (CC) in mid-sagittal brainmagnetic resonance images (MRI).

Categories and Subject DescriptorsI.4.6 [Image Processing and Computer Vision]: Seg-mentation

General TermsAlgorithms

KeywordsGenetic algorithms, deformable models, segmentation, med-ical imaging

1. INTRODUCTIONMedical image segmentation remains a daunting task, but

one whose solution will allow for the automatic extraction ofimportant structures, organs and diagnostic features, withapplications to computer-aided diagnosis, statistical shapeanalysis, and medical image visualization. Several classifica-tions of segmentation techniques exist including edge, pixel,

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and region-based techniques, in addition to clustering, graphtheoretic, and model correlation-based approaches [32, 24,28]. However, the unreliability of traditional purely pixelbased methods in the face of shape variation and noise hascaused recent trends to focus on incorporating prior-knowledgeabout the location, intensity, and shape of the target anatomy[13]. One method that has been of particular interest tomeeting these requirements is deformable models due totheir inherent smoothness properties and ability to fit ill-defined boundaries, compared to say region-growing approaches.

Deformable models for medical image segmentation gainedpopularity since the 1987 introduction of snakes by Ter-zopoulos et al [29, 16]. In addition to physics-based ex-plicit deformable models [20, 21], geometry-based implicitimplementations also attracted attention [22, 3, 26]. Sev-eral techniques were proposed to improve segmentation re-sults by controlling model deformations [4, 31]. However,with only smoothness and image-based constraints on theirdeformations these models were highly susceptible to signif-icant gradient noise, and local minima (Figure 1).

In many applications, prior knowledge about object shapevariability is available or can be obtained by studying atraining set of shape examples. This allows only feasibledeformations to be produced through the incorporation ofshape knowledge [7, 8, 17, 30, 11]. However, these methodsoften used globally calculated statistics which don’t gener-alize to specific locations, and scales on the model. Con-sequently, the deformations are constrained to be global innature and, hence, can not adapt to local variations in shapewhich are often of high interest. Moreover, the deformationsthemselves are un-intuitively defined and as such it is notclear what deforming along a particular mode of global vari-ation will accomplish (whether it will bulge, stretch, or bendthe shape).

However, there are many main issues common to thesedeformable model-based techniques. The first is that themodel still needs to be initialized to some optimal targetarea of an image, with some shape, orientation, and scale.Secondly, the deformations they exhibit are boundary based,non-intuitive, do not follow the geometry of the object, arenot properly spatially constrained, and their extent is notspatially localized (to a location or scale). These prob-lems motivate the use of the hierarchical regional princi-pal component analysis [10] (HRPCA) of a medial basedshape representation, in combination with GA driven defor-mations to obtain: intuitive deformations (via medial-basedshape rep.), that are statistically-based and spatial localized(via HRPCA), with automatic parameter setting and robust

Figure 1: (top) The corpus callosum is the band ofnerve fiber tissue connecting the left and right hemi-spheres of the brain (bottom) Incorrect progress ofa snake (red) segmenting the CC in an MRI image.Leaking occurs due to weak edge strength and in-correct parameter setting. Both of these problemsare addressed by our proposed method.

model initialization (via GA).The fitting of deformable models to image data is typically

performed through the minimization of a particular energyfunctional or some higher-level, for example user driven, con-trol process [13].

Energy-functional minimization can be carried out in avariety of ways. One solution is to perform explicit differ-entiation using the Euler-Lagrange where each new applica-tion requiring a modified energy functional must be accom-panied by one such derivation. Subsequently, the roots ofthe derivative (minima/maxima of the functional) can be lo-cated using Newton’s method, or some modification thereof.However, as the number of variables increases these meth-ods become to expensive to compute and must, therefore,be approximated. One such technique based on a first-orderderivative estimate is gradient descent. Common amongstthese root-finding methods is that as the number of depen-dant variables (shape, location, scale, orientation, etc.) in-creases as does the complexity of the search space, whichoften increases the amount of local minima and requires thecalculation of an increasingly large number of derivatives(Figure 2).

What is, therefore, required is a method that avoids calcu-lating or estimating the energy functionals derivative, whileallowing the exploration of the search space in a manner thatstill converges towards an optimal solution. Thus retainingspeed by avoiding gradient calculations, allowing the explo-ration to be carried out from a variety of initial locations,and enabling it be done in a way that reflects the learnedvariations of shape in terms of bends, bulges and stretches.GA are an example of one such method.

