Local Geometry Processing for Deformations of Non-Rigid 3D Shapes

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Geometry processing and in particular spectral geometry processing deal with many different deformations that complicate shape analysis problems for non-rigid 3D objects. Furthermore, point-wise description of surfaces has increased relevance for several applications such as shape correspondences and matching, shape representation, shape modelling and many others. In this thesis we propose four local approaches to face the problems generated by the deformations of real objects and improving the point-wise characterization of surfaces. Differently from global approaches that work simultaneously on the entire shape we focus on the properties of each point and its local neighbour. Global analysis of shapes is not negative in itself. However, having to deal with local variations, distortions and deformations, it is often challenging to relate two real objects globally. For this reason, in the last decades, several instruments have been introduced for the local analysis of images, graphs, shapes and surfaces. Starting from this idea of localized analysis, we propose both theoretical insights and application tools within the local geometry processing domain. In more detail, we extend the windowed Fourier transform from the standard Euclidean signal processing to different versions specifically designed for spectral geometry processing. Moreover, from the spectral geometry processing perspective, we define a new family of localized basis for the functional space defined on surfaces that improve the spatial localization for standard applications in this field. Finally, we introduce the discrete time evolution process as a framework that characterizes a point through its pairwise relationship with the other points on the surface in an increasing scale of locality. The main contribute of this thesis is a set of tools for local geometry processing and local spectral geometry processing that could be used in standard useful applications. The overall observation of our analysis is that localization around points could factually improve the geometry processing in many different applications.