Shape-based discriminative analysis of combined bilateral hippocampi using multiple object alignment

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1 Shape-based discriminative analysis of combined bilateral hippocampi using multiple object alignment Li Shen a, Fillia Makedon a, and Andrew Saykin b a Dartmouth Experimental Visualization Laboratory, Department of Computer Science, Dartmouth College, Hanover, NH 03755, USA b Brain Imaging Laboratory, Departments of Psychiatry and Radiology, Dartmouth Medical School, Lebanon, NH 03756, USA ABSTRACT Shape analysis of hippocampi in schizophrenia has been preformed previously using the spherical harmonic SPHARM description. In these studies, the left and right hippocampi are aligned independently and the spatial relation between them is not explored. This paper presents a new SPHARM-based technique which examines not only the individual shape information of the two hippocampi but also the spatial relation between them. The left and right hippocampi are treated as a single shape configuration. A ploy-shape alignment algorithm is developed for aligning configurations of multiple SPHARM surfaces as follows: (1) the total volume is normalized; (2) the parameter space is aligned for creating the surface correspondence; (3) landmarks are created by a uniform sampling of multiple surfaces for each configuration; (4) a quaternion-based algorithm is employed to align each landmark representation to the mean configuration through the least square rotation and translation iteratively until the mean converges. After applying the poly-shape alignment algorithm, a point distribution model is applied to aligned landmarks for feature extraction. Classification is performed using Fisher s linear discriminant with an effective feature selection scheme. Applying the above procedure to our hippocampal data (14 controls versus 25 schizophrenics, all right-handed males), we achieve the best cross-validation accuracy of 92%, supporting the idea that the whole shape configuration of the two hippocampi provides valuable information in detecting schizophrenia. The results of an ROC analysis and a visualization of discriminative patterns are also included. Keywords: Shape analysis, surface modeling, shape alignment, classification, hippocampus, MRI 1. INTRODUCTION Object classification via shape analysis is an important and challenging problem in computer vision and medical image analysis. In the brain imaging domain, the goal is to identify shape abnormalities in a structure of interest that are associated with a particular condition to aid diagnosis and treatment. In this paper, we concentrate on detecting hippocampal shape changes in schizophrenia. The hippocampus is a critical structure of the human limbic system, which is involved in learning and memory processing. Several shape classification studies have been conducted for discovering hippocampal shape abnormality in the neuropsychiatric disease of schizophrenia. In these studies, classification accuracies are all estimated using leave-one-out cross-validation. Csernansky et al. 1 studied hippocampal morphometry using an image-based deformation representation, and achieved 80% classification accuracy through principal component analysis (PCA) and a linear discriminant. Golland, Timoner et al. 2 4 conducted amygdala-hippocampus complex studies using distance transformation maps and displacement fields as shape descriptors, and achieved best accuracies of 77% and 87%, respectively, This work was supported by NSF IDM Mining Human Brain Data: Analysis, Classification, and Visualization. Send correspondence to Li Shen. s: {li,makedon}@cs.dartmouth.edu, saykin@dartmouth.edu Copyright 2004 Society of Photo-Optical Instrumentation Engineers. This paper will be published in SPIE Medical Imaging 2004: Image Processing, Proc and is made available as an electronic reprint (preprint) with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.

