Multiple Shape Registration using Constrained Optimal Control

Lagrangian particle formulations of the large deformation diffeomorphic metric mapping algorithm (LDDMM) only allow for the study of a single shape. In this paper, we introduce and discuss both a theoretical and practical setting for the simultaneous study of multiple shapes that are either stitched to one another or slide along a submanifold. The method is described within the optimal control formalism, and optimality conditions are given, together with the equations that are needed to implement augmented Lagrangian methods. Experimental results are provided for stitched and sliding surfaces

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

Doctor of Philosophy

dissertationStatistical analysis of time dependent imaging data is crucial for understanding normal anatomical development as well as disease progression. The most promising studies are of longitudinal design, where repeated observations are obtained from the same subjects. Analysis in this case is challenging due to the difficulty in modeling longitudinal changes, such as growth, and comparing changes across different populations. In any case, the study of anatomical change over time has the potential to further our understanding of many dynamic processes. What is needed are accurate computational models to capture, describe, and quantify anatomical change over time. Anatomical shape is encoded in a variety of representations, such as medical imaging data and derived geometric information extracted as points, curves, and/or surfaces. By considering various shape representations embedded into the same ambient space as a shape complex, either in 2D or 3D, we obtain a more comprehensive description of the anatomy than provided by an single isolated shape. In this dissertation, we develop spatiotemporal models of anatomical change designed to leverage multiple shape representations simultaneously. Rather than study directly the geometric changes to a shape itself, we instead consider how the ambient space deforms, which allows all embedded shapes to be included simultaneously in model estimation. Around this idea, we develop two complementary spatiotemporal models: a flexible nonparametric model designed to capture complex anatomical trajectories, and a generative model designed as a compact statistical representation of anatomical change. We present several ways spatiotemporal models can support the statistical analysis of scalar measurements, such as volume, extracted from shape. Finally, we cover the statistical analysis of higher dimensional shape features to take better advantage of the rich morphometric information provided by shape, as well as the trajectory of change captured by spatiotemporal models