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.
This projects provides new theoretical and computational approaches for the comparison of images and forms, within a framework called metamorphosis, in which objects are compared based on smooth changes in their shapes (deformations) combined with more abrupt variations, due, for example, to the emergence of new structures in one of the objects. Even if the research remained at a mathematical level, this setting can have important applications in computer vision, and in the analysis of medical images (computational anatomy), in which variations in shape or structure of internal organs can often be related to pathologies. We have studied two types of objects, with very different properties. The first type, corresponding to the mathematical notion of measures, allows for singular structures, the simplest example being configurations of points in space. Using this example to simplify, metamorphosis compares such configurations by optimizing a continuous process that transforms one configuration into the other. During this process, points may be displaced (inducing shape changes), removed or added (inducing structure changes). Studying the kind of transformation that occurs during this process turned out to be rather challenging and was carried out based on the theory of pseudo-differential operators. These results were published in 2012 in the Journal of Geometric Mechanics. The second problem, which was considered in the last year of the grant, addressed another context in which the considered objects are images that can evolve via deformations combined by transformations of their content as if obtained by a blurry paintbrush. In this case, optimal transformations can be represented in the form of moving points that deform the image during their motion, while also repainting it. A new algorithm, implementing a so-called particle method, has been designed in order to optimize their evolution, subject to the constraint of transforming a source image into a target one. A research paper describing this approach and results is being completed at the time of writing this report. The supporting images, using the LEAFSNAP database, illustrate an experiment in which a source and a target images are provided. The metamorphosis algorithm optimizes a continous transformation between the source and the target, and an intermediate position in this process is provided. This intermediate leaf is purely synthetic: it is an interpolation between the source and the target.