2 using support vector machines (SVMs). 5 Saykin el al. 6 studied hippocampal shape classification, using a symmetric alignment model and binary images, and achieved 96% accuracy using only the second principal component after PCA. In addition to the voxel-based or image-based approaches mentioned above, there are also surface-based shape analysis studies using the SPHARM description. This approach has several advantages, including inherent interpolation, implicit correspondence, accurate scaling and so on. The SPHARM description 7 is a parametric surface description using spherical harmonics as basis functions, with applications in model-based segmentation 8 and 3D medial shape (m-rep) modeling. 9 Gerig, Styner and colleagues have done numerous studies 10 identifying statistical shape abnormalities of different neuroanatomical structures using SPHARM and m-rep. They have used SPHARM for calculating hippocampal asymmetry, 11 combined it with volume, and achieved 87% accuracy using SVM. In our previous studies of detecting schizophrenia using hippocampal shape, we have also employed the SPHARM model, combined it with a set of effective pattern classification, feature selection, evaluation and visualization techniques, and obtained an improved accuracy of over 90%. However, in each of the SPHARM-based studies mentioned above, the left and right hippocampi are aligned independently and the spatial relation between them is not explored. In this paper, we treat the two hippocampi as a single shape configuration, and examine not only the individual shape information of the two hippocampi but also the spatial relation between them. We aim to investigate if the shape information extracted from this configuration can help discriminate between schizophrenics and controls. To achieve this goal, we develop a new SPHARM-based technique using a newly designed poly-shape alignment algorithm for multiple object alignment. We describe our approach in Section 2, present the results in Section 3, and conclude the paper in Section METHODS Our dataset includes hippocampi from 14 healthy controls and 25 schizophrenic patients, all right-handed males. Hippocampi are manually segmented from magnetic resonance (MR) scans. The left and right hippocampi in each MR image are identified and segmented by manual tracing in each acquisition slice using the BRAINS software package. 15 A 3D binary image is reconstructed from each set of 2D hippocampus segmentation results, with isotropic voxel values corresponding to whether each voxel is excluded or included. The surface of this 3D binary image is composed of a mesh of square faces; please see Figure 1(a) for sample hippocampal surfaces. In this section, we present (1) how to describe each of such surfaces using SPHARM parameterization; (2) how to normalize each pair of left and right hippocampi into a common reference system so as to establish surface correspondence across different subjects as well as extract only shape information by excluding translation, rotation and scaling; (3) how to use a point distribution model (PDM) to extract shape features; (4) how to select significant features; and (5) how to perform classification using selected features Shape description We adopt the SPHARM expansion technique 7 to create a shape description for each left or right hippocampal surface. The first step is to create a spherical parameterization, which is a continuous and uniform mapping from the object surface to the surface of a unit sphere. It is formulated as a constrained optimization problem for preserving topology and area and minimizing distortion. The result is a bijective mapping between each point v on a surface and spherical coordinates θ and φ. Based on the spherical parameterization, the second step is to expand the object surface into a complete set of SPHARM basis functions Yl m, where Yl m is the spherical harmonic of degree l and order m. The expansion takes the form: l v(θ, φ) = c m l Yl m (θ, φ), (1) where l=0 m= l c m l = (c m xl, c m yl, c m zl) T. (2) The coefficients c m l up to a user-desired degree can be estimated by solving a set of linear equations in a leastsquares fashion. The object surface can be reconstructed using these coefficients, and using more coefficients

