A Statistical Model of Riemannian Metric Variation for Deformable Shape Analysis

The analysis of deformable 3D shape is often cast in terms of the shape's intrinsic geometry due to its invariance to a wide range of non-rigid deformations. However, object's plasticity in non-rigid transformation often result in transformations that are not completely isometric in the surface's geometry and whose mode of deviation from isometry is an identifiable characteristic of the shape and its deformation modes. In this paper, we propose a novel generative model of the variations of the intrinsic metric of de formable shapes, based on the spectral decomposition of the Laplace-Beltrami operator. To this end, we assume two independent models for the eigenvectors and the eigenvalues of the graph-Laplacian of a 3D mesh which are learned in a supervised way from a set of shapes belonging to the same class. We show how this model can be efficiently learned given a set of 3D meshes, and evaluate the performance of the resulting generative model in shape classification and retrieval tasks. Comparison with state-of-the-art solutions for these problems confirm the validity of the approach

The Deformable Universe

The concept of smooth deformations of a Riemannian manifolds, recently evidenced by the solution of the Poincar\'e conjecture, is applied to Einstein's gravitational theory and in particular to the standard FLRW cosmology. We present a brief review of the deformation of Riemannian geometry, showing how such deformations can be derived from the Einstein-Hilbert dynamical principle. We show that such deformations of space-times of general relativity produce observable effects that can be measured by four-dimensional observers. In the case of the FLRW cosmology, one such observable effect is shown to be consistent with the accelerated expansion of the universe.Comment: 20 pages, LaTeX, 3 figure

Soliton Dynamics in Computational Anatomy

Computational anatomy (CA) has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms. Among these geometric structures, landmarks and image outlines in CA are shown to be singular solutions of a partial differential equation that is called the geodesic EPDiff equation. A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta. The momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as well as new data representation.Comment: published in NeuroImag

Diffeomorphic Metric Mapping of High Angular Resolution Diffusion Imaging based on Riemannian Structure of Orientation Distribution Functions

In this paper, we propose a novel large deformation diffeomorphic registration algorithm to align high angular resolution diffusion images (HARDI) characterized by orientation distribution functions (ODFs). Our proposed algorithm seeks an optimal diffeomorphism of large deformation between two ODF fields in a spatial volume domain and at the same time, locally reorients an ODF in a manner such that it remains consistent with the surrounding anatomical structure. To this end, we first review the Riemannian manifold of ODFs. We then define the reorientation of an ODF when an affine transformation is applied and subsequently, define the diffeomorphic group action to be applied on the ODF based on this reorientation. We incorporate the Riemannian metric of ODFs for quantifying the similarity of two HARDI images into a variational problem defined under the large deformation diffeomorphic metric mapping (LDDMM) framework. We finally derive the gradient of the cost function in both Riemannian spaces of diffeomorphisms and the ODFs, and present its numerical implementation. Both synthetic and real brain HARDI data are used to illustrate the performance of our registration algorithm

Bridge Simulation and Metric Estimation on Landmark Manifolds

We present an inference algorithm and connected Monte Carlo based estimation procedures for metric estimation from landmark configurations distributed according to the transition distribution of a Riemannian Brownian motion arising from the Large Deformation Diffeomorphic Metric Mapping (LDDMM) metric. The distribution possesses properties similar to the regular Euclidean normal distribution but its transition density is governed by a high-dimensional PDE with no closed-form solution in the nonlinear case. We show how the density can be numerically approximated by Monte Carlo sampling of conditioned Brownian bridges, and we use this to estimate parameters of the LDDMM kernel and thus the metric structure by maximum likelihood

