Topological Aspects of Maps Between Surfaces
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2022
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Abstract
The generation of high-quality maps between surfaces of 3D shapes is a fundamental task with countless applications in geometry processing. There is a particular demand for maps that offer strict validity properties such as continuity and bijectivity, i.e. surface homeomorphisms. Such maps not only define a geometric one-to-one correspondence between surface points, but also a matching of topological features: an identification of handles and tunnels and how the map wraps around them. Finding a natural, low-distortion surface homeomorphism between a given pair of shapes is a challenging design task that involves both combinatorial (topological) and continuous (geometric) degrees of freedom. However, while powerful methods exist to improve existing homeomorphisms through continuous modifications, these are limited to merely geometric updates, and hence cannot alter map topology. In this light, it is quite surprising that most existing techniques for the initial construction of homeomorphisms do not systematically deal with questions of map topology and instead relegate these issues to user input or ad-hoc solutions. Unfortunately, this lack of reliable and automatic methods for the critically important topological initialization has so far prevented a further automation of homeomorphic surface map generation. In this thesis, we aim to close this practical gap by devising new algorithms that specifically address the map-topological issues underlying the construction of surface homeomorphisms. Our theoretical foundation is the study of the mapping class group, an algebraic structure which characterizes the entire topological design space. We approach the task of map topology generation from two different angles, based on different mapping class representations: We propose a robust method for the construction of maps from sparse landmark correspondences, based on compatible layout embeddings. Our robust embedding strategy systematically searches for short, natural embeddings and therefore reliably avoids a range of sporadic topological initialization errors which can occur with previous heuristic approaches. Additionally, we introduce a novel algorithm to extract topological map descriptions from approximate, non-homeomorphic input maps. Such a purely abstract description of map topology may then be used to guide the construction of a proper homeomorphism. As our inference method is highly robust to a wide range of map defects and imperfect map representations, this effectively allows to delegate the difficult task of finding a natural map topology to specialized shape matching methods, which have grown increasingly capable. These advancements promote the further automation of map generation techniques in two regards: They vastly reduce the need for human supervision, and make the results of automatic shape matching methods accessible for topological initialization.
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