Our proposed method utilizes prior knowledge to produce

F(r)

r

F(r)

r

undesiredboundarywithdark intensity

targetboundarywithdarkest intensity

(a) (b) (c)

Figure 2: Example of single parameter deformablemodel with local minima. The circular deformablemodel’s only parameter is its radius r. The energyfunction F (r) is minimal for the circle with darkestaverage intensity. The input image is shown in (a)with the darkest outmost boundary representing theglobal minima. In (b) traditional deformable modelsare initialized at a single point shown in red whileGA-based statistical deformable models are initial-ized at multiple locations (green) and perform mu-tations on r. In (c) the GA converges to the globalminima (darkest), while the deformable model getsstuck in a less dark local minima.

feasible deformations while also controlling the scale and lo-cation of these deformations. Moreover, through the use ofGAs we solve the initialization problem, improve resistivityto local minima, and allow the optimization of highly cus-tomizable energy functionals while avoiding costly derivativeestimations. Thus allowing us to explore a high-degree of si-multaneous solutions in a highly-multivariate search-space,while decomposing the shape deformations into intuitive andlocalized constituents (bulge, bend, and stretch) that renderthe results more interpretable by clinicians (e.g. how muchbend was needed in a particular part to fit to the new patientanatomy).

Other works have used GAs to drive traditional deformablemodels [1, 2, 18]. However, their lack of statistical con-straints forces the reliance on traditional energy functionalsthat incorporate internal smoothness constraints. In [15]GAs were used with statistically-based Active Shape Mod-els (ASMs) (section 1.2), where the parameter space consistsof possible ranges of values for the pose and shape parame-ters of the model. The objective function to be maximizedreflects the similarity between the gray levels related to theobject in the search stage and those found from training.

In what follows we develop our method for medical imagesegmentation using GAs to drive statistically-based, con-trolled (location/scale) and intuitive (bend, bulge, and stretch),deformations towards optimal solutions of a problem-specificenergy functional. We begin with a summary of how clas-sical deformable models optimize their energy functionals(section 1.1) then describe how statistical shape constraintscan be added to restrict segmentation (section 1.2). Wedescribe how to obtain statistically constrained and intu-itive deformations (section 1.3), resulting in a shape repre-sentation that allows for statistically constrained deforma-tion types at specific locations and scales. In section 1.4we present an overview of GAs. Finally, we detail how we

employ GAs to drive our statistically based deformationsin section 2, present results in section 3, and conclude insection 4.

1.1 Classical Deformable Shape ModelsClassical deformable shape models [29], are represented as

a dynamic 2D parametric contour v (s, t) = (x (s, t) , y (s, t)),where s ∈ [0, 1] traverses the contour, t denotes time, andv is deformed to fit to image data by minimizing an energyterm ξ,

ξ (v (s, t)) = α (v (s, t)) + β (v (s, t)) . (1)

that depends on both the shape of the contour and the imagedata I(x, y) reflected via the internal and external energyterms, α (v (s, t)) and β (v (s, t)), respectively.

The internal energy term is given as

α (v (s, t)) =

10

w1 (s) ∂v (s, t)

∂s2

+ w2 (s) ∂2v (s, t)

∂s22

ds.

(2)Whereas the external energy term is given as

β (v (s, t)) =

10

w3 (s) P (v (s, t)) ds. (3)

The weighting functions w1 and w2 control the tension andflexibility of the contour, respectively, and w3 controls theinfluence of image data. wi’s can depend on s but aretypically set to different constants. For the contour to beattracted to image features, the function P (v (s, t)) is de-signed such that it has minima where the features have max-ima. For example, for the contour to be attracted to highintensity changes (high gradients) we can choose

P (v (s, t)) = P (x (s, t) , y (s, t)) =−‖∇ [Gσ ∗ I (x (s, t) , y (s, t))]‖

(4)

where Gσ ∗I denotes the image convolved with a smoothing(e.g. Gaussian) filter with a parameter σ controlling theextent of the smoothing (e.g. variance of Gaussian).

Traditionally, the dynamic contour v that minimizes theenergy ξ must, according to the calculus of variations [9], sat-isfy the vector-valued partial differential (Euler-Lagrange)equation

µ (s) ∂2v

∂t2+ γ (s) ∂v

∂t− ∂

∂s w1∂v

∂s + ∂2

∂s2 w2∂2

v

∂s2 +w3∇P (v (s, t)) = 0

(5)

where µ (s) and γ (s) are mass and damping densities, re-spectively. Solving for ∂v

∂tyields a first-order iterative op-

timization method. Though other optimizations have beenexplored using simulated physical dynamics [12], or morerecently GAs [1, 2, 18].

