Modeling relations between components of 3D objects is essential for many geometry editing tasks. In this talk, I will present two approaches for structure-aware 3D shape modeling, for shape completion and synthesis. In the first, we proposed a framework for analyzing uncurated collections of 3D models and learning two important types of semantic relations among partial shapes: complementarity and interchangeability. To do so, we embed partial shapes as fuzzy sets in dual embedding spaces, and model the two relations as fuzzy set operations performed across the embedding spaces, and within each space, respectively. In the second approach, we proposed a neural network architecture for semantic structure-aware 3D shape modeling. It produces a factorized shape embedding space, where the semantic structure of the shape collection translates into a data-dependent sub-space factorization. In this space, shape composition and decomposition become simple linear operations on the embedding coordinates. We further model shape assembly using an explicit learned part deformation module, utilizing a 3D spatial transformer network. Together, these two components allow us to perform part-level shape decomposition and composition, unattainable by previous approaches.
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