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Abstract

In this thesis, we investigate a certain type of local similarities between geometric shapes. We analyze the surface of a shape and find all points that are contained inside identical, spherical neighborhoods of a radius r. This allows us to decompose surfaces into canonical sets of building blocks, which we call microtiles. We show that the microtiles of a given object can be used to describe a complete family of related shapes. Each of these shapes is locally similar to the original, meaning that it contains identical r-neighborhoods, but can have completely different global structure. This allows for using r-microtiling for inverse modeling of shape variations and we develop a method for shape decomposi
tion into rigid, 3D manufacturable building blocks that can be used to physically assemble shape collections. We obtain a small set of constructor pieces that are well suited for manufacturing and assembly by a novel method for tiling grammar simplification: We consider the connection between microtiles and noncontext-free tiling grammars and optimize a graph-based representation, finding a good balance between expressiveness, simplicity and ease of assembly. By changing the objective function, we can re-purpose the grammar simplification method for mesh compression. The microtiles of a model encode its geometrically redundant parts, which can be used for creating shape representations with minimal memory footprints. Altogether, with this work we attempt to give insights into how rigid partial symmetries can be efficiently computed and used in the context of inverse modeling of shape families, shape understanding, and compression.