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

CS - Bryn Mawr - USA

Dianna Xu

Joint work with

Jean Gallier

CIS - UPenn - USA

Marcelo Ferreira Siqueira

UFMS - Brazil

Dimas Morera

ICMC - USP - Brazil

Luis Gustavo Nonato

ICMC - USP - Brazil

IMPA - Brazil

Luiz Velho

Problem Statement

2

ST Given a simplicial surface, ST , in R3, with an empty boundary, a positive integer k, and a positive real number ǫ, we want to . . .

Problem Statement

S ⊂ R3

3

find a Ck surface, S, in R3 such that . . .

|ST |

Problem Statement

4

there exists a homeomorphism, h : |ST | → S, satisfying h(v) − v ≤ ǫ for every vertex v in ST . S

Problem Statement

REMARK: ST is expected to be “very large” (∼ 106 vertices).

5

An Adaptive Fitting Approach

6

Step 1: ST S′

T

Simplify ST using the Four-Face Clusters algorithm. See [Velho, 2001]

An Adaptive Fitting Approach

Each vertex of S′

T is a vertex of ST .

ST S′

T

S′

T is also a hierarchical multiresolution mesh.

7

Algorithm preserves topology.

An Adaptive Fitting Approach

Step 2: Map the edges of S′

T to ST using geodesics.

8

An Adaptive Fitting Approach

Re-triangulate ST so that the geodesics are covered by edges. Step 2: (continuation...) 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 R2 and containing the vertex (0,1).

v

10

An Adaptive Fitting Approach

Step 3: (continuation...) Map the vertices of ST to the regular polygons.

v

11

We use Floater’s parametrization for each “macro triangle”.

An Adaptive Fitting Approach

Step 4:

w′

ST

w

12

For each vertex v ∈ S′

T , we define a C∞ function,

γv : R2 → R3 through a least squares fitting using the parameter points in the polygon associated with v and their corresponding vertices in ST . Non-polynomial convex combination of B´ ezier patches!

An Adaptive Fitting Approach

(continuation...) Step 4: 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. Compute the approximation error: γv(w′) − w. If γv(w′) − w ≥ ǫ then S′

T must be locally refined.

13

14

An Adaptive Fitting Approach

S′

T

Step 5: We define a parametric pseudo-manifold, M, in R3 using the topology of S′

T , the vertices of ST , and the parametrizations

computed in Step 3.

M

M is the image of M in R3.

Gluing Data and PPM’s

Ω2 Rn Ω1 θ2 θ1 Ω12 Ω21 ϕ12 ϕ21 Rm See [Grimm and Hughes, 1995]

15

Gluing Data and PPM’s

Ω2 Rn Ω1 θ2 θ1 Ω12 Ω21 ϕ12 ϕ21 Rm p θi(p) θj ◦ ϕ21(p)

16

17

Gluing Data and PPM’s

See [Siqueira, Xu, and Gallier, 2008]

Concluding Remarks

18

The overall idea (mesh simplification + mesh parametrization)

• f 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 ST is not stable.

Concluding Remarks

Code for computing parametric pseudo-surfaces is finished.

19

References

• M. Siqueira, D. Xu, and J. Gallier.

Construction of C∞ Surfaces from Triangular Meshes Using Parametric Pseudo-Manifolds. Techni- cal Report MS-CIS-08-14, Department of Computer and Information Science, University of Pennsylvania, 2008.

• 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-
• gy Using Manifolds. In Proceedings of the ACM SIGGRAPH, 1995.

20