3 (a) Original surfaces (b) SPHARM reconstructions Figure 1. (a) Volumetric hippocampal surfaces from two subjects, where each subject has a pair of left and right hippomcapi. (b) The corresponding SPHARM reconstructions. Figure 2. Two SPHARM reconstructions are aligned in the parameter space but not in the object space, where one configuration is in red color and the other is in blue color. The parameter space is indicated by the surface mesh. leads to a more detailed reconstruction. Figure 1(a) shows volumetric hippocampal surfaces from two subjects, where each subject has a pair of left and right hippomcapi. Figure 1(b) shows their SPHARM reconstructions using coefficients up to degree Multiple object alignment As mentioned before, we treat the bilateral hippocampi as a single shape configuration. To create a shape descriptor for each configuration, we need to remove the effects of scaling, rotation and translation. Based on the SPHARM representation, scaling invariance can be achieved by dividing all the SPHARM coefficients by a scaling factor f. We choose f so that the total volume of both hippocampi is normalized. After normalizing for the total volume, we perform multiple object alignment across configurations to remove the effects of rotation and translation and create a shape descriptor for each configuration. To align one configuration to another, we first create surface correspondence between them in the parameter space and then align them in the object space. The correspondence between two shape configurations is established by aligning the parameter space for both left and right hippocampi using the degree one SPHARM

4 reconstruction, which is always an ellipsoid. The parameter net on this ellipsoid is rotated to a canonical position such that the north pole is at one end of the longest main axis, and the crossing point of the zero meridian and the equator is at one end of the shortest main axis. Figure 2 shows such an example of aligning two subjects in the parameter space, where the parameter space is indicated by the surface mesh. The aligned parameter space creates the surface correspondence between two configurations. For each shape configuration, we have two sets of SPHARM coefficients, one for the left hippocampus and the other for the right one. After the alignment of the parameter space, we can convert these SPHARM coefficients to a dual landmark representation using a uniform icosahedron subdivision of spherical surfaces; see Kelemen et al. 8 for an example. The landmark representation of the whole configuration simply includes all the sampling points from both hippocampi. Compared with the SPHARM coefficients, the landmark representation is more intuitive. We use icosahedron subdivision level 3, resulting in n = = 1284 landmarks for each configuration. Note that the surface correspondence established before implies the correspondence for landmarks between two different configurations in our case. Thus, let P = { p 1,, p n } and X = { x 1,, x n } be two landmark configurations, where p i corresponds to x i for i {1,, n}. To align P to X, we attempt to minimize the following objective function by using a rotation matrix R and a translation vector T: f(r, T) = 1 n n x i R p i T 2. (3) i=1 A quaternion-based algorithm has been presented in Besl et al. 16 to achieve the above least square rotation and translation for the problem of the corresponding point set registration. This algorithm is simple and effective. We employ it in our study and refer the reader to Besl et al. 16 for the implementation details. The above describes how to align two configurations together. In our shape classification study, we need to align all the configurations so that they can be compared with any of the others according to a common reference system. This is achieved by creating a template and aligning all the configurations to the template. We use the following procedure to create the template and do the alignment: 1. Initialize the template to be the shape of a designated control. 2. Align all shapes to the template using the quaternion-based algorithm. 3. Calculate the mean of newly aligned shapes and use the mean as the new template. 4. Repeat steps (2-3) until the mean converges to the previous template. For convenience, we call the above multiple object alignment procedure as poly-shape alignment. We apply the poly-shape alignment algorithm to our hippocampal data set. It takes only three iterations for the mean shape to converge. Figure 3 shows some sample results, where the least square rotation and translation is achieved between each shape and the mean shape (i.e., the template) Feature extraction After applying the poly-shape alignment algorithm, the landmarks of each configuration are aligned to the template and form a high dimensional shape representation for the configuration. In this section, we discuss how to convert this high dimensional landmark representation to a low dimensional feature vector and how to select useful features for classification. We use icosahedron subdivision level 3 for selecting landmarks, resulting in n = 642 landmarks for each surface and 3n 2 = 3852 feature elements for each pair of hippocampal surfaces. Clearly, we have many more dimensions than training objects. Principal component analysis (PCA) 17 is applied to reduce dimensionality to make classification feasible. This involves eigenanalysis of the covariance matrix Σ of the data: ΣP = DP, (4)

