Fast Approximation of Distance Between Elastic Curves using Kernels

Abstract Elastic shape analysis on non-linear Riemannian manifolds provides an efficient and elegant way for simultaneous comparison and registration of non-rigid shapes. In such formulation, shapes become points on some high dimensional shape space. A geodesic between two points corresponds to the optimal deformation needed to register one shape onto another. The length of the geodesic provides a proper metric for shape comparison. However, the computation of geodesics, and therefore the metric, is computationally very expensive as it involves a search over the space of all possible rotations and reparameterizations. This problem is even more important in shape retrieval scenarios where the query shape is compared to every element in the collection to search. In this paper, we propose a new procedure for metric approximation using the framework of kernel functions. We will demonstrate that this provides a fast approximation of the metric while preserving its invariance properties

Elastic Shape Models for Face Analysis Using Curvilinear Coordinates

International audienceThis paper studies the problem of analyzing variability in shapes of facial surfaces using a Rie- mannian framework, a fundamental approach that allows for joint matchings, comparisons, and deformations of faces under a chosen metric. The starting point is to impose a curvilinear coordinate system, named the Darcyan coordinate system, on facial surfaces; it is based on the level curves of the surface distance function measured from the tip of the nose. Each facial surface is now represented as an indexed collection of these level curves. The task of finding optimal deformations, or geodesic paths, between facial surfaces reduces to that of finding geodesics between level curves, which is accomplished using the theory of elastic shape analy- sis of 3D curves. Elastic framework allows for nonlinear matching between curves and between points across curves. The resulting geodesics provide optimal elastic deformations between faces and an elastic metric for comparing facial shapes. We demonstrate this idea using examples from FSU face databas

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

Statistical shape analysis in a Bayesian framework; The geometric classification of fluvial sand bodies.

We present a novel shape classification method which is embedded in the Bayesian paradigm. We focus on the statistical classification of planar shapes by using methods which replace some previous approximate results by analytic calculations in a closed form. This gives rise to a new Bayesian shape classification algorithm and we evaluate its efficiency and efficacy on available shape databases. In addition we apply our results to the statistical classification of geological sand bodies. We suggest that our proposed classification method, that utilises the unique geometrical information of the sand bodies, is more substantial and can replace ad-hoc and simplistic methods that have been used in the past. Finally, we conclude this work by extending the proposed classification algorithm for shapes in three-dimensions

Modelling shape fluctuations during cell migration

Cell migration is of crucial importance for many physiological processes, including embryonic development, wound healing and immune response. Defects in cell migration are the cause of chronic in ammatory diseases, mental retardation and cancer metastasis. Cell movement is driven by actin-mediated cell protrusion, substrate adhesion and contraction of the cell body. The emergent behaviour of the intracellular processes described above is a change in the morphology of the cell. This inspires the main hypothesis of this work which is that there is a measurable relationship between cell morphology dynamics and migratory behaviour, and that quantitative models of this relationship can create useful tools for investigating the mechanisms by which a cell regulates its own motility. Here we analyse cell shapes of migrating human retinal pigment epithelial cells with the aim to map cell shape changes to cellular behaviour. We develop a non-linear model for learning the intrinsic low-dimensional structure of cell shape space and use the resultant shape representation to analyse quantitative relationships between shape and migration behaviour. The biggest algorithmic challenge overcome in this thesis was developing a method for efficiently and appropriately measuring the shape difference between pairs of cells that may have come from independent image scenes. This difference measure must be capable of coping with the widely varying morphologies exhibited by migrating epithelial cells. We present a new, rapid, landmark-free, shape difference measure called the Best Alignment Metric (BAM). We show that BAM performs highly within our framework, generating a shape space representation of a very large dataset without any prior information on the importance of any given shape feature. We demonstrate quantitative evidence for a model of cell turning based on repolarisation and discuss the impact our proposed framework could have on the continued study of migratory mechanisms

Structural abnormality of the corticospinal tract in major depressive disorder

BACKGROUND: Scientists are beginning to document abnormalities in white matter connectivity in major depressive disorder (MDD). Recent developments in diffusion-weighted image analyses, including tractography clustering methods, may yield improved characterization of these white matter abnormalities in MDD. In this study, we acquired diffusion-weighted imaging data from MDD participants and matched healthy controls. We analyzed these data using two tractography clustering methods: automated fiber quantification (AFQ) and the maximum density path (MDP) procedure. We used AFQ to compare fractional anisotropy (FA; an index of water diffusion) in these two groups across major white matter tracts. Subsequently, we used the MDP procedure to compare FA differences in fiber paths related to the abnormalities in major fiber tracts that were identified using AFQ. RESULTS: FA was higher in the bilateral corticospinal tracts (CSTs) in MDD (p’s < 0.002). Secondary analyses using the MDP procedure detected primarily increases in FA in the CST-related fiber paths of the bilateral posterior limbs of the internal capsule, right superior corona radiata, and the left external capsule. CONCLUSIONS: This is the first study to implicate the CST and several related fiber pathways in MDD. These findings suggest important new hypotheses regarding the role of CST abnormalities in MDD, including in relation to explicating CST-related abnormalities to depressive symptoms and RDoC domains and constructs