Numerical Computation of Geodesics in the Framework of Metamorphosis

Johns Hopkins University, Baltimore MD

Investigators

Abstract

The proposed research focuses on metamorphosis for shape analysis, which relies on a shape transformation model within which shape variation is coupled with other transformations of the data, permitting topological changes, or partial advection of attributes attached to the deformed objects. This results in a versatile framework in which many different models can be devised, based on any mathematical structure that can both be advected by diffeomorphisms and embedded in a Hilbert or Riemannian space. This construction equips the space of deformable objects of interest with a new Riemannian metric, allowing for the comparison of these objects, and for the use of tools associated to data analysis in Riemannian manifolds, like the representation of data sets in exponential charts. The research will involve models of metamorphosis in which the deformable structures are represented by images, densities, or measures, in two or three dimensions. One of the main issues in this context is the computation of geodesics, either as a variational problem (shortest path between two points in the manifold) or as an initial value problem (solving the Euler-Lagrange equation for the evolution of geodesics). The numerical analysis of both problems is challenging, especially when one adds the requirement for the two solutions to be numerically consistent, in the sense that discrete solutions of the first problem coincide with discrete solutions of the second one, which is important for applications. This research will address these issues, by developing variational integrators for the initial value problems, and shooting methods for the boundary value problems, in contexts that will involve solutions that combine smooth and singular components. The PI and collaborators will also deploy and extend of a comprehensive software that provides a collection of algorithms associated to diffeomorphic matching. The goal of shape analysis is to understand and represent variations of shapes in data sets of deformable objects (like collections of landmarks, images, curves or surfaces). This issue is important, in particular, for the characterization of anatomical variations in medical images, and of their relation with pathologies. One of the main areas of applications in this context is known as Computational Anatomy, and methods from mathematical shape analysis have already been used for  several successful applications. Examples of developments in this domain include collaborations of the PI with researchers at the Kennedy Krieger Institute in Baltimore, or at the Institute for Computational Medicine at  Johns Hopkins University, on the analysis of brain disease and of cardiac failure.  The theory and tools that will be developed in this research will enable the analysis of situations that cannot be  handled by previous methods, which work under the assumption that anatomical variation can be essentially described by smooth changes of shape. The proposed approach, called metamorphosis, will be able to address cases for which these assumptions are not satisfied, and make possible, for example, the analysis of images that include dramatic changes between subjects. This includes the analysis of datasets measuring the evolution of tumors, or describing brain recovery after a major stroke. The research will contribute to the emergence of new solutions in such contexts, and make the related software available to the scientific community.