slides - DIMAp

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, ST , in R3 , with an empty boundary,
a positive integer k, and a positive real number , we want to
...
ST
2
Problem Statement
find a C k surface, S, in R3 such that . . .
S ⊂ R3
3
Problem Statement
there exists a homeomorphism, h : |ST | → S, satisfying
h(v) − v ≤ for every vertex v in ST .
|ST |
S
4
Problem Statement
REMARK:
ST is expected to be “very large” (∼ 106 vertices).
5
An Adaptive Fitting Approach
Step 1:
Simplify ST using the Four-Face Clusters algorithm.
ST
ST
See [Velho, 2001]
6
An Adaptive Fitting Approach
Algorithm preserves topology.
Each vertex of ST is a vertex of ST .
ST
ST
ST is also a hierarchical multiresolution mesh.
7
An Adaptive Fitting Approach
Step 2:
Map the edges of ST to ST using geodesics.
8
An Adaptive Fitting Approach
Step 2: (continuation...)
Re-triangulate ST 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 ST 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
We use Floater’s parametrization for each “macro triangle”.
11
An Adaptive Fitting Approach
Step 4:
∞
,
we
define
a
C
function,
For
each
vertex
v
∈
S
Non-polynomial convex
combination of
Bézier patches!
T
γv : R2 → R3
through a least squares fitting using the parameter points in
the polygon associated with v and their corresponding vertices
in ST .
ST
w
w
12
An Adaptive Fitting Approach
Step 4: (continuation...)
Compute the approximation error: γv (w ) − w.
If γv (w ) − w ≥ then ST must be locally refined.
Refinement is simple: we take advantage of the hierarchical
and multiresolusion structure of ST . 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 R3 using the
topology of ST , the vertices of ST , and the parametrizations
computed in Step 3.
M
ST
M is the image of M in R3 .
14
Gluing Data and PPM’s
Rm
θ1
θ2
Rn
ϕ12
Ω1
Ω21
Ω12
ϕ21
See [Grimm and Hughes, 1995]
15
Ω2
Gluing Data and PPM’s
Rm
θ1
θ2
θi (p)
Rn
θj ◦ ϕ21 (p)
ϕ12
Ω1
p
Ω21
Ω12
ϕ21
16
Ω2
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 ST 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 Proceedings of the International Conference on Shape Modeling & Applications (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 Topology Using Manifolds. In Proceedings of the ACM SIGGRAPH, 1995.
• M. Siqueira, D. Xu, and J. Gallier. Construction of C ∞ Surfaces
from Triangular Meshes Using Parametric Pseudo-Manifolds. Technical Report MS-CIS-08-14, Department of Computer and Information
Science, University of Pennsylvania, 2008.
20