The automatic creation of man-made 3D objects is an active area in computer graphics. Computer-assisted mixing and blending of components or subcomponents from existing example shapes can help users quickly produce interesting and creative designs. A key factor for automating this task is using algorithms that can match compatible parts between objects of different shape and structure. However, due to the coarse correspondence computed by current matching algorithms, automatic shape blending is mainly limited to the substitution of large compatible part sets. In this thesis, we address the problem of relating 3D shapes of different geometry and topology, with applications in shape synthesis. Our goal is to compute a fine-grained mapping between two shapes differing in the geometry, cardinality, and connectivity of their parts, and to use this mapping for continuous shape interpolation. First, we propose a framework for shape matching using a joint geometric and topological transformation. The framework follows the assumption that the best mapping for a pair of shapes is one that results from a shape transformation that minimally distorts the structural properties of a shape. We establish meaningful correspondences between shapes with large topological discrepancy by going beyond shape deformations and incorporating topological operations such as part split, duplication, and merging. We evaluate our correspondence algorithm on a diverse set of shape classes and compare the results to state-of-the-art methods. Second, we propose an algorithm for synthesizing interpolations between structurally different 3D shapes. Our algorithm produces a continuous and plausible shape transformation that gradually morphs the geometry of the individual parts, as well as performs any necessary topology-changing operations. We further demonstrate the utility of our framework by developing intuitive shape creation tools. We show how these tools can allow novice users to synthesize new 3D models from continuous blends of topologically different shapes.
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