40-Issue 2

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https://diglib.eg.org/handle/10.2312/2633016

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Browsing 40-Issue 2 by Subject "Computational geometry"

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    Orthogonalized Fourier Polynomials for Signal Approximation and Transfer
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Maggioli, Filippo; Melzi, Simone; Ovsjanikov, Maks; Bronstein, Michael M.; Rodolà, Emanuele; Mitra, Niloy and Viola, Ivan
    We propose a novel approach for the approximation and transfer of signals across 3D shapes. The proposed solution is based on taking pointwise polynomials of the Fourier-like Laplacian eigenbasis, which provides a compact and expressive representation for general signals defined on the surface. Key to our approach is the construction of a new orthonormal basis upon the set of these linearly dependent polynomials. We analyze the properties of this representation, and further provide a complete analysis of the involved parameters. Our technique results in accurate approximation and transfer of various families of signals between near-isometric and non-isometric shapes, even under poor initialization. Our experiments, showcased on a selection of downstream tasks such as filtering and detail transfer, show that our method is more robust to discretization artifacts, deformation and noise as compared to alternative approaches.
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    Restricted Power Diagrams on the GPU
    (The Eurographics Association and John Wiley & Sons Ltd., 2021) Basselin, Justine; Alonso, Laurent; Ray, Nicolas; Sokolov, Dmitry; Lefebvre, Sylvain; Lévy, Bruno; Mitra, Niloy and Viola, Ivan
    We propose a method to simultaneously decompose a 3D object into power diagram cells and to integrate given functions in each of the obtained simple regions.We offer a novel, highly parallel algorithm that lends itself to an efficient GPU implementation. It is optimized for algorithms that need to compute many decompositions, for instance, centroidal Voronoi tesselation algorithms and incompressible fluid dynamics simulations. We propose an efficient solution that directly evaluates the integrals over every cell without computing the power diagram explicitly and without intersecting it with a tetrahedralization of the domain. Most computations are performed on the fly, without storing the power diagram. We manipulate a triangulation of the boundary of the domain (instead of tetrahedralizing the domain) to speed up the process. Moreover, the cells are treated independently one from another, making it possible to trivially scale up on a parallel architecture. Despite recent Voronoi diagram generation methods optimized for the GPU, computing integrals over restricted power diagrams still poses significant challenges; the restriction to a complex simulation domain is difficult and likely to be slow. It is not trivial to determine when a cell of a power diagram is completely computed, and the resulting integrals (e.g. the weighted Laplacian operator matrix) do not fit into fast (shared) GPU memory. We address all these issues and boost the performance of the state-of-the-art algorithms by a factor 2 to 3 for (unrestricted) Voronoi diagrams and ax50 speed-up with respect to CPU implementations for restricted power diagrams. An essential ingredient to achieve this is our new scheduling strategy that allows us to treat each Voronoi/power diagram cell with optimal settings and to benefit from the fast memory.