5 (a) Before alignment (b) After alignment Figure 3. Shape alignment in the object space: The plot shows the spatial relation between two subjects (blue circles) and the final template (red dots) before (a) and after (b) the alignment procedure. where the columns of P hold eigenvectors, and the diagonal matrix D holds eigenvalues of Σ. The eigenvectors in P can be ordered decreasingly according to respective eigenvalues, which are proportional to the variance explained by each eigenvector. Now any shape x in the data can be obtained using x = x + Pb, (5) where b is a vector containing the components of x in basis P, which are called principal components. Since eigenvectors are orthogonal, b can be obtained using b = P T (x x). (6) Given a dataset of m objects, the first m 1 principal components are enough to capture all the data variance. Thus, b becomes an m 1 element vector, which can be thought of as a new and more compact representation of the shape x in the new basis of the deformation modes. This model is a point distribution model (PDM). 8 We apply PDM to our hippocampal data set to obtain a b (referred to as a feature vector hereafter) for each shape configuration Feature selection It is often helpful to select a subset of the most useful features, which may improve classification accuracy by reducing the number of parameters that need to be estimated. In our study, features are principal components, and we feel that some components are more useful than others for classification, but not necessarily matching the ordering of the variance amounts they explain; please refer to Shen et al. 14 for such an example. To rank the effectiveness of features, we employ the PCTt ordering scheme introduced in Shen et al.. 14 use a simple two-sample t-test 18 on each feature and obtain a p-value associated with the test statistic T = We Ȳ 1 Ȳ2 s 2 1 /N 1 + s 2 2 /N, (7) 2

6 where N 1 and N 2 are the sample sizes, Ȳ1 and Ȳ2 are the sample means, s 2 1 and s 2 2 are the sample variances, and the samples are two sets of feature values in two respective classes. The p-value indicates the probability that the observed value of T could be as large or larger by chance under the null hypothesis that the means of Y 1 and Y 2 are the same. Thus, a lower p-value implies stronger group difference statistically and corresponds to a more significant feature. We hypothesize that more significant features can help more in classification. In our feature selection scheme, we select the first n features according to a certain ordering of principal components, where varying values of n are also considered. Note that we will use a leave-one-out cross-validation method to estimate the classification accuracy, see Section 3.1. To match that, we employ the PCTt ordering scheme 14 : principal components are ordered by p-value associated with t-test applied to each leave-one-out training set, increasingly, where different leave-one-out trials could have different PCTt orderings Classification We employ Fisher s linear discriminant (FLD) on selected features for classification, as we did for the singleobject cases. 14 Linear techniques are simple and well-understood. Once they succeed in real applications, the results can then be interpreted more easily than those derived from complicated techniques. FLD projects a training set (consisting of c classes) onto c 1 dimensions such that the ratio of between-class and within-class variability is maximized, which occurs when the FLD projection places different classes into distinct and tight clumps. This optimal projection W opt is calculated as follows. Assume that we have a set of n d-dimensional samples x 1,..., x n, n i in the subset D i labeled ω i, where n = c k=1 n k and i {1,..., c}. Define the between-class scatter matrix S B and the within-class scatter matrix S W as S B = S W = c D i (m i m)(m i m) T, (8) i=1 c i=1 x D i (x m i )(x m i ) T, (9) where m is the mean of all samples and m i the mean of class ω i. If S W is nonsingular, the optimal projection W opt is chosen by W opt = argmax W W T S B W W T S W W = [w 1w 2... w m ] (10) where {w i i = 1, 2,..., m} is the set of generalized eigenvectors of S B and S W corresponding to set of decreasing generalized eigenvalues {λ i i = 1, 2,..., m}, i.e., S B w i = λ i S W w i. (11) Note that an upper bound on m is c 1; please see Duda et al. 17 for a detailed explanation. We have only two classes, and the FLD basis W opt becomes a column vector w. New feature vectors can be compared to the training set, and thus classified, by projecting them onto w. In the FLD space, we use the nearest mean approach for classification, which assigns a new object to the class having the nearest mean Cross-validation results 3. RESULTS We use leave-one-out cross-validation to estimate the classification accuracy. Figure 4 shows both the training and cross-validation accuracies for our data using the FLD approach together with the PCTt feature selection. The number of features used in the experiments according to the PCTt ordering is plotted on the x-axis. The y-axis shows the accuracy rate. From the figure, we can see that the training accuracy can achieve 100% if we include a sufficient number of features. However, in the cross-validation experiment, the classes are not

