Mapping the Cerebral Sulci: Application to Morphological Analysis of the Cortex and to Non-rigid Registration

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1 Mapping the Cerebral Sulci: Application to Morphological Analysis of the Cortex and to Non-rigid Registration Marc Vaillant and Christos Davatzikos Neuroimaging Laboratory, Department of Radiology, Johns Hopkins University School of Medicine, 600 N. Wolfe Street, Baltimore MD emalh hristos~welchlink.welch.jhu.edu Abstract. We propose a methodology for extracting parametric representations of the cerebral sulci from magnetic resonance images, and we consider its application to two medical imaging problems: quantitative morphological analysis and spatial normalization and registration of brain images. Our methodology is based on deformable models utilizing characteristics of the cortical shape. Specifically, a parametric representation of a sulcus is determined by the motion of an active contour along the medial surface of the corresponding cortical fold. The active contour is initialized along the outer boundary of the brain, and deforms toward the deep edge of a sulcus under the influence of an external force field restricting it to lie along the medial surface of the particular cortical fold. A parametric representation of the surface is obtained as the active contour traverses the sulcus. In this paper we present results of this methodology and its applications. 1 Introduction A great deal of attention has been given during the past several years to the quantitative analysis of the morphology of the human brain. Much progress has been made in the development of methodologies for analyzing the shape of subcortical structures [1, 2, 3]. However, the complexity of the cortical shape has been an obstacle in applying these methods to this highly convoluted structure. In this paper we present a methodology for finding a representation of the shape of the deep cortical folds from MR images, which extends our previously reported work [4]. The region between two juxtaposed sides of a cortical fold, called a sulcus, is a thin convoluted ribbon embedded in 3D. The cerebral sulci are important brain structures, since they are believed to be associated with functionally distinct cortical regions. In particular, during the development of the brain in the embryo, connections between specific cortical regions and connections between cortical regions and subcortical structures, in conjunction with an overall growth process constrained by the skull, are believed to induce the sharp inward folding of the cortex, resulting in the formation of the sulci. The roots of the sulci often demarcate the boundaries between functionally and structurally different

2 142 cortical regions [5]. Most notably, the central sulcus is the boundary between its posteriorly located primary somatosensory cortex and its anteriorly located primary motor cortex. Along both sides of the central sulcus is a fairly consistent somatotopic distribution of motor and somatosensory functional regions of the cortex, known as Penfield's homunculus. In order to study the cerebral sulci and the distribution of the function along their associated cortical folds, we first need to establish a quantitative methodology for describing the shape of these ribbon-like structures. In this paper we present a methodology for finding parametric representations of the sulcal ribbons. Fig. 1. A 2D schematic diagram of the Related to our work is the work in force field acting on the active contour. [5], which studies the cortical topography by determining a graph of the sulci. Although some global characteristics of the sulcal shape, including depth and orientation, were considered in [5], no attempt was made to characterize the local geometric structure of the sulci. The latter is the focus of our work. Also related is the work in [6, 7], which determines the medial axis of arbitrary shapes at multiple scales, the work in [8], which is based on manually outlining the sulci and the work in [9, 10] which focuses on segmentation of the sulci. The method we propose in this paper is a physically-based algorithm utilizing the particular characteristics of the cortical shape. Specifically, our algorithm uses the outer cortical surface as a starting point. Based on the principal curvatures of this surface, we identify the outer edges of the sulcal ribbons. By placing active contours [11, 12] along these edges, which are then let free to move inward following the cortical gyrations, our method obtains a parametric representation of the sulcal ribbons. Two main applications are considered in this paper. The first is the quantitative morphological analysis of the sulci. In particular, we use the parametric representation of the sulcal ribbons to determine the average shape of a sulcus and the inter-individual variability around this average. Inter-subject comparisons can be readily formulated based on this representation. The second application is in the spatial normalization and registration of brain images, in which a key issue is often obtaining a parametric representation of distinct surfaces which can then be matched based on their geometric characteristics [13, 14, 15, 16]. The sulcal ribbons of distinct cortical folds can be used for this purpose in shape transformation methodologies [16, 15].

