Merging, extending and learning representations for 3D shape matching

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In the last decades, researchers devoted considerable attention to shape matching. Correlating surfaces unlocks otherwise impossible applications and analysis. However, non-rigid objects (like humans) have an enormous range of possibilities to deform their surfaces, making the correspondence challenging to obtain. Computer Graphics and Vision has developed many different representations, each with its peculiarities, conveying different properties and easing different tasks. In this thesis, we exploit, extend, and propose representations to establish correspondences in the non-rigid domain. First, we show how the latent representation of a morphable model can be combined with the spectral embedding, acting as regularization of registration pipelines. We fill the gap in unconstrained problems like occlusion in RGB+D single view or partiality and topological noise for 3D representations. Furthermore, we define a strategy to densify the morphable model discretization and catch variable quantities of details. We also analyze how different discretizations impact correspondence computation. Therefore, we combine intrinsic and extrinsic embeddings, obtaining a robust representation that lets us transfer triangulation among the shapes. Data-driven techniques are particularly relevant to catch complex priors. Hence, we use deep learning techniques to obtain a new high-dimensional embedding for point clouds; in this representation, the objects align with a linear transformation. This approach shows resilience to sparsity and noise. Finally, we connect super-compact latent representations by linking autoencoder latent codes with Laplace-Beltrami operator spectra. This strategy lets us solving a complicated historical problem, enriching the learning framework with geometric properties, and matching objects regardless of their representations. The main contributions of this thesis are the theoretical and practical studies of representations, the advancement in shape matching, and finally, the data and code produced and publicly available.