7 100 FLD Classification Accuracy Percent correct Cross Validation Training Shape feature number Figure 4. Classification results for hippocampal data, where the total volume of the left and right hippocampi are normalized. The number of features used in the classification is plotted on the x axis, where the features are ordered according to the PCTt scheme. The classification accuracy is plotted on the y axis. Both training accuracies and leaveone-out cross-validation accuracies are shown for FLD classification, where the nearest mean classification is performed in the FLD space. separated well if there are insufficient features, while using too many introduces extra noise. We achieve the best cross-validation accuracy of 92%, supporting the idea that the whole shape configuration of the two hippocampi provides valuable information in detecting schizophrenia ROC analysis In medical classification problems, the terms sensitivity and specificity are defined as follows: sensitivity is the probability of predicting disease given the true state is disease; specificity is the probability of predicting nondisease given the true state is non-disease. The receiver operating characteristic (ROC) curve 19 is a commonly used summary for assessing the tradeoff between sensitivity and specificity. It is a plot of the sensitivity versus specificity as we vary the parameters of a classification rule. In the case of a linear classifier, this can be done by setting the decision boundary at various points. The ROC approach overcomes the problem of possible bias introduced by a fixed threshold or different size classes and is often used in visualizing the behavior of diagnostic systems. We perform ROC analysis using our PCA and FLD framework. However, in our leave-one-out experiments, each trial corresponds to an independent FLD projection. Therefore, test objects may be projected differently according to different bases, which makes the resulting scalar values incomparable across different trials. We use the following procedure to normalize these values into a standard range: for each trial, we scale and shift all the projections so that the class means for each trial s training set fall on -1 and 1, and all projections are sign-flipped, if necessary, so that the mean of the patient class is positive. Now test subject projections can then be combined, since they correspond to identically aligned training sets in the FLD space. For each leave-one-out experiment, we calculate normalized test subject projections as described above. Given a decision threshold, a test subject is classified as a control if its normalized projection is less than the threshold; otherwise it is a patient. By varying the threshold, an ROC curve can be constructed based on the sensitivity and specificity that result at each threshold. In addition, the area under the ROC curve (AUROC) can be used as a performance measure for a classifier, 20 because it is the average sensitivity over all possible specificities. Figure 5 shows the results of the ROC analysis for leave-one-out cross-validation using FLD and PCTt feature selection, where one can see how specificity trades off sensitivity. Here, we consider both situations of normalizing and not normalizing data for volume. In the best case, the normalized version gets better results, indicating shape information alone predicts better than shape and size.

8 Sensitivity S no, 0.93 auroc, 20 Fs S vm, 0.97 auroc, 20 Fs Area under ROC S no S vm Specificity (a) ROC curve Feature number (b) Area under the ROC curve Figure 5. ROC analysis for leave-one-out cross-validation using FLD and PCTt feature selection. S vm and S no are two scaling schemes, where S vm denotes that the total volume of the left and right hippocampi is normalized by isotropic scaling, and S no denotes no scaling at all. (a) ROC curves show the sensitivity (the percentage of patients being predicted correctly) versus specificity (the percentage of controls being predicted correctly)) as the threshold for classification is shifted. The legend shows the scaling scheme, the area under the ROC curve (AUROC) and the number of features used. (b) AUROC is an alternative for evaluating the performance of a classifier. It is plotted on the y axis against the number of features used in each experiment on the x axis Discriminative patterns In this section, we discuss how to visualize the discriminative patterns captured by our classifier. Based on the PCA and FLD framework, we can visualize the deformation showing class differences by backprojecting the vector representing the normal to the separating hyperplane onto the mean surface. 