Variational Shape Approximation SIGGRAPH 2004 David CohenSteiner Pierre

Variational Shape Approximation SIGGRAPH 2004 David Cohen-Steiner Pierre Alliez Mathieu Desbrun Duke U. INRIA U. of So. Cal. 22/02/2021

Outline - Introduction - Shape Approximation - Optimizing Shape Proxies - Results - 22/02/2021 2

Introduction • A method concise, faithful approximation of complex 3 D datasets is key to reducing the computational cost of graphics application. • Using the concept of geometric proxies. 22/02/2021 400 K-face 5 K-face 3

Related Work • Partitioning – Greedy – Some of the metric used for clustering can be proven asymptotically optimal (i. e. , for infinitesimal triangles) for the L 2 metric. [Heckbert and Garland 1999. Optimal Triangulation and Quadric-Based Surface Simplification] • Global optimization – Energy Functional [Hoppe et al. 1993] –. . . 22/02/2021 4

Shape Approximation • Approximation Theory a) Functional Setting b) Height Field Approximation c) Arbitrary Geometry 22/02/2021 5

Shape Approximation • Approximation Theory a) Functional Setting p Strong results in Lp metric. p Relies on parameterization b) Height Field Approximation c) Arbitrary Geometry 22/02/2021 6

Shape Approximation • Approximation Theory a) Functional Setting b) Height Field Approximation p Obvious Parameterization. p Few results about optimality in Lp metric. c) Arbitrary Geometry 22/02/2021 7

Shape Approximation • Approximation Theory a) Functional Setting b) Height Field Approximation c) Arbitrary Geometry p 22/02/2021 Metric commonly used Lp. 8

Shape Approximation • Propose to reformulate the problem of surface approximation by introducing the notions of shape proxies and variational partitions. – Shape proxies: ① A mesh partition R = {Ri} ② A shape proxy is a pair Pi = ( Xi , Ni ); Xi = point, Ni = normal – Variational partitions: iteratively seek a partition that minimize a given error. 22/02/2021 9

Shape Approximation • Metrics of Proxies p L 2 metric 22/02/2021 10

Shape Approximation • Metrics of Proxies p L 2, 1 metric 22/02/2021 11

Optimizing Shape Proxies • Background on Lloyd’s Clustering Algorithm. p Voronoi Diagram + finds the centroid of each set in the partition • Lloyd’s method hinges on the two phases 22/02/2021 13

Optimizing Shape Proxies • Background on Lloyd’s Clustering Algorithm. p Voronoi Diagram + finds the centroid of each set in the partition • Lloyd’s method hinges on the two phases 22/02/2021 14

Optimizing Shape Proxies • Background on Lloyd’s Clustering Algorithm. p Voronoi Diagram + finds the centroid of each set in the partition (1) (2) (3) (4) • Lloyd’s method hinges on the two phases 22/02/2021 15

Optimizing Shape Proxies • Background on Lloyd’s Clustering Algorithm. • Lloyd’s method hinges on the two phases: p. Geometry Partitioning It is used an error-minimizing region-growing algorithm. The object is segmented in non-overlapping regions. p. Proxy Fitting For each partition it is computed an optimal local representative. 22/02/2021 16

Results 300 proxies, 346 triangles 22/02/2021 50 proxies, 100 triangles 17

Results 22/02/2021 18

Results 700 K triangles 5 min 10 min 22/02/2021 19