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Johns Hopkins University
This thesis is concerned with the theory and applications of varifolds to the representation, approximation, and diffeomorphic registration of shapes. Originating from geometric measure theory, the theory of varifolds provides a convenient way to represent geometric shapes like curves, surfaces or, submanifolds both in continuous and discrete settings. Previous works in shape analysis have made use of this representation as a surrogate to design numerically tractable fidelity terms for curve and surface registration problems. So far, these approaches have primarily focused on processing submanifold data and were not designed to handle more general structures. The varifold representation however provides a very flexible framework that is not restricted to submanifolds but its generality has not yet been exploited to its full extent in shape analysis. In this work, we take a step in this direction by considering deformations acting on general varifolds, and propose a mathematical model for diffeomorphic registration of varifolds under a natural group action which we formulate as an optimal control problem. This new framework allows us to tackle diffeomorphic registration problems for a much wider class of geometric objects and lead to a more versatile algorithmic pipeline. Varifold matching frameworks heavily rely on the kernel metrics defined on the varifolds spaces. However, the properties of this type of metrics and their relationships with the classical metrics/topologies on measure spaces have not been investigated thoroughly yet. In this work, we study in detail the construction of kernel metrics on the space of varifold and the resulting topological properties of those metrics. Based on these results, we address the problem of optimal finite approximations (quantization) for kernel metrics, propose a projection-based approach for varifold representation, and show a $\Gamma$-convergence property for the discrete registration functionals. In the last part of this thesis, we tackle the imbalanced shape matching problems, namely the situation in which the source and target shapes involve considerable variations of mass or density which cannot be entirely described by diffeomorphic transformations. We extend our varifold matching model by augmenting the diffeomorphic component with a global or local density changes. Based on the optimality conditions provided by the Pontryagin maximum principle, we derive a shooting algorithm to numerically estimate solutions and illustrate the practical interest of this model for several types of geometric data such as fiber bundles with inconsistent fiber densities or partially observed and incomplete surfaces.
varifolds, diffeomorphic registration, reproducing kernels, optimal control, density variations