1.2 Statistically Constrained DeformationsIn order to constrain the deformations according to some

learned shape variation, a training set of shapes in differentconfigurations must be collected. The training set is typi-cally created by labelling corresponding landmark points ineach shape example. A classical approach is to perform prin-cipal component analysis (PCA) on the boundary points of atraining set to obtain a point distribution model (PDM)[7].

The PDM describes the average positions of the land-marks, the main modes of variation of landmark positions,

and the amount of variation each mode explains. To empha-size how our method differs from the traditional statisticalmethods used with GAs in [15] we summarize the steps in-volved in generating a PDM and the use of ASMs for imagesegmentation.

First to construct the PDM L landmarks are chosen todescribe the training shapes [5]. After global PCA an objectshape is represented by the sum of a mean shape and a linearcombination of, say t, principal components, i.e.,

x = x + Pb (6)

where x is the vector of landmark coordinates, x is themean shape, P is the matrix of principal components, b isa vector of weighting parameters, also called shape parame-ters. x and x are each of length 2L. P is 2L × t and b is avector of length t.

Constraining the entries of b to ± a few standard de-viations along each principle component ensures deformedshapes lie in the Allowable Shape Domain (ASD).

ASMs find the proposed movement of the landmarks of acurrent shape estimate to new and better locations, whichrequires a model of gray level information (image data). Themodel can be obtained by examining the intensity profilesat each landmark and normal to the boundary created bythe landmark and its neighbors. Then the intensity profilesare used to derive a normalized intensity difference (gra-dient, or derivative) profile giving invariance to the offsetsand uniform scaling of the gray levels [6]. With L land-marks representing each shape, N training shapes, and Ntraining images, they derive N profiles for each landmark,one from each image, and calculate the mean profile for eachlandmark using

yj =1

N

Ni=1

yij (7)

where yij is the normalized derivative profile for the jth

landmark in the ith image and yj is the mean normalizedderivative profile for the jth landmark.

Given a new image, the basic idea is to start with an initialestimate, then examine the neighborhood of the landmarksaiming at finding better locations for the landmarks. Theshape and the pose of the current estimate are hence changedto better fit the new locations of the landmarks while pro-ducing in the process a new acceptable or allowable shape.

The pose parameters are first found by aligning the cur-rent estimate to the new proposed shape. The remaininglandmark position modifications generally span 2L-dimensions,whereas the shape variations obtained from the model areonly t-dimensional. A least-squares solution can be used tosolve the following equation for the changes in shape param-eters db (with orthonormal column of P we have PT P = I)

dx = Pdb ⇒ db = (PT P)−1PT dx = PT dx (8)

where dx is a vector containing the remaining landmarkposition modifications, db is a vector of changes in the shapeparameters, and P is the matrix of principal components.

Finally, the shape variations are limited to obtain an ac-ceptable or allowable shape within the ASD by applying theconstraints on the shape parameters. They obtain new es-timates and re-iterate until approximate convergence (whenthe parameter changes are insignificant).

However, a significant drawback of this approach is thatthe result of varying the weight of a single variation modegenerally causes all the landmark positions to change. Inother words, although the original PDM model producesfeasible shape deformations only, a desirable trait, it gen-erally produces global deformations over the entire object.In the next section we detail how HRPCA can be used toobtain localized deformations describing variation of specificanatomical regions, and allowing finer control over the de-formation process.

1.3 Statistically Constrained Localized and In-tuitive Deformations using HRPCA

We use our multi-scale (hierarchical) and multi-location(regional) PCA method introduced in [10] on our publiclyavailable training set of medial shape profiles computed us-ing 51 mid-sagittal CC images provided in citeshenton1992.We will first give an overview of medial shape profiles andthen proceed to describe how HRPCA can be applied.

Medial-axis based 2D shape representations enable suchdeformations by describing the object’s shapes in terms ofan axis positioned in the middle of the object along withthickness values assigned to each point on the axis that im-ply the shape of the boundary. We therefore describe theshapes as a mapping x : R → R

4. The domain of whichis a parameter m that traverses the medial axis. We use asingle primary medial axis. Though secondary medial axesare needed to represent more complex structures. The rangeof the mapping consists of 4 scalar values for each m, calledmedial profiles. These are a length profile L(m), an orien-tation profile R(m), a left (with respect to the medial axis)thickness profile T l(m), and a right thickness profile T r(m),where m = 1, 2, ..., N , N is the number of medial nodes,and nodes 1 and N are the terminal nodes. The lengthprofile represents the distances between consecutive pairs ofmedial nodes, and the orientation profile represents the an-gles between segments connecting consecutive pairs of me-dial nodes. The thickness profiles represent the distancesbetween medial nodes and their corresponding boundarypoints on both sides of the medial axis (Figure 3-bottom).Corresponding boundary points are calculated by computingthe intersection of a line passing through each medial nodein a direction normal to the medial axis, with the boundaryrepresentation of the object. Example medial profiles areshown in Figure 3-top.