3 143 2 Extraction of the Sulcal Ribbons 2.1 Overall Idea The overall idea behind our algorithm is based on the particular shape of the sulci. Specifically, since a sulcus resembles a thin ribbon embedded in 3D, it can be extracted from an MR volumetric image as an active contour slides along the sulcal groove in between two juxtaposed sides of the associated cortical fold (see Fig. 1). The collection of the deformed configurations of the active contour readily provides a parameterization of the sulcal ribbon. The initial active contour is along the outer cortical surface; its final configuration is along the root of the sulcus. The inward motion of the active contour is determined by two force fields (see Fig. 1). The first one restricts the motion of the active contour to be along the medial surface of a sulcus. The second one resembles a gravitational force and is responsible for the inward motion of the active contour. 2.2 Initial Placement of the Active Contours As mentioned earlier in the paper, one active contour for each sulcus is initially placed along the outer boundary of the brain. In order to determine this initial configuration of the active contour, we first find a parametric representation of the outer cortical surface using a deformable surface algorithm described elsewhere [17]. A deformable surface, initially having a spherical configuration surrounding the brain, deforms like a contracting elastic membrane attracted by the outer cortical Fig. 2. (a) Outer cortical surface with the initial active contour superimposed. (b) Flattened outer cortical map of the minimum (principal) curvature with the initial active contour superimposed. boundary and eventually adapts to its shape. We determine the outer cortical boundary through a sequence of three operations. A morphological erosion is first used to detach the brain tissue from the nearby dura, skull, and bone marrow. A seeded region growing then extracts the brain tissue. Finally, a conditional morphological dilation recaptures the brain tissue lost in the erosion step. In some cases manual editing of the final result is required, if the erosion step does not fully detach the parenchyma. Resulting from the deformable surface algorithm is a parametric representation of the outer cortical surface, denoted by b(u, v), where (u, v) takes values in a planar domain in which the deformable surface is parameterized. The intersections of this surface with the sulcal ribbons are the outer edges of the sulci,

4 144 and are identified along the outer cortical surface using the minimum (principal) curvature of b(u,v), which has high value along them [17] (see Fig. 2). On these high curvature curves we then initialize active contours having evenly spaced points. The initial configuration of an active contour will be denoted by x(s,0) = x(s,t)lt=0. Here, the parameter t e [0,1] denotes time. As s 9 [0, 1] sweeps the unit interval, x(s, t) runs along the active contour at its configuration at time t. Equivalently, x(s, t) and x(s, t+ At) are two consecutive deformed configurations of the active contour. 2.3 External Force Fields After its initial placement along the outer edge of a sulcus, the active contour slides along the medial surface of the sulcus, in between two opposite sides of a cortical fold, under the influence of its internal elastic forces and two external force fields which are described next. Center-of-Mass Force. The first force field, F1, acting on the active contour restricts its motion to be on the sulcal medial surface. This force is based on the observation that the points lying on the medial surface of a sulcus satisfy the following condition (see Fig. 3): X = c(x), (1) where c(x) is the center of the cortical mass included in a spherical neighborhood, Af(X), centered on X 1. Accordingly, we define the force acting on a point located at X as Ft(X) = c(x) - X p(x) ' (2) where p(x) is the radius of A/'(X) and is spatially varying, as described below. If the cortex had exactly uniform thickness throughout its extent, and if the two juxtaposed sides of the cortical folds were N:... always in contact with each other, then p in (2) would be fixed to a value equal to the cortical thickness. In that case, Ft would be exactly zero on the sulci. However, this is often not the case. In order to account for Fig. 3. Schematic diagram of the variations in the cortical and sulcal thickcross-section of a sulcus and its sur- ness, we allow p(x) to vary throughout the rounding cortex. cortex; at each point X, Af(X) adapts its size to encompass the cortical gray matter in its full width. In particular, at each point X, p(x) is defined as the radius of the smallest spherical neighborhood intersecting the cortical boundaries on both sides of a cortical fold. Now consider an active contour point located at X. If X is exactly on the medial axis of the sulcus then F1 (X) = 0; in this case Ft does not affect this point. Otherwise, Ft moves the point towards the sulcal medial axis (see Fig. 3). After a sequence of incremental movements, the point balances at a position satisfying 1 In the experiments herein, the cortical mass is the indicator function of the cortex, and is obtained at a pre-segmentation step using a fuzzy C-means algorithm, courtesy of Dzung Pham.

