Central Sets of Simplicial Complexes
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2021
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University of Hawaii at Manoa
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The notion of a central set, that is, a subset of a shape located in its "center", has been studied (under various disguises) for quite some time. It became well known in the computer science community under the name of medial axis and found numerous applications in such areas as shape recognition and motion planning despite being unstable under even $C^1$ perturbations.A lot of work has been done in an attempt to fix this issue, with the notion of a $\lambda$-medial axis providing, arguably, the most general and mathematically rigorous approach. It defines a subset of a medial axis that is stable under relatively small Hausdorff perturbations. However, choosing an appropriate value of the $\lambda$ parameter for a $\lambda$-medial axis may be a challenge in certain situations, especially considering that the existing stability result does not strengthen when only homotopic perturbations are considered, which is an important use case in many applications. In this dissertation, a new approach to defining a central set of a simplicial complex is presented. The focus on simplicial complexes is not arbitrary and stems from the fact that a vast majority of medial axis computations are done using simplicial (or polygonal) approximations. It is demonstrated that our notion of a central set is well defined, is efficiently computable, and has several useful properties. In particular, it is proven that a central set of a simplicial manifold satisfies certain stability requirements under homotopic perturbations. Furthermore, new central sets are considered in two separate contexts, based on the type of perturbations allowed -- homotopic perturbations, and general, non-homotopic ones. Algorithms for computing central sets in these two cases have been developed and implemented, and a variety of computational experiments have been performed. The results suggest that both notions may be stable at least under some restrictions. It is important to mention that, in the context of homotopic perturbations, our new approach allows us to define a central set of a simplicial complex of any codimension. Consequently, one can construct a 1-dimensional representation of a possibly high dimensional shape that preserves certain geometric features, which is illustrated on a 4-dimensional example.
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