These profiles are rotation- and translation-invariant andcapture intuitive measure of shape: length, orientation, andthickness. Altering these profiles produces intuitive, con-trolled deformations: stretching, bending, and bulging, re-spectively.

Here the principal component analysis is a function of thelocation, scale, and type of shape profile (length, orientation,or thickness) (Figure 4). Hence we obtain an average medialsub-profile, the main modes of variation, and the amount ofvariation each mode explains for each location, scale, andshape profile type.

Global PCA becomes a special case of HRPCA by speci-fying loc = 1 and scl = N , where N is the number of shapevariables covering the whole extent of the object. Hence weobtain N modes of variation of length N . Whereas, loc = land scl = s will produce s×1 values, say thickness values forthe Tr profile, and, as such, result in an s×s covariance ma-trix for the s modes of variation of length s. Consequently,

0 50 1000.7

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length profile L(m) orientation profile R(m)

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left thickness profile Tl(m) right thickness profile T

r(m)

fornix

23N-2

1

genuspleniumrostrum

body

NN-1

right boundaryleft boundary fornix

23N-2

1

genuspleniumrostrum

body

NN-1

right boundaryleft boundary

2rx

2lx

rmx

lmx

mx

1x

2x

1mx −

1mx +

( )rT m

( )lT m

( )L m

( )R m

2rx

2lx

rmx

lmx

mx

1x

2x

1mx −

1mx +

( )rT m

( )lT m

( )L m

( )R m

Figure 3: Top: Example medial shape profiles usedto reconstruct the CC. Middle: Anatomically la-beled CC shape reconstruction resulting from themedial profiles. Bottom: Details of outlined recon-struction showing medial profiles shape representa-tion. Medial nodes shown in black, left and rightboundary nodes shown in dark and light gray, re-spectively. xm, xl

m and xrm are the mth medial, left

boundary and right boundary nodes, respectively.L(m), R(m), T l(m) and T r(m) are the length, orienta-tion, left and right thickness profile values, respec-tively (adapted from [10]).

we can now generate a statistically feasible stretch, bend, orbulge deformation at a specific location and scale in termsof the corresponding main modes of variation.

Generally with HRPCA, for a single deformation, loca-tion, and scale specific PCA we obtain the following modelof medial profile variations,

pdls = pdls + Mdlswdls (9)

location

scale

deformation

stre

tch

bend bu

lge

PCA(d,l,s)

1 2 3 4 5 6 7

2

3

4

5

6

7

1

location

scale

deformation

stre

tch

bend bu

lge

PCA(d,l,s)

1 2 3 4 5 6 7

2

3

4

5

6

7

1

Figure 4: Hierarchical regional principal componentanalysis is a function of the deformation (d), location(l), and scale (s) (Adapted from [10]).

where p is the shape profile, d is the deformation profiletype, l and s are the location and scale values, pdls is theaverage medial profile, Mdls describes the main variationmodes, and wdls are weights of the variation modes and aretypically limited to ±3 standard deviations.

Note that for any shape profile type multiple variationmodes can be activated by setting the corresponding weight-ing factors to non-zero values. Since each variation modeacts at a certain location and scale we obtain

pd = pd +

l

s

Mdlswdls. (10)

In summary, varying the weights of one or more of thevariation modes alters the length, orientation, or thicknessprofiles and generates, upon reconstruction, statistically fea-sible stretch, bend, or bulge deformations at specific loca-tions and scales.

1.4 Genetic AlgorithmsGAs are a special form of local search that models our

own understanding of evolution. In essence a number of si-multaneous agents (the population) each having an encodedstate (the chromosome) perform a random walk (mutations)around the search space, while forming new solutions fromcombinations of existing solutions (crossover) and, thus, ad-justing and refocusing the efforts of the search on exception-ally good areas once located. A few important choices aremade during any application of genetic algorithms, involvinghow to encode the population (binary, integer, decimal, etc),how to mutate the population (mutate all genes, some genes,etc), how to select the parents for crossovers (roulette wheel,tournament selection), how to perform those crossovers (uni-form, single-point), and finally what fitness function to usefor evaluation. Though these choices seem complex, in sit-uations where the energy functional has hundreds or eventhousands of dependent variables and parameters these fewchoices can yield nearly-optimal values for all variables andparameters concerned.