Doctor of Philosophy

dissertationThe statistical study of anatomy is one of the primary focuses of medical image analysis. It is well-established that the appropriate mathematical settings for such analyses are Riemannian manifolds and Lie group actions. Statistically defined atlases, in which a mean anatomical image is computed from a collection of static three-dimensional (3D) scans, have become commonplace. Within the past few decades, these efforts, which constitute the field of computational anatomy, have seen great success in enabling quantitative analysis. However, most of the analysis within computational anatomy has focused on collections of static images in population studies. The recent emergence of large-scale longitudinal imaging studies and four-dimensional (4D) imaging technology presents new opportunities for studying dynamic anatomical processes such as motion, growth, and degeneration. In order to make use of this new data, it is imperative that computational anatomy be extended with methods for the statistical analysis of longitudinal and dynamic medical imaging. In this dissertation, the deformable template framework is used for the development of 4D statistical shape analysis, with applications in motion analysis for individualized medicine and the study of growth and disease progression. A new method for estimating organ motion directly from raw imaging data is introduced and tested extensively. Polynomial regression, the staple of curve regression in Euclidean spaces, is extended to the setting of Riemannian manifolds. This polynomial regression framework enables rigorous statistical analysis of longitudinal imaging data. Finally, a new diffeomorphic model of irrotational shape change is presented. This new model presents striking practical advantages over standard diffeomorphic methods, while the study of this new space promises to illuminate aspects of the structure of the diffeomorphism group

Activity Representation from Video Using Statistical Models on Shape Manifolds

Activity recognition from video data is a key computer vision problem with applications in surveillance, elderly care, etc. This problem is associated with modeling a representative shape which contains significant information about the underlying activity. In this dissertation, we represent several approaches for view-invariant activity recognition via modeling shapes on various shape spaces and Riemannian manifolds. The first two parts of this dissertation deal with activity modeling and recognition using tracks of landmark feature points. The motion trajectories of points extracted from objects involved in the activity are used to build deformation shape models for each activity, and these models are used for classification and detection of unusual activities. In the first part of the dissertation, these models are represented by the recovered 3D deformation basis shapes corresponding to the activity using a non-rigid structure from motion formulation. We use a theory for estimating the amount of deformation for these models from the visual data. We study the special case of ground plane activities in detail because of its importance in video surveillance applications. In the second part of the dissertation, we propose to model the activity by learning an affine invariant deformation subspace representation that captures the space of possible body poses associated with the activity. These subspaces can be viewed as points on a Grassmann manifold. We propose several statistical classification models on Grassmann manifold that capture the statistical variations of the shape data while following the intrinsic Riemannian geometry of these manifolds. The last part of this dissertation addresses the problem of recognizing human gestures from silhouette images. We represent a human gesture as a temporal sequence of human poses, each characterized by a contour of the associated human silhouette. The shape of a contour is viewed as a point on the shape space of closed curves and, hence, each gesture is characterized and modeled as a trajectory on this shape space. We utilize the Riemannian geometry of this space to propose a template-based and a graphical-based approaches for modeling these trajectories. The two models are designed in such a way to account for the different invariance requirements in gesture recognition, and also capture the statistical variations associated with the contour data

Numerical inversion of SRNFs for efficient elastic shape analysis of star-shaped objects.

The elastic shape analysis of surfaces has proven useful in several application areas, including medical image analysis, vision, and graphics. This approach is based on defining new mathematical representations of parameterized surfaces, including the square root normal field (SRNF), and then using the L2 norm to compare their shapes. Past work is based on using the pullback of the L2 metric to the space of surfaces, performing statistical analysis under this induced Riemannian metric. However, if one can estimate the inverse of the SRNF mapping, even approximately, a very efficient framework results: the surfaces, represented by their SRNFs, can be efficiently analyzed using standard Euclidean tools, and only the final results need be mapped back to the surface space. Here we describe a procedure for inverting SRNF maps of star-shaped surfaces, a special case for which analytic results can be obtained. We test our method via the classification of 34 cases of ADHD (Attention Deficit Hyperactivity Disorder), plus controls, in the Detroit Fetal Alcohol and Drug Exposure Cohort study. We obtain state-of-the-art results