In this thesis we extend signal processing techniques originally formulated in the context of image processing to techniques that can be applied to signals on arbitrary triangles meshes. We develop methods for the two most common representations of signals on triangle meshes: signals sampled at the vertices of a finely tessellated mesh, and signals mapped to a coarsely tessellatedmesh through texture maps.Our first contribution is the combination of Lagrangian Integration and the Finite Elements Method in the formulation of two signal processing tasks: Shock Filters for texture and geometry sharpening, and Optical Flow for texture registration.Our second contribution is the formulation of Gradient-Domain processing within the texture atlas. We define a function space that handles chart discontinuities, and linear operators that capture the metric distortion introduced by the parameterization.Our third contribution is the construction of a spatiotemporal atlas parameterization for evolving meshes. Our method introduces localized remeshing operations and a compact parameterization that improves geometry and texture video compression. We show temporally coherent signal processing using partial correspondences.
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