Regression analysis is a powerful tool for the study of changes in a dependent variable as a function of an independent regressor variable, and in particular it is applicable to the study of anatomical growth and shape change. When the underlying process can be modeled by parameters in a Euclidean space, classical regression techniques [13, 34] are applicable and have been studied extensively. However, recent work suggests that attempts to describe anatomical shapes using flat Euclidean spaces undermines our ability to represent natural biological variability [9, 11]. In this paper we develop a method for regression analysis of general, manifold-valued data. Specifically, we extend Nadaraya-Watson kernel regression by recasting the regression problem in terms of Fréchet expectation. Although this method is quite general, our driving problem is the study anatomical shape change as a function of age from random design image data. We demonstrate our method by analyzing shape change in the brain from a random design dataset of MR images of 89 healthy adults ranging in age from 22 to 79 years. To study the small scale changes in anatomy, we use the infinite dimensional manifold of diffeomorphic transformations, with an associated metric. We regress a representative anatomical shape, as a function of age, from this population.

The tensors produced by diffusion tensor magnetic resonance imaging (DT-MRI) represent the covariance in a Brownian motion model of water diffusion. Under this physical interpretation, diffusion tensors are required to be symmetric, positive-definite. However, current approaches to statistical analysis of diffusion tensor data, which treat the tensors as linear entities, do not take this positivedefinite constraint into account. This difficulty is due to the fact that the space of diffusion tensors does not form a vector space. In this paper we show that the space of diffusion tensors is a type of curved manifold known as a Riemannian symmetric space. We then develop methods for producing statistics, namely averages and modes of variation, in this space. We show that these statistics preserve natural geometric properties of the tensors, including the constraint that their eigenvalues be positive. The symmetric space formulation also leads to a natural definition for interpolation of diffusion tensors and a new measure of anisotropy. We expect that these methods will be useful in the registration of diffusion tensor images, the production of statistical atlases from diffusion tensor data, and the quantification of the anatomical variability caused by disease. The framework presented in this paper should also be useful in other applications where symmetric, positive-definite tensors arise, such as mechanics and computer vision. 1

Abstract. High angular resolution diffusion imaging (HARDI) permits the computation of water molecule displacement probabilities over a sphere of possible displacement directions. This probability is often referred to as the orientation distribution function (ODF). In this paper we present a novel model for the diffusion ODF namely, a mixture of von Mises-Fisher (vMF) distributions. Our model is compact in that it requires very few variables to model complicated ODF geometries which occur specifically in the presence of heterogeneous nerve fiber orientation. We also present a Riemannian geometric framework for computing intrinsic distances, in closed-form, and performing interpolation between ODFs represented by vMF mixtures. As an example, we apply the intrinsic distance within a hidden Markov measure field segmentation scheme. We present results of this segmentation for HARDI images of rat spinal cords – which show distinct regions within both the white and gray matter. It should be noted that such a fine level of parcellation of the gray and white matter cannot be obtained either from contrast MRI scans or Diffusion Tensor MRI scans. We validate the segmentation algorithm by applying it to synthetic data sets where the ground truth is known. 2 1

An important goal of statistical shape analysis is the discrimination between populations of objects, exploring group differences in morphology not explained by standard volumetric analysis. Certain applications additionally require analysis of objects in their embedding context by joint statistical analysis of sets of interrelated objects. In this paper, we present a framework for discriminant analysis of populations of 3-D multi-object sets. In view of the driving medical applications, a skeletal object parametrization of shape is chosen since it naturally encodes thickening, bending and twisting. In a multi-object setting, we not only consider a joint analysis of sets of shapes but also must take into account differences in pose. Statistics on features of medial descriptions and pose parameters, which include rotational frames and distances, uses a Riemannian symmetric space instead of the standard Euclidean metric. Our choice of discriminant method is the distance weighted discriminant (DWD) because of its generalization ability in high dimensional, low sample size settings. Joint analysis of 10 subcortical brain structures in a pediatric autism study demonstrates that multi-object analysis of shape results in a better group discrimination than pose, and that the combination of pose and shape performs better than shape alone. Finally, given a discriminating axis of shape and pose, we can visualize the differences between the populations. 1.