5 145 (1) (within some preset tolerance factor) where F1 vanishes. It is important to note that p(.) adapts continuously to the cortical thickness during this restoring motion. At equilibrium, the spherical neighborhood balances between the two opposite boundaries of the cortical fold. Inward Force.The second force field; F2, acting on the active contour is responsible for its inward motion toward the deepest edge of a sulcus. At the active contour's initial configuration x(s, 0), F2 at each contour point is in the direction of the inward normal to the outer cortical surface, which is denoted by FN[X(S, 0)]. As mentioned earlier, the orientation of the sulci tend to deviate from this normal direction, especially deep in the brain. Accordingly, we adapt F2 as the active contour moves inward. If x(s, t) is the position of an active contour point at time t, then xt(s,t), and xtt(s,t) respectively represent speed and acceleration components of the active contour's dynamics. These terms effectively provide a directional estimation of the active contour's motion. We now define F2(s, t) as F2(s, t) = axt + ~xtt + 7FN[X(S, 0)], (3) where subscripts denote partial derivatives. The first two terms in (3) give rise to damping and inertial influences which have the effect of averaging the previously traveled direction of the active contour with the initial inward direction. This actively adapts the inward force to the shape of the sulcus. Under the influence of the two force fields, F1 and F2, the active contour deforms elastically, sliding along the medial surface of a sulcus towards its deepest edge. The inward trajectory of each active contour point is terminated when the magnitude of the total force acting on it becomes less than a threshold ~ E [0, 1]. This typically occurs at the bottom parts of the cortical folds where F1 and F2 have almost opposite directions (see Fig. 1). 2.4 Refinement of the Sulcal Surface The collection of the deformed configurations of the active contour constitute a surface, parameterized by x(s, t). This surface tends to be smooth along one family of isoparametric curves, the one obtained by fixing t, because of the internal elastic forces of the active contour which are along these curves. However, it is not necessarily smooth along the other family of isoparametric curves, the one obtained by fixing s, which are oriented along the direction of the inward motion. In order to obtain a smooth surface, y(s, t), from x(s, t) we solve the following regularization problem: Minimize//" Ily(s, t) - x(s, t)ll2dsdt + K//(llYs (s, t)ll 2 + llyt (s, t)ll 2) dsdt,(4) where subscripts denote partial derivatives. In (4), the surface y(s, t) is an elastic membrane which adapts to the shape of the sulcal surface x(s, t), while maintaining a certain degree of smoothness

6 146 determined by the elasticity parameter K. Equation (4) is discretized and solved iteratively, resulting in a smoother surface parameterized by y(s,t), which is defined in the domain Z) = [0, 1] x [0, 1]. Fig. 4 illustrates the effect of this procedure on an original sulcus before smoothing. The resulting sulcus Fig. 4b is clearly smoother, but maintains the shape of the original sulcus. 2.5 Fixed-Point Algorithm Fig. 4. A sulcus, (a) before, and (b) af- ter the smoothing procedure, The solution of (4) is a smooth parameterization of a sulcal ribbon. Although there is an infinite number of such parameterizations of a sulcus (or of any surface, as a matter of fact), any of them is adequate for visualization purposes, since the shape of a surface does not depend on the way that the surface is parameterized. However, for all the applications considered in the following section, it is at least desirable, if not necessary, to find a Unique parameterization with specific properties. This is especially important in a procedure determining the average shape of a sutcus in which all the sulci whose average form is sought must be parameterized in a consistent way, so that corresponding points are averaged together. For example, Fig. 5 shows two surfaces, corresponding to two hypothetical sulci. The arrows show corresponding points, ie. points with the same (s, t) parametric coordinates. The surfaces in (b) and (c) have identical shape, but clearly the parameterization in (b) is a much more reasonable match to that in (a). One way to obtain a consistent, across subjects, surface parameterization, which is natural in the absence of landmarks along the sulcus, is to find a parameterization x(u, v) that has the following first fundamental form: E:~s, f=o, a:x~ (5) where )~s and At are constants. Intuitively, such a parameterization can be obtained by stretching the parametric domain T~ by a factor of )~s horizontally, a factor At vertically, and folding it in 3D while preserving the angles between its isoparametric curves. If As -- At, then the parameterization is homothetic, i.e. an isometry together with a uniform scaling factor. Equation (5) implies that the surface with this first fundamental form must have isoparametric curves of constant speed (although the speeds of the u-curves and the v-curves differ if As r )~t). Moreover, the isoparametric curves intersect at, right angles. In order to find such a parameterization, we apply a fixed point algorithm described in detail in [17], which iteratively reparameterizes the curves

