Provably Good Quality Multi-triangulation of Surfaces Esdras Medeiros (Departamento de Matemática - UFC) and Marcelo Siqueira (DIMAP - UFRN) Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 1
Manifold What is a Manifold? Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 2
Manifold What is a Manifold?...Manifold is hard to define, but you know one when you see one. S. Weinberger Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 3
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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 4
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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 5
Basic Construction Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 6
Problem Statement Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 7
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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 8
Possible Approaches Re-Sampling and Reconstruction Provides good triangle quality. Hard to handle topology and sampling conditions.. Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 9
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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 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. 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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 11
Method Overview Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 12
Key Concepts Localized Restricted r-neighborhood. Localized Restricted VD/DT. Localized Restricted Poisson Disk Sampling (PDS). Hierarchical Localized Restricted PDS. Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 13
Localized Restricted r-neighborhood Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 14
Localized Restricted VD/DT Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 15
Localized Restricted VD/DT Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 16
Localized Restricted VD/DT Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 17
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 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 18
Localized Restricted PDS Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 19
Hierarchical LR-PDS Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 20
Real-Time Stippling Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 21
Simplification - The simplification algorithm follows the hierarchy of Points by applying local Delaunay Modifications as much as possible. Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 22
Simplification Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 23
Multi-triangulation DAG Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 24
Guarantees Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 25
Guarantees Misplaced pair Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 26
Min Angle - Results Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 27
A few results Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 28
Adaptive Meshes Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 29
Adaptive Meshes Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 30
Dense-to-Coarse Parametrization Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 31
Dense-to-Coarse Parametrization Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 32
Thank you! Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 33
Parabéns Luiz Velho! Luiz Velho 60-2017 IMPA, Rio de Janeiro, RJ, Brazil 34