Stochastic Filtering on Shape Manifolds

Abstract

This thesis addresses the problem of learning the dynamics of deforming objects in image time series. In many biomedical imaging and computer vision applications it is important to satisfy certain geometric constraints which traditional time series methods are not capable of handling. We focus on building topology-preserving spatio-temporal stochastic models for shape deformation, which we combine with the observed images to obtain robust object tracking. The shape of the object is modeled as obtained through the action of a group of diffeomorphisms on the initial object boundary. We formulate a state space model for the diffeomorphic deformation of the object, and implement a particle filter on this shape space to estimate the state of the shape in each video frame. We use a practical method for sampling diffeomorphic shapes in which we generate deformations via flows of finitely generated vector fields. Based on the observations and the proposed samples we obtain an approximate estimate for the posterior distribution of the shape. We present the performance of this framework on various image sequences under different scenarios.

We extend the random perturbation models to diffusion models on the manifold of planar (discretized) shapes whose drift component represents a trend in the shape deformation. To obtain trends intrinsic to the shape, we define the drift as a gradient of appropriate functions defined over the boundary of the shape. Given a sequence of observations from the path of the suggested stochastic differential equations, we propose a likelihood-ratio-based technique to estimate the missing parameters in the drift terms. We show how to reduce the computational burden and improve the robustness of the estimators by constraining the motion of the shapes to a lower-dimensional submanifold equipped with a sub-Riemannian metric. We further discuss how to apply this methodology to obtain estimates when we have only a limited number of observations.