7 147 Fig. 5. Surfaces (a), and (b) have similar parameterizations. (c) shows the same surface as in (b) with a dramaticmly different parameterization. Points with similar parametric coordinates (s, t) on the surfaces in (a) and (b) tend to fall in consistent locations along the two sulci; this is not true for (a) and (c). x(s, j/n) and x(i/n, t) for each i, j E [0, 1,..., N]. This procedure typically converges in a few iterations. 3 Applications of Sulcal Mapping 3.1 Morphological Analysis of the Sulci One of the fundamental problems in the emerging field of computational neuroanatomy is determining average shapes of brain structures and measuring the anatomical variability around them [2, 1, 18, 19, 8]. The parametric representation of the shape of the sulci presented in the previous section can form the basis for shape analysis of the sulcal structure and for determining average sulcal shapes_ In order to obtain the average of a number of sum, we apply the Procrustes fit, described in more detail in [2]. In this algorithm, the sulci are first translated to have a common centroid. Then, they are scaled so that the sum of the magnitudes of the vectors from the centroid of each sulcus to the points along the sulcus is equal to unity. Subsequently, the sulci are rotated iteratively until their distance from their average becomes minimal. This is a very fast algorithm, so typically a few iterations are sufyicient for convergence. If the resulting sulcal parameterizations of N subjects in a group after the Procrustes fit are Pl (s, t), p2(s, t), -.., pn(s, t), (s, t) C l), then the average sulcus is N 1 ~ pi(s,t). (6) p(s, t) = The covariance tensor around this mean can be readily obtained. Obtaining a measure of the variance or the covariance tensor around the average shape inevitably depends on the specific parameterization scheme that was used. Although our parameterization is a sensible choice in the absence of specific landmarks along the sulcus, still it is one of many possible ones. In order to minimize the effects of the parameterization on the measurement of the variance around the average shape of a salons, in the experiments of Section 4 i:l

8 148 we measure the variance of the component of the residual vector Pi(S, t) - ~(s, t) that is perpendicular to the tangent plane of the sulcus. This component depends primarily on the shape of the sulci and not on the way it is parameterized. 3.2 Spatial Normalization and Registration of 3D Images Finding parametric representations of the sulcal ribbons also finds application in the spatial normalization and registration of brain images. We have previously reported a methodology [16, 13] that elastically warps one brain image to another, by mapping a number of surfaces in one brain to their homologous surfaces in the other brain. In [16, 13] the warping was based on the outer cortical and the ventricular surfaces, which can be extracted fairly easily from MR images. This method is extended in this paper to include the sulcal surfaces as additional features driving the 3D elastic warping. (A similar approach was described in [15], which used a different warping transformation as well as manually outlined sulci.) Let xl(s, t) and x2(s, t) be the parameterizations of two homologous sulci. In our elastic warping algorithm we deform one sulcus to the other so that xl(s,t) > x~(s,t)v(s,t) 9 l). (7) This effectively stretches one sulcus uniformly along each family of isoparametric curves so that it matches the other, while maintaining the angles at which the horizontal and vertical isoparametric curves intersect. The rest of the 3D image deforms following the equations of an elastic solid deformation. Non-Uniform Sulcal Stretching. In the absence of either anatomical of functional landmarks, the directionally uniform stretching described above is the best guess for defining point-to-point correspondences between two sulci. However, often the structure of the sulci allows us to obtain a more accurate map from one sulcus to another. A notable example is the central sulcus, which has a fairly consistent folding pattern forming an S-like shape. This folding pattern is reflected by the principal curvatures along the sulcus, which can be therefore used for determining a nonuniform stretching of one sulcus to another that brings the curvatures of the stretched and the target sulci in best agreement. In order to estimate the principal curvatures, nl and n2, of the (discretized) surface x(s, t), we find a least-squares quadratic patch fit around each point of the sulcal ribbon and calculate the curvatures as described in [17]. Having defined the curvatures, we then find the 2D elastic transformation of 7P to itself, which is driven by a force field attempting to maximize the similarity of the curvatures along two sulci. This is essentially a reparameterization of one sulcus so that points of similar geometric structure in two homologous sulci have the same parametric coordinates (s, t). The boundary conditions of this transformation fix the four corners of the sulcal ribbon, and allow the points along its four sides to slide freely under the influence of the external and the internal elastic forces. The details of this 2D elastic deformation are described in [20, 21].

