Marcelo Ferreira Siqueira UFMS - Brazil Joint work with Jean - PowerPoint PPT Presentation

Marcelo Ferreira Siqueira UFMS - Brazil Joint work with Jean Gallier Dimas Morera Luis Gustavo Nonato CIS - UPenn - USA ICMC - USP - Brazil ICMC - USP - Brazil Dianna Xu Luiz Velho CS - Bryn Mawr - USA IMPA - Brazil

Problem Statement Given a simplicial surface, S T , in R 3 , with an empty boundary, a positive integer k , and a positive real number ǫ , we want to . . . S T 2

Problem Statement find a C k surface, S , in R 3 such that . . . S ⊂ R 3 3

Problem Statement there exists a homeomorphism, h : | S T | → S , satisfying � h ( v ) − v � ≤ ǫ for every vertex v in S T . | S T | S 4

Problem Statement REMARK : S T is expected to be “very large” ( ∼ 10 6 vertices). 5

An Adaptive Fitting Approach Step 1 : Simplify S T using the Four-Face Clusters algorithm. S ′ S T T See [Velho, 2001] 6

An Adaptive Fitting Approach Algorithm preserves topology. Each vertex of S ′ T is a vertex of S T . S ′ S T T S ′ T is also a hierarchical multiresolution mesh. 7

An Adaptive Fitting Approach Step 2 : Map the edges of S ′ T to S T using geodesics. 8

An Adaptive Fitting Approach Step 2 : (continuation...) Re-triangulate S T so that the geodesics are covered by edges. Adapted from the algorithm in [Morera, Carvalho and Velho, 2005] 9

An Adaptive Fitting Approach Step 3 : Parametrize the star of each vertex v of S ′ T over a regular polygon inscribed in a unit circle in R 2 and containing the vertex (0,1). v 10

An Adaptive Fitting Approach Step 3 : (continuation...) Map the vertices of S T to the regular polygons. v We use Floater’s parametrization for each “macro triangle”. 11

An Adaptive Fitting Approach Step 4 : T , we define a C ∞ function, For each vertex v ∈ S ′ Non-polynomial convex combination of B´ ezier patches! γ v : R 2 → R 3 through a least squares fitting using the parameter points in the polygon associated with v and their corresponding vertices in S T . S T w w ′ 12

An Adaptive Fitting Approach Step 4 : (continuation...) Compute the approximation error: � γ v ( w ′ ) − w � . If � γ v ( w ′ ) − w � ≥ ǫ then S ′ T must be locally refined. Refinement is simple : we take advantage of the hierarchical and multiresolusion structure of S ′ T . This comes from the simplification algorithm. After all faces are refined, we go back to Step 2. 13

An Adaptive Fitting Approach Step 5 : We define a parametric pseudo-manifold , M , in R 3 using the topology of S ′ T , the vertices of S T , and the parametrizations computed in Step 3. M S ′ T M is the image of M in R 3 . 14

Gluing Data and PPM’s R m θ 1 θ 2 R n ϕ 12 Ω 21 Ω 1 Ω 2 Ω 12 ϕ 21 See [Grimm and Hughes, 1995] 15

Gluing Data and PPM’s R m θ 1 θ 2 θ j ◦ ϕ 21 ( p ) θ i ( p ) R n ϕ 12 Ω 21 Ω 1 Ω 2 Ω 12 p ϕ 21 16

Gluing Data and PPM’s See [Siqueira, Xu, and Gallier, 2008] 17

Concluding Remarks The overall idea (mesh simplification + mesh parametrization) of the previous adaptive fitting is not new, but the components (i.e., geodesics and parametric pseudo-manifolds ) used in our solution make it simpler and/or more powerful than similar approaches. The work is still in progress... Code for computing geodesics and re-triangulate S T is not stable. 18

Concluding Remarks Code for computing parametric pseudo-surfaces is finished. 19

References • L. Velho. Mesh Simplification Using Four-Face Clusters. In Proceed- ings of the International Conference on Shape Modeling & Applica- tions (SMI), 2001. • D. Morera, L. Velho, and P.C. Carvalho. Computing Geodesics on Triangular Meshes, Computer & Graphics , 29(5): 667-675, 2005. • C. M. Grimm and J. F. Hughes. Modeling Surfaces of Arbitrary Topol- ogy Using Manifolds. In Proceedings of the ACM SIGGRAPH , 1995. Construction of C ∞ Surfaces • M. Siqueira, D. Xu, and J. Gallier. from Triangular Meshes Using Parametric Pseudo-Manifolds . Techni- cal Report MS-CIS-08-14, Department of Computer and Information Science, University of Pennsylvania, 2008. 20