2. METHODSIn this work we use GAs to solve the typical initialization,

local minima and parameter sensitivity problems associatedwith traditional energy-minimization techniques. Moreover,the medial shape representation provides an intuitive wayto synthesize/control deformations, while HRPCA enableslocalized statistics thereby localizing the variations and de-formations to specific anatomical regions of a shape (Figure3-middle).

Here we describe our representation of individuals, our en-coding of the model into chromosomes (deformation weights)to be optimized, our method of mutating (deforming) themodel, our selection and crossover methods, and our fitnessfunction (energy functional to minimize).

2.1 Population RepresentationIn medical image segmentation using GAs, the individuals

forming the population represent potential shapes of the tar-get structure, each having some level of accuracy measuredby the fitness function (section 2.6). Consequently, we re-quire a shape representation consistent across models andcapable of intuitive mutations. One straightforward way ofrepresenting shapes is using boundary nodes (section 1.1).However, intuitive aspects of shape variation (such as bend-ing, thickness, and elongation) can not be easily capturedand, therefore, not properly represented in the mutations,selection and crossover phases. We require a shape represen-tations that allows us to describe and control the shape de-formations intuitively and in-terms of our calculated shapestatistics. Consequently, we represent each individual by itsassociated stretching, bending and thickness profiles alongwith its global orientation, base location, and scale (section1.3).

2.2 Encoding the Weights for GAWe use chromosomes to represent a set of all the weights

of the principal components as obtained from the HRPCA,where each gene represents a weight for a particular defor-mation, location, scale and mode of variation (Figure 5-top).

In total there are4

d=1

N

lc=1

N−lc+1

sc=1

sc

w=1

1 weights available for

mutation since for each of the four deformations, d, we haveN = 27 different locations, but for each location, lc, we canonly have up to N − lc + 1 scales, sc, each of which has scweights for the sc principle components. (Figure 4). In ourapplication this adds up to 14616 dependent variables forour model, which motivates the use of GAs to search thehighly-multivariate space.

2.3 MutationsIn order to somewhat guide the solution in the right di-

rection, and, thereby, facilitate faster initial convergence, westart by constraining the mutations to the affine transfor-mation parameters: base-node position (translation values)(tx, ty), model orientation angle θ, and scale values (sx, sy)(Figure 5-top). Since our initial shape is the mean CC (ob-tained by setting all weights to zero) it can be expected toprovide a reasonably strong fitness value when an accept-able position, orientation, and scale are set. In essence, weeliminate the possibility of getting a low score for a goodlocation, scale and orientation, simply because of a bad ran-dom shape mutation.

With an adequate location, scale, and orientation ob-

wd,1,2 wd,1,3 wd,1,27… wd,27,1…

Location 1, all scales Location 27, only scale 1…

tytx sysx

Affine transformation parameters Statistical shape deformation parameters

wd,1,1wd,1,2 wd,1,3 …

Mutation

wd,1,1 wd,1,2 wd,1,3 wd,1,27… wd,27,1…

…

Cross over

wd,1,1θ

wd,27,1wd,2,26 wd,3,1…wd,1,27 wd,2,1…

wd,1,2 wd,1,3 wd,1,27… wd,27,1…

Location 1, all scales Location 27, only scale 1…

tytx sysx

Affine transformation parameters Statistical shape deformation parameters

wd,1,1wd,1,2 wd,1,3 …

Mutation

wd,1,1 wd,1,2 wd,1,3 wd,1,27… wd,27,1…

…

Cross over

wd,1,1θ

wd,27,1wd,2,26 wd,3,1…wd,1,27 wd,2,1…

Figure 5: Segmenting an anatomical structureamounts to finding the optimal set of shape param-eters. In our GA implementation (section 2.1) werepresent each shape as a chromosome with genesencoding affine and statistical shape deformation pa-rameters (top). Mutation (section 2.3) is performedby altering the weights of the HRPCA (middle).Crossover amounts to swapping a set of weights be-tween two individuals (bottom).

tained we allow the mutations to begin including shape de-formations (Figure 5-mid). Dynamic mutation of a singlegene amounts to altering the corresponding weight by sam-pling it from a uniform random variable under the constraintthat the total weight lie within ±2 standard deviations ofthe corresponding mode of variation (square root of the ex-plained variance obtained in HRPCA). Modifying a weightwill change the medial profiles and hence the reconstructedshape boundary (section 2.6.1).