9 149 4 Experiments Morphological Analysis. Because of the functional importance of the central sulcus and the known distribution of function along it, mentioned previously as Penfield's homunculus, it is our primary interest to quantify its morphology. Therefore, we have focused our preliminary experiments on extracting and analyzing the central sulcus of five subjects. The sequence of steps, described in Section 2 were used initialize each active contour. A sulcal ribbon generated from the resultant trajectories is shown in Fig. 6. The parametric domain, sampled at regular intervals is superimposed on the sulcus in 3D, illustrating smooth uniform spacing along the t-isoparametric and s-isoparametric curves. Figure 8 shows that the ribbon obtained by the algorithm lies along the me- Fig. 6. A resultdial surface of the sulcus, ing sulcal surface. After extracting the sulcal surfaces of the five subjects, we then applied the Procrustes fit algorithm as described in 3.1. The average sulcus, shown in 7, reveals two prominent folds, which further suggest a consistency in the folding pattern of the central sulcus in these five subjects. Next, we applied our algorithm to three additional sulci: pre-central, superior frontal, and inferior frontal of the left hemisphere. The resulting sulci of two subjects are shown in Fig. 9 and a visual comparison suggests that sulci other than the central sulcus have fairly consistent gyrations. Most notably the superior frontal sulcus which appears to have one or two consistent folds, similar to the Fig. 7. shape consistencies found in the central sulcus. The Procrustes av- Validation. The performance of our algorithm was evalu- erage of the central ated by quantitatively comparing the sulci obtained from sulcus from five subour algorithm, for each of the five subjects, with a manual jects. tracing of the same sulci. The central sulcus was traced on the axial slices; the outlines were then stacked to a 3D volume. Error measurements were obtained by recording the maximum distance in millimeters between the traced sulcus and the resulting sulcal representation from our algorithm. We obtained a maximum and average error for each central sulcus of the five subjects. The distance measure is calculated by growing a spherical neighborhood around each point in the parameterized surface until it intersects a manually traced point. The radius in mm of the resulting neighborhood is the error measure which we refer to as Error 1. This error measure, however, does not reflect deep parts of the actual sulcus which were not reached by the active contour. Specifically, each point in our parameterized surface y(s, t) is close to some manually traced sulcal point. However, there might be parts of the traced sulci which are not reached by the active contour. In order to measure such errors and test

10 150 the ability of our algorithm to cover a sulcus in its full extent, we also obtained a second error measure. This measure is similar to the one described above except that it measures the distance between each point in the manually traced sulcus and its nearest point in the parameterized sulcus. Ie. the neighborhood grows and is centered about each manually traced point rather than each point in the pararneterized sulcus. At a deep region of the sulcus that is not reached by the active contour, this error measure is high. We refer to this error as Error 2. Fig. 8. Cross-sections of a resulting sulcal surface superimposed on the corresponding MR slices. Fig. 9. (a) Four sulci(central, pre-central, superior frontal, and inferior frontal of the left hemisphere). (b) The same four sulci of another subject. Each sulcus was manually outlined by two different raters; the errors are shown in Tables 10a and 10b. We also computed the differences between the two manual tracings by using the second error measure. The results show that the average errors fall roughly between.7mm and 1.6mm which are comparable to the average difference between the two raters which fall between.6mm and.9mm and have similar variance. We conclude that manual tracing would result in only a marginal improvement, at the expense of irreproducibility, subjectivity, and excessive human effort. In considering the variability along the sulcus, it is necessary to correlate variability with error because variability measurements at regions of high error are dubious. The most meaningful variability information should be considered