14 Applying PCA and FLD with feature selection as detailed above to a shape set, we get a value v for each shape x: v = x T B pca B fld = x T D w, (12) where B pca consists of a subset of eigenvectors, depending on which principal components are selected, and B fld is the corresponding FLD basis. Thus D w is a column vector that weights the contribution of each element in x to v. In fact, D w is also the normal to the separating hyperplane of the corresponding linear classifier. We can map these weights onto the mean surface to show significant discriminative regions. Note that each landmark on the mean surface is associated with three weights, which form a local weight vector. A local weight vector with a large magnitude indicates the location has discriminative power. In addition, the local weight vector suggests a deformation direction towards the positive class (i.e., the class with more positive discriminative values). In our experiment, the patient class is the positive one. In the visualization, we use color to code the magnitude of each local weight vector and map it onto the mean surface; we also show the direction of each local weight vector in a separate plot. Figure 6 shows such a visualization of discriminative patterns. The medial anterior and medial posterior parts of both hippocampi turn out to be quite significant in terms of discrimination. According to the deformation direction plots, the schizophrenics tend to have more curved tails. Note that these discriminative patterns are similar to the discriminative patterns we obtained for singleobject analysis in Shen et al.. 13 This, to some extent, validates the effectiveness of the single-object alignment approach, which is based on aligning the first order ellipsoids in both the parameter space and the object space. 4. CONCLUSIONS Many previous studies have been done for the analysis of hippocampal shape changes in schizophrenia using the SPHARM surface representation 9, 11 14, , 6, 25 and also voxel-based or image-based representations.

9 (a) Top view (b) Bottom view Figure 6. Discriminative patterns shown in both top (a) and bottom (b) views, where nine significant features are used in the PDM/FLD framework to achieve a perfect discrimination for all 39 subjects, and the schizophrenic class has more positive projections. The projection vector, which is normal to the decision hyperplane, is backprojected onto the mean surfaces and indicates the deformation that makes the mean surfaces more schizophrenic. On the left, the direction of the deformation is shown, where the length of each vector is scaled for better visualization. The schizophrenics tend to have more curved tails. On the right, the color codes the magnitude of the deformation. The medial anterior and medial posterior parts of both hippocampi seem to form the significant discriminative region. Compared with those voxel-based and image-based representations, the SPHARM approach has several advantages, including inherent interpolation, implicit correspondence, accurate scaling and so on. However, in prior SPHARM studies, the left and right hippocampi are aligned independently and the spatial relation between them is ignored. In this paper, we have developed a new SPHARM-based technique which examines not only the individual shape information of the two hippocampi but also the spatial relation between them. This technique is based on our previous 3D surface classification framework 14 and extends it from the single object case to the multiple object case. The new contribution is the poly-shape alignment algorithm, which first aligns the surfaces in the parameter space to create correspondence and then use a quaternion-based algorithm to align the surface configurations in the object space. Applying this technique to our data, we have achieved 92% cross-validation accuracy, which is competitive with the best results in prior studies and also demonstrates the whole shape configuration of the two hippocampi distinguishes schizophrenics from controls quite well. We have also visualized the discriminative patterns, indicating that the signification discriminative region lies in the medial anterior and medial posterior parts of both hippocampi and the schizophrenics tend to have more curved tails. There are several future directions related to this research. We observe that the current SPHARM description technique works only for the surface of a voxel volume and is not applicable to general triangle meshes, since its spherical parameterization algorithm exploits the uniform quadrilateral structure of the voxel surface. One future direction is to design new spherical parameterization algorithms for general triangle meshes. Another direction is to make this parameterization process more efficient and scalable so that it can deal with large scale surfaces such as the brain cortex. We are now working on these topics.

