Provably Good Quality Multi-triangulation of Surfaces

Transcrição

1 Provably Good Quality Multi-triangulation of Surfaces Esdras Medeiros (Departamento de Matemática - UFC) and Marcelo Siqueira (DIMAP - UFRN) Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 1

2 Manifold What is a Manifold? Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 2

3 Manifold What is a Manifold?...Manifold is hard to define, but you know one when you see one. S. Weinberger Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 3

4 Manifold What is a Manifold? It is a topological space that is locally Euclidean. Equivalently, any object that can be charted is a manifold. Smooth manifolds are manifolds for which overlapping charts relate smoothly to each other. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 4

5 Parametric Pseudo-Manifolds Manifold is not constructive. It assumes what a manifold is by assuming that the the space already exists. A parametric pseudo-manifold (PPM) is a topological space defined from a sets of gluing data. Under certain conditions, PPM's are manifolds in the Euclidean space. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 5

6 Basic Construction Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 6

7 Problem Statement Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 7

8 Problem Statement Desired properties for base mesh: As simplified as possible. Good approximation from dense mesh. Good triangle quality (e.g. aspect ratio, max-min angle). Valence ~ 6 for each vertex. Low variance of areas among neighbor vertices. Randomized distribution of vertices (e.g. Blue Noise). Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 8

9 Possible Approaches Re-Sampling and Reconstruction Provides good triangle quality. Hard to handle topology and sampling conditions.. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 9

10 Possible Approaches Re-Sampling and Reconstruction Provides good triangle quality. Hard to handle topology and sampling conditions. Mesh Simplification Can easily handle topology. Do not provide guarantees on good triangle quality. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 10

11 Possible Approaches Re-Sampling and Reconstruction Provides good triangle quality. Hard to handle topology and sampling conditions. Mesh Simplification Can easily handle topology. Do not provide guarantees on good triangle quality. We build a hybrid approach Our approach combines the advantages of the methods above. Under certain conditions, we guarantee topology preservation, good triangle quality and asap base mesh. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 11

12 Method Overview Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 12

13 Key Concepts Localized Restricted r-neighborhood. Localized Restricted VD/DT. Localized Restricted Poisson Disk Sampling (PDS). Hierarchical Localized Restricted PDS. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 13

14 Localized Restricted r-neighborhood Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 14

15 Localized Restricted VD/DT Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 15

16 Localized Restricted VD/DT Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 16

17 Localized Restricted VD/DT Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 17

18 RDT vs LRDT LRDT is more resilient to low resolution samplings. It can correctly represent the topology of the surface where RDT fails. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 18

19 Localized Restricted PDS Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 19

20 Hierarchical LR-PDS Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 20

21 Real-Time Stippling Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 21

22 Simplification - The simplification algorithm follows the hierarchy of Points by applying local Delaunay Modifications as much as possible. Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 22

23 Simplification Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 23

24 Multi-triangulation DAG Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 24

25 Guarantees Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 25

26 Guarantees Misplaced pair Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 26

27 Min Angle - Results Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 27

28 A few results Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 28

29 Adaptive Meshes Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 29

30 Adaptive Meshes Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 30

31 Dense-to-Coarse Parametrization Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 31

32 Dense-to-Coarse Parametrization Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 32

33 Thank you! Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 33

34 Parabéns Luiz Velho! Luiz Velho IMPA, Rio de Janeiro, RJ, Brazil 34