We present a fully-automated technique for visualizing localized cerebral ventricle shape differences between large clinical subject groups who have received a magnetic resonance (MR) image scan. The technique combines a robust, automated technique for ventricular segmentation with a 3D surfacebased radial thickness mapping approach that allows spatiallylocalized statistical tests of relative shape differences between clinical groups. The technique is used to analyze localized ventricular expansion in Alzheimer’s Disease (AD) and mild cognitive impairment (MCI) in a large cohort of communitydwelling elderly individuals (N=339). The resulting maps are the first to chart localized ventricular dilation in a cohort of this size. Besides showing patterns of ventricular expansion that may be consistent with the spatial progression of ADrelated pathology, the maps reveal new information about localized ventricular atrophy that may have been overlooked to date. A detailed understanding of spatial atrophy patterns may be useful for early disease detection or for patient monitoring in drug trials. 1.

Abstract. We present a method for two-sample hypothesis testing for statistical shape analysis using nonlinear shape models. Our approach uses a true multivariate permutation test that is invariant to the scale of different model parameters and that explicitly accounts for the dependencies between variables. We apply our method to m-rep models of the lateral ventricles to examine the amount of shape variability in twins with different degrees of genetic similarity. 1

Building of atlases representing average and variability of a population of images or of segmented objects is a key topic in application areas like brain mapping, deformable object segmentation and object classification. Recent developments in image averaging, i.e. constructing an image which is central within the population, focus on unbiased atlas building with nonlinear deformations. Groupwise nonlinear image averaging creates images which appear sharper than linear results. However, volumetric atlases do not explicitely carry a notion of statistics of embedded shapes. This paper compares population-based linear and non-linear image averaging on 3D objects segmented from each image and compares voxelbased versus surface-based representations. Preliminary results suggest improved locality of group average differences for the nonlinear scheme, which might lead to increased significance for hypothesis testing. Results from a clinical MRI study with sets of subcortical structures of children scanned at two years with follow-up at four years are shown. 1.

Abstract. We present a method to extract principal deformation modes from a set of articulated models describing the human spine. The spine was expressed as a set of rigid transforms that superpose local coordinates systems of neighbouring vertebrae. To take into account the fact that rigid transforms belong to a Riemannian manifold, the Fréchet mean and a generalized covariance computed in the exponential chart of the Fréchet mean were used to construct a statistical shape model. The principal deformation modes were then extracted by performing a principal component analysis (PCA) on the generalized covariance matrix. Principal deformations modes were computed for a large database of untreated scoliotic patients and the obtained results indicate that combining rotation and translation into a unified framework leads to an effective and meaningful method of dimensionality reduction for articulated anatomical structures. The computed deformation modes also revealed clinically relevant information. For instance, the first mode of deformation appeared to be associated with patients ’ growth, the second is a double thoraco-lumbar curve and the third is a thoracic curve. 1

Communicated by (Silvia Biasotti) We present a novel medial-based, multi-scale approach to shape representation and controlled deformation. We use medial-based profiles for shape representation, which follow the geometry of the structure and describe general, intuitive, and independent shape measures (length, orientation, and thickness). Controlled shape deformations (stretch, bend, and bulge) are obtained either as a result of applying deformation operators at certain locations and scales on the medial profiles, or by varying the weights of the main variation modes obtained from a new hierarchical (multi-scale) and regional (multi-location) principal component analysis of the medial profiles. We demonstrate the ability to produce controlled shape deformations on a medial-based representation of the corpus callosum. We show how this control of shape deformations facilitates the design of a layered framework for image segmentation and present results of segmenting the corpus callosum from 2D mid-sagittal magnetic resonance images of the human ∗ Corresponding author. 1 2 Hamarneh et al brain. Furthermore we show how the medial-based representation facilitates hierarchical, deformation-specific statistical shape analysis of segmented corpora callosa.

We describe a new approach for elucidating the nonlinear degrees of freedom in a distribution of shapes depicted in digital images. By combining a deformation-based method for measuring distances between two shape configurations together with multidimensional scaling, a method for determining the number of degrees of freedom in a shape distribution is described. In addition, a method for visualizing the most representative modes of variation (underlying shape parameterization) in a nuclei shape distribution is also presented. The novel approach takes into account the nonlinear nature of shape manifolds and is related to the ISOMAP algorithm. We apply the method to the task of analyzing the shape distribution of HeLa cell nuclei and conclude that approximately three parameters are responsible for their shape variation. Excluding differences in size, translation, and orientation, these are: elongation, bending (concavity), and shifts in mass distribution. In addition, results show that, contrary to common intuition, the most likely nuclear shape configuration is not symmetric. Index Terms — Nuclear shape analysis, nonlinear, dimension reduction, image registration. 1.