Planar Minimization Diagrams via Subdivision with Applications to Anisotropic Voronoi Diagrams

• Published in: Computer graphics forum Vol. 35; no. 5; pp. 229 - 247
• Main Authors: Bennett, H., Papadopoulou, E., Yap, C.
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.08.2016
• Subjects: Algorithms; Anisotropy; Categories and Subject Descriptors (according to ACM CCS); F.2.2 [Analysis of Algorithms and Problem Complexity]: Nonnumerical Algorithms and Problems-Geometrical problems and computations; G.1.0 [Numerical Analysis]: General-Interval arithmetic; G.1.2 [Numerical Analysis]: Approximation-Approximation of surfaces and contours; G.4 [Mathematical Software]: -Algorithm design and analysis; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Geometric algorithms; Image processing systems; Mathematical analysis; Mathematical models; Minimization; Optimization; Scalars; Studies; Subdivisions
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12979

Let X = {f1, …, fn} be a set of scalar functions of the form fi : ℝ2 → ℝ which satisfy some natural properties. We describe a subdivision algorithm for computing a clustered ε‐isotopic approximation of the minimization diagram of X. By exploiting soft predicates and clustering of Voronoi vertices, our algorithm is the first that can handle arbitrary degeneracies in X, and allow scalar functions which are piecewise smooth, and not necessarily semi‐algebraic. We apply these ideas to the computation of anisotropic Voronoi diagram of polygonal sets; this is a natural generalization of anisotropic Voronoi diagrams of point sites, which extends multiplicatively weighted Voronoi diagrams. We implement a prototype of our anisotropic algorithm and provide experimental results.