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Approximating Functions on a Mesh with Restricted Voronoi Diagrams

Vincent Nivoliers 1 Bruno Lévy 2
2 ALICE - Geometry and Lighting
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : We propose a method that computes a piecewise constant approximation of a function defined on a mesh. The approximation is associated with the cells of a restricted Voronoi diagram. Our method optimizes an objective function measuring the quality of the approximation. This objective function depends on the placement of the samples that define the restricted Voronoi diagram and their associated function values. We study the continuity of the objective function, derive the closed-form expression of its derivatives and use them to design a numerical solution mechanism. The method can be applied to a function that has discontinuities, and the result aligns the boundaries of the Voronoi cells with the discontinuities. Some examples are shown, suggesting potential applications in image vectorization and compact representation of lighting.
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Vincent Nivoliers, Bruno Lévy. Approximating Functions on a Mesh with Restricted Voronoi Diagrams. Computer Graphics Forum, Wiley, 2013, 32 (5), pp.83-92. ⟨10.1111/cgf.12175⟩. ⟨hal-00929994⟩



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