11 151 along the regions of the sulcus which correspond to low error. Interestingly, our results have shown that some of the most variable regions have low error. One fairly distinct region corresponds to a fold along the center of the sulcus, from which we have made two inferences. First, the low error indicates that our algorithm has been successful in following a seemingly difficult, high curvature region of the sulcus. Second, according to the homunculus, the facial regions of the body would somatotopically map to what we have found as a highly variable region of the central sulcus. One might expect such shape variability because of the high variability in facial expressions, which are partly controlled by that region of the primary motor cortex, as compared to the feet, hands, etc. We note that the parameters in 3 and 4 were determined empirically. Detailed experiments validating the robustness of our method to them are reported in [22]. 3D Warping. In this experiment we warped one volumetric MR image to another, using the outer cortical surface, the ventricular surface, and 4 sulci (central, pre-central, superior frontal, and inferior frontal, only of the left hemisphere) as features(see Fig. 9). A uniform stretching was applied to corresponding sulci to map them to each other. A cross-section from the warped image is shown in Fig. lla, and in Fig. llb overlayed on the cortical outline of the target image, which is shown in Fig. llc. ISubjecql Error 1 I a [ Err~ 21 ~ IIS~bjectll Error 11 ~ I Err~ 21 cr [ mm mm mm mm (a) (b) [Subjectl[ Error 2[ a [ mm (c) Fig. 10. (a) Errors for first rater, (b) Errors for second rater, (c) Differences between first and second rater. It is apparent from these images that a better registration was achieved at the left hemisphere (right side in the images, according to the radiology convention), which is the hemisphere of which the 4 sulci were used as features. Most notable is the registration around the central sulcus. Experiments demonstrating the elastic reparameterization of the sulci, based on their principal curvatures are reported in [22].

12 152 Fig D elastic warping using the outer cortical surface and 4 sulci of the left hemisphere (right side in the images) as landmarks. (a) A cross-section of the warped image. (b) The same cross-section overlayed on the cortical outline of the target image. (c) The corresponding cross-section of the target image. The arrows a and b highlight the sulci where a better registration was achieved. A poorer registration, in which the 4 sulci were not used as features, is seen along the analogous sulci, c and d on the other hemisphere. 5 Discussion We have developed a quantitative methodology for describing the shape of the sulci, utilizing particular characteristics of the cortical folds. A force field guides a set of active contour points along the medial surfaces of sulci, thereby parameterizing the sulcal "ribbons". This technique was applied to five subjects, and its application was demonstrated in quantitative morphological analysis and spatial normalization and registration of brain images. The advantage of our method is that a parameterized representation of the cortical folds is automatically obtained. The limitation of previous methods for medial axis finding is the difficulty in parameterization of the result. Traditional axis-width descriptors as described in [23, 6, 7] often generate noisy medial axes, or more importantly disconnected axes, making parameterization and subsequent shape analysis very difficult. Moreover, axes in the whole image are extracted, most of which do not correspond the sulci and therefore must be manually removed at a post-processing step. Several extensions of this basic approach are possible. In particular, a current limitation of our algorithm is the requirement for the manual initialization of the active contour. The automation of this procedure is a very difficult task, primarily because of the complexity of the cortical structure. Specifically, although sulcal edges can be identified from the peaks of the absolute value of the minimum curvature of the outer cortical surface [24, 25, 26], the differentiation

13 153 between the sulci eventually requires a higher level model. Our current and future work in this direction focuses on the use of prior probability distributions which reflect our expectation for finding a particular sulcal edge at a given coordinate (u, v) of the outer cortical surface b(u, v). These priors, together with geometric properties of the outer cortex, such as curvatures, can potentially lead to the automatic identification of specific sulci, and the initial placement of the active contour along them. Another extension of our algorithm is in the refinement procedure of Section 2.4. In particular, one could constrain the surface y(s, t) to the center of mass of the sulcus rather than x(s, t), therefore allowing it to freely slide along the medial surface rather than "being tied" to x(s, t). For example, the first term in equation (4) could be replaced by, ff Ily(s, t) -c(s,t)ll2dsdt where c(s, t) is the center of mass of a neighborhood at y(s, t). As well, an elastic warping of the sulci based on matching curvature can help to achieve a better map between subjects[21]. A current problem with the algorithm occurs when the active contour encounters an interruption, which frequently occurs in sulci such as the pre-central sulcus. If several active contour points exceed the threshold 4, as described in Section 2.3, the entire inward motion of the active contour terminates. In this case, the deepest edge of the sulcus will not be reached. If instead, the active contour continues, the fixed-point procedure of Section 2.5 will not parameterize the sulci consistently. We are currently pursuing a refinement of the algorithm which divides the active contour into two parts when an interruption is encountered. The parameterization will be applied separately to each section of the active contour. References 1. C. Davatzikos, M. Vaillant, S. Resnick, J.L. Prince, S. Letovsky, and R.N. Bryan. A computerized approach for morphological analysis of the corpus callosum. Y. of Comp. Assisted Tomography, 20:88-97, Jan./Feb F.L. Bookstein. Biometrics, biomathematics, and the morphometric synthesis. Bulletin of Mathematical Biology, 58: , G. Szekely, A. Kelemen, C. Brechbuhler, and G. Gerig. Segmentation of 2-D and 3- D objects from MRI volume data using constrained deformations of flexible Fourier contour and surface models. Medical Image Analysis, 1:19-34, M. Vaillant, C. Davatzikos, and R.N. Bryan. Finding 3D parametric representations of the deep cortical folds. Proe. of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, pages , June J.-F. Mangin, J. Regis, I. Bloch, V. Frouin, Y. Samson, and J. L5pez-Krahe. A MRF based random graph modeling the human cortical topography. In Proc. First Int. Conf., CVRMed, pages , Nice, France, S.M. Pizer, J.M. Coggins, C.A. Burbeck, B.S. Morse, and D. Fritsch. Object shape before boundary shape: Scale-space medial axes. Journal of Mathematical Imaging and Vision, 4: , B.S. Morse, S.M. Pizer, and A. Liu. Multiscale medial analysis of medical images. Image and Vision Computing, 12(6): , July/August 1994.