10 ACKNOWLEDGMENTS We thank NSF IDM , NARSAD, NH Hospital and Ira DeCamp Foundation for support. We are grateful to Hany Farid for recommending the quaternion-based algorithm, and to Annette Donnelly, Laura A. Flashman, Tara L. McHugh and Molly B. Sparling for creating hippocampal traces. REFERENCES 1. J. G. Csernansky, S. Joshi, L. Wang, J. W. Halleri, M. Gado, J. P. Miller, U. Grenander, and M. I. Miller, Hippocampal morphometry in schizophrenia by high dimensional brain mapping, Proc. National Academy of Sciences USA 95, pp , September P. Golland, B. Fischl, M. Spiridon, N. Kanwisher, R. L. Buckner, M. E. Shenton, R. Kikinis, A. Dale, and W. E. L. Grimson, Discriminative analysis for image-based studies, in Proc. of MICCAI 2002: 5th International Conference on Medical Image Computing And Computer Assisted Intervention, LNCS 2488, pp , (Tokyo, Japan), September 25 28, P. Golland, W. E. L. Grimson, M. E. Shenton, and R. Kikinis, Small sample size learning for shape analysis of anatomical structures, in Proc. MICCAI 2000: 3th International Conference on Medical Image Computing and Computer-Assisted Intervention, LNCS 1935, pp , (Pittsburgh, Pennsylvania, USA), October 11 14, S. J. Timoner, P. Golland, R. Kikinis, M. E. Shenton, W. E. L. Grimson, and W. M. W. III, Performance issues in shape classification, in Proc. MICCAI 2002: 5th International Conference on Medical Image Computing and Computer-Assisted Intervention, LNCS 2488, pp , (Tokyo, Japan), September 25 28, N. Cristianini and J. Shawe-Taylor, An Introduction to Support Vector Machines (and other kernel-based learning methods), Cambridge University Press, Cambridge, U.K. ; New York, A. J. Saykin, L. A. Flashman, T. McHugh, C. Pietras, T. W. McAllister, A. C. Mamourian, R. Vidaver, L. Shen, J. C. Ford, L. Wang, and F. Makedon, Principal components analysis of hippocampal shape in schizophrenia, in International Congress on Schizophrenia Research, (Colorado Springs, Colorado, USA), March 29 April 2, C. Brechbühler, G. Gerig, and O. Kubler, Parametrization of closed surfaces for 3D shape description, Computer Vision and Image Understanding 61(2), pp , A. Kelemen, G. Szekely, and G. Gerig, Elastic model-based segmentation of 3-D neuroradiological data sets, IEEE Transactions on Medical Imaging 18, pp , M. Styner, G. Gerig, S. Pizer, and S. Joshi, Automatic and robust computation of 3D medial models incorporating object variability, International Journal of Computer Vision 55(2/3), pp , G. Gerig, Selected Publications, gerig/pub.html, G. Gerig and M. Styner, Shape versus size: Improved understanding of the morphology of brain structures, in Proc. MICCAI 2001: 4th International Conference on Medical Image Computing and Computer-Assisted Intervention, LNCS 2208, pp , (Utrecht, The Netherlands), October 14 17, L. Shen, J. Ford, F. Makedon, and A. Saykin, Hippocampal shape analysis: Surface-based representation and classification, in Medical Imaging 2003: Image Processing, Proc. of the SPIE, M. Sonka and J. M. Fitzpatrick, eds., 5032, pp , (San Diego, CA, USA), February L. Shen, J. Ford, F. Makedon, Y. Wang, T. Steinberg, S. Ye, and A. Saykin, Morphometric analysis of brain structures for improved discrimination, in MICCAI 2003, Medical Image Computing and Computer Assisted Intervention, (Montreal, Quebec, CANADA), 15st 18th November L. Shen, J. Ford, F. Makedon, and A. Saykin, A surface-based approach for classification of 3D neuroanatomic structures, Intelligent Data Analysis, An International Journal 8(5), Iowa MHCRC Image Processing Lab, Brains Software, P. J. Besl and N. D. McKay, A method for registration of 3-D shapes, IEEE Transactions on Pattern Analysis and Machine Intelligence 14(2), pp , R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification (2nd ed), Wiley, New York, NY, NIST/SEMATECHR, e-handbook of Statistical Methods,

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