During the initial phases of the evolution each member ofthe population undergoes a random deformation with globalscale, and at random amplitudes set in multiples of the cor-responding standard deviations. Thus resulting in a newshape. The initial constraint to global deformations is well-suited for our statistical deformations as localized deforma-tions (say bulging the splenium in Figure 3-middle) will nothelp until an acceptable global fit is obtained. Consequently,after a particular number of generations has passed, we allowthe deformations to begin varying in both position and scaleto include at first larger deformations (those correspondingto an entire anatomical region, and, hence, a primary areaof variation) then smaller deformations which surmount tosmall variations in local regions.

2.4 SelectionGenetic algorithms require a method of determining which

members of a generation will reproduce, and which will sur-vive. We use roulette wheel selection, where each member iof a population P has a probability of selection equal to

Pselection(i) = Fit(i)/j∈P

Fit(j) (11)

to randomly select members for reproduction, where Fit(i)is the fitness function (Section 2.6). We also employ an “eli-

tist” strategy under-which the best member of the popula-tion is always maintained, and the weakest are replaced bythe new individuals resulting from the crossover operation.

2.5 CrossoverGenetic algorithms use crossover to combine the informa-

tion from two existing “parents” into a single “offspring”,that contains genes from each parent. We used uniformcrossover, which makes an independent random decision foreach gene whereunder both parents have an equal probabil-ity of making the contribution (Figure 5-bottom).

2.6 FitnessOur fitness function is specifically designed for segmen-

tation of the corpus callosum, though as noted in section1 the use of GAs allow us to easily adapt the function toany given task including both prior shape and image-basedknowledge; something traditional deformable models do notallow for. For example, we have adopted the fitness func-tion Fit(i) to consider mean image intensity, edge strength,standard deviation, and anatomical size of the CC.

Fit(i) = αS(i) + β

1 − e −E(i)

χ +φ

1 − e −µ(i)

ϕ + δe −σ(i)ε (12)

where

S(i) =|Ωinternal|

$(13)

E(i) =1

|Ωcontour|

Ωcontour

‖∇I‖ (14)

µ(i) =1

|Ωinternal|

Ωinternal

I (15)

σ(i) = 1

|Ωinternal|

Ωinternal

(I − µ(i))2 (16)

|Ωinternal| is the area enclosed by the reconstructed bound-ary (section 2.6.1), |Ωcontour| is the length of the bound-ary, and $, η are the average size and standard deviationof the CC learned from the training set, respectively. I isthe image, and χ, ε, ϕ are learned edge strength, standarddeviation, and mean intensity. Hence, S(i) represents theshape’s area, E(i) the average gradient magnitude at theshape’s boundary, µ(i) the image intensity enclosed by theshape’s boundary, σ(i) standard deviation. Finally, α, β, φ,and δ are scalar weights controlling the importance of eachterm in the segmentation process. For all experiments inthis paper α = 0.1, β = 0.05, φ = 0.80, and δ = 0.05, andparameter learning is performed using leave-one-out valida-tion.

2.6.1 Shape Reconstruction for Fitness CalculationIn order to evaluate the fitness, the boundary of the CC

shape must be reconstructed from the set of affine parame-ters and medial profiles specified by the shape weights. Toreconstruct the object’s shape given its set of medial pro-files, we calculate the positions of the medial and boundarynodes from a known reference node at location x1 = (tx, ty).The next node at position x2 = x1 + v1 is specified usingan offset vector v whose angle is specified by the orientation

Figure 6: Two example segmentation results pro-gressing left to right, showing fittest individual afterautomatic initialization (left), global deformations(middle), and local deformations (right).

profile plus the base angle θ, and length is specified by thestretch profile scaled by (sx, sy). The corresponding bound-ary nodes xl

2 and xr2 (Figure 3) are then orthogonal to the

medial axis, at a distance specified by the thickness profilescaled by (sx, sy). This process is repeated recursively, gen-erating x3 = x2+v2, and so on. For details see [10]. Finally,with the medial profiles of Figure 3-top, for example, as aninput we reconstruct the corpus callosum (CC) structure inFigure 3-middle.

3. RESULTSWe validate our method through the segmentation of the

corpus callosum, which is the largest white-matter tract inthe human brain. Specifically, it serves as the primary meansof communication between the two cerebral hemispheres andmediates the integration of cortical processes from oppositesides of the brain. The presence of morphological differencesin the corpus callosum in schizophrenics has been the subjectof intense investigation [25]. The corpus callosum may alsobe involved in Alzheimer’s disease [23], mental retardation[19], and other disorders.