14 P.M. Thompson, C. Schwartz, and A.W. Toga. High resolution random mesh algorithms for creating a probabilistic 3D surface atlas of the human brain. Neuroimage, 3:19-34, G. Le Goualher, C. Barillot, L. Le Briquer, J.C. Gee, and Y. Bizais. 3D detection and representation of cortical sulci. Proc. Computer Assisted Radiology, symposium on Computer and Communication Systems for Image Guided Diagnosis and Therapy, pages , June F. Kruggel. Automatical adaption of anatomical masks to the neocortex. Proc. Int. Conf. on Computer Vision, Virtual Reality and Robotics in Medicine, pages , April M. Kass, A. Witkin, and D. Terzopoulos. Snakes: Active contour models. International Journal of Computer Vision, 1: , C.A. Davatzikos and J.L. Prince. An active contour model for mapping the cortex. IEEE Trans. on Medical Imaging, 14:65-80, C. Davatzikos. Spatial normalization of 3D images using deformable models. J. Comp. Assist. Tomogr., 20: , July/August S.R. Sandor. Atlas-Guided Deformable Models for Automatic Labeling of Magnetic Resonance Brain Images. PhD thesis, Un. of Southern California, P. Thompson and A.W. Toga. A surface-based technique for warping threedimensional images of the brain. IEEE Trans. on Med. Imaging, 15: , C. Davatzikos. Nonlinear registration of brain images using deformable models. Proe. of the Workshop on Math. Meth. in Biota. Image Anal., pages , June C. Davatzikos and P~.N. Bryan. Using a deformable surface model to obtain a shape representation of the cortex. IEEE Trans. on Med. Imaging, 15: , Dec M.I. Miller, G.E. Christensen, Y. Amit, and U. Grenander. Mathematical textbook of deformable neuroanatomies. Proc. of the National Academy of Sciences, 90: , A.C. Evans, D.L. Collins, S.R. Mills, E.D. Brown, R.L. Kelly, and T.M. Peters. 3D statistical neuroanatomical models from 305 MRI volumes. Proe. of the IEEE Nucl. Se. Symposium and Med. Imaging Conf., 3: , C. Davatzikos and J.L. Prince. Brain image registration based on curve mapping. Proe. of the IEEE Workshop on Biomedical Image Analysis, pages , C. Davatzikos, J.L. Prince, and R.N. Bryan. Image registration based on boundary mapping. IEEE Trans. on Med. Imaging, 15(1): , Feb M. Vaillant and C. Davatzikos. Finding parametric representations of the cortical sulci using an active contour model. Medical Image Analysis, accepted. 23. H. Blum and R.N. Nagel. Shape description using weighted symmetric axis features. Part. Recog., 10: , C. Davatzikos and P~.N. Bryan. Using a deformable surface model to obtain a mathematical representation of the cortex. Proc. of the IEEE Comp. Vision Symp., pages , Nov J.P. Thirion, O. Monga, S. Benayoun, A. Gueziec, and N. Ayache. Automatic registration of 3-D images using surface curvature. SPIE Proc., Mathematical Methods in Medical Imaging, 1768: , G. Subsol, J.P. Thirion, and N. Ayache. First steps towards automatic building of anatomical atlases. INRIA, Technical Report N ~ 2216, 1994.