Table 1: Error comparison between statistical-baseddeformations with a hand-crafted schedule, and ourGA based models. Error ε = (S∪M − S∩M)/M isused, where S and M denote the area enclosedwithin the result of the automatic segmentation andthe manual expert delineation, respectively.

Error mean median min max std

Hand-crafted

0.1834 0.1706 0.1095 0.4526 0.0576

GA 0.1719 0.1501 0.0732 0.5464 0.0868

We present qualitative as well as quantitative results ofthe fully-automatic segmentation of 46 corpus callosum inmid-sagittal magnetic resonance images [27] using our GAdriven, statistically-constrained deformable models (Figures6, 7). We compare our results to those previously obtainedin [14], where statistically-constrained, physically-based de-formations are controlled by a CC specific hand-crafted sched-ule and initialization method (Table 1). Here our use of GAsenabled us to obtain superior accuracy without dependingon an application-specific schedule, which strongly empha-sizes the extendability of our method.

0 100 200 300 400 5000.85

0.86

0.87

0.88

0.89

0.9

0.91

Generations

Fitn

ess(

best

)

Figure 7: Plot of generation number versus fitnessof the best individual.

4. CONCLUSIONSWe have developed a novel segmentation technique by ad-

dressing the main concerns associated with both traditionaland statistical-based deformable models. Firstly, by usingGAs to solve the initialization, local minima and parame-ter sensitivity problems associated with traditional energy-minimization techniques. Secondly, our medial shape repre-sentation provides a powerful way to synthesize and analyzedeformations thus decomposing deformations into differenttypes that are intuitively controlled and are more easily com-municated to medical experts than boundary based defor-mations. Finally, our use of HRPCA enables localized statis-tics thereby localizing the variations and deformations tospecific anatomical regions that render the results more in-terpretable by clinicians and enable regional statistical anal-ysis.

Furthermore, we have demonstrated how GAs can be com-bined with constrained shape deformations to effectivelyexplore the search space of a complex energy functional,thereby incorporating prior-knowledge into the solution whileretaining multiple simultaneous searches of the space. Inessence, we have constrained the random-walks of the GAto lie within the allowable shape domain thus greatly reduc-ing the search space traditionally associated with deformablemodels.

Our method is also extendible to other segmentation prob-lems. Specifically, given a training set of shapes for a dif-ferent anatomical structure, one can perform skeletoniza-tion followed by medial profile extraction and, subsequently,HRPCA. Further, the components of the fitness functionspresented here can apply to other anatomical structures,with possible minor modifications as the application war-rants them. Nevertheless other terms can easily be addedrelated to texture, colour or other image features. Finally,we are working on extending these ideas to 3D, where thegenes become the weights of 3D shape representation pa-rameters.

Though other works have used GAs to drive deformablemodels [1, 2, 18, 15]. To the best of our knowledge, noworks have combined GAs with statistical-based deforma-tions in a way that yields intuitively constrained deforma-tions, nor have they employed fitness functions as well-suitedto the problem domain. Furthermore, by comparison ourmethod retains speed by avoiding gradient calculations, al-lows search space exploration to be carried out from a vari-ety of initial locations, and enables it be done in a way thatintuitively reflects the learned variations of shape (bends,bulges and stretches).

5. ACKNOWLEDGMENTSWe wish to thank Dr. Martha Shenton of the Harvard

Medical School for providing the MRI data, and Peter Plettfor assisting with code development.

6. REFERENCES[1] L. Ballerini. Genetic snakes for medical image

segmentation. Proc. SPIE, Mathematical Modeling

and Estimation Techniques in Computer Vision, 3457,1998.

[2] L. Ballerini. Genetic snakes for color imagessegmentation. Lecture Notes in Computer Science,2037:268, 2001.

[3] V. Caselles, R. Kimmel, and G. Sapiro. Geodesicactive contours. In IJCV, volume 22(1), pages 61–79,1997.

[4] L. D. Cohen. On active contour models and balloons.CVGIP: Image Underst., 53(2):211–218, 1991.

[5] T. Cootes, C. Taylor, D. Cooper, and J. Graham.Training models of shape from sets of examples.BMVC, pages 9–18, 1992.

[6] T. Cootes, C. Taylor, A. Hill, and J. Halsam. The useof active shape models for locating structures inmedical images. Proceedings of the 13th International

Conference on Information Processing in Medical

Imaging, pages 33–47, 1993.

[7] T. F. Cootes, D. Cooper, C. J. Taylor, andJ. Graham. Active shape models- their training andapplication. In Computer Vision and Image

Understanding, volume 61, pages 38–59, 1995.

[8] T. F. Cootes, G. J. Edwards, and C. J. Taylor. Activeappearance models. In IEEE Transactions on Pattern

Analysis and Machine Intelligence, volume 23(1),pages 681–685, 2001.

[9] L. Elsgolc. Calculus of Variations. Pergamon PressLtd., 1962.

[10] G. Hamarneh, R. Abu-Gharbieh, and T. McInerney.Medial profiles for modeling and statistical analysis ofshape. International Journal of Shape Modeling,10(2):187–209, 2004.

[11] G. Hamarneh and T. Gustavsson. Statisticallyconstrained snake deformations. In IEEE

International Conference on Systems, Man, and

Cybernetics, volume 3, pages 1610–1615, 2000.

[12] G. Hamarneh and T. McInerney. Physics-based shapedeformations for medical image analysis. volume 5014,pages 354–362. SPIE, 2003.

[13] G. Hamarneh, T. McInerney, and D. Terzopoulos.Deformable organisms for automatic medical imageanalysis. In MICCAI, pages 66–76, 2001.

[14] G. Hamarneh and C. McIntosh. Physics-baseddeformable organisms for medical image analysis.Proceedings of SPIE Medical Imaging: Image

Processing, 5747:326–335, 2005.

[15] A. Hill and C. Taylor. Model-based imageinterpretation using genetic algorithms. Image and

Vision Computing, 10(5):295–300, 1992.

[16] M. Kass, A. Witkin, and D. Terzopoulos. Snakes:Active contour models. volume 1(4), pages 321–331,1987.

[17] M. Leventon, E. Grimson, and O. Faugeras. Statistical

shape influence in geodesic active contours. In CVPR,2000.

[18] L. MacEachern and T. Manku. Genetic algorithms foractive contour optimization. IEEE Proceedings of the

International Symposium on Circuits and Systems,4:229–232, 1998.

[19] E. Marszal, E. Jamrow, J. Pilch, and et. al. Agenesisof corpus callosum: Clinical description and etiology.Journal of Child Neurology, 15:401–405, 2000.

[20] T. McInerney and D. Terzopoulos. Deformable modelsin medical image analysis: A survey. In Medical Image

Analysis, volume 1(2), pages 91–108, 1996.

[21] J. Montagnat, H. Delingette, and N. Ayache. A reviewof deformable surfaces: topology, geometry anddeformation. In Image and Vision Computing, volume19(14), pages 1023–1040, 2001.

[22] S. Osher and N. Paragios. Geometric Level Set

Methods in Imaging Vision and Graphics. SpringerVerlag, 2003.

[23] J. Pantel, J. Schroder, M. Jauss, and et. al.Topography of callosal atrophy reflects distribution ofregional cerebral volume reduction in alzheimer’sdisease. Psychiatry Research, 90:180–192, 1999.

[24] R. A. Robb. Biomedical Imaging, Visualization, and

Analysis. Wiley-Liss Inc., 2000.

[25] R. Rosenthal and L. Bigelow. Quantitative brainmeasurements in chronic schizophrenia. British

Journal of Psychiatry, 121:259–64, 1972.

[26] J. A. Sethian. Level set methods; evolving interfacesin geometry, fluid mechanics. In Computer Vision and

Material Sciences. Cambridge University Press, 1996.

[27] M. Shenton, R. Kikinis, F. Jolesz, S. Pollak,M. LeMay, C. Wible, H. Hokama, J. Martin,D. Metcalf, M. Coleman, and et al. Abnormalities inthe left temporal lobe and thought disorder inschizophrenia: A quantitative magnetic resonanceimaging study. In New England J. Medicine, volume327, pages 604–612, 1992.

[28] M. Sonka and J. Fitzpatrick. Handbook of Medical

Imaging, Volume 2: Medical Image Processing and

Analysis. SPIE-International Society for OpticalEngine, 2000.

[29] D. Terzopoulos. On matching deformable models toimages. Topical Meeting on Machine Vision, Technical

Digest Series, 12:160–167, 1987.

[30] S. K. Warfield, M. Kaus, F. A. Jolesz, and R. Kikinis.Adaptive, template moderated, spatially varyingstatistical classification. In Medical Image Analysis,volume 4(1), pages 43–55, 2000.

[31] C. Xu and J. Prince. Snakes, shapes, and gradientvector flow. IEEE Transactions on Image Processing,7(3):359–369, 1998.

[32] T. S. Yoo. Insight into Images: Principles and

Practice for Segmentation, Registration, and Image

Analysis. A k Peters Ltd., suite 230, 888 Worcester St.Wellesey, MA, USA, 2004.