SLIDE 1
SMI 2012
As-Conformal-As-Possible Discrete Volumetric Mapping
Gilles-Philippe Paillé Pierre Poulin
University of Montreal, Canada May 23, 2012
2
SMI 2012 As-Conformal-As-Possible Discrete Volumetric Mapping - - PowerPoint PPT Presentation
SMI 2012 As-Conformal-As-Possible Discrete Volumetric Mapping - - PowerPoint PPT Presentation
SMI 2012 As-Conformal-As-Possible Discrete Volumetric Mapping Gilles-Philippe Paill Pierre Poulin University of Montreal, Canada May 23, 2012 Context Parameterization 2D 3D 2 Context Parameterization Some popular codomains Cube
SLIDE 2
SLIDE 3
3
Context
Some popular codomains
Parameterization Cube Sphere Polycube
SLIDE 4
4
Context
- Energy
– Distortion measure
- Measured transformation
– Rotation – Uniform scale – Non-uniform scale – Shear
Parameterization
SLIDE 5
5
Context
- Energy
– Distortion measure
- Measured transformation
– Rotation – Uniform scale – Non-uniform scale – Shear
Parameterization
SLIDE 6
6
Context
- Energy
– Distortion measure
- Desired transformation
– Rotation – Uniform scale – Non-uniform scale – Shear
Parameterization
SLIDE 7
7
Context
- Conformal maps
– Local transformation = Uniform scale * Rotation
Parameterization
SLIDE 8
8
Context
Conformal maps – Jacobian matrix
SLIDE 9
9
Context
Conformal maps – Jacobian matrix
SLIDE 10
10
Context
Conformal maps – Jacobian matrix
SLIDE 11
11
Context
Conformal maps – Jacobian matrix
SLIDE 12
12
Context
- Problem
– 2D conformality is possible – 3D conformality is not possible
- Our solution
– Be as-conformal-as-possible (ACAP) – Linear method based on Cauchy-Riemann
Conformal maps – 3D?
SLIDE 13
13
Context
- Problem
– 2D conformality is possible – 3D conformality is not possible
- Our solution
– Be as-conformal-as-possible (ACAP) – Linear method based on Cauchy-Riemann
Conformal maps – 3D?
SLIDE 14
14
Context
Harmonic way
2D 3D First order Cauchy-Riemann ? Second order Harmonic Harmonic
Conformal maps – 3D?
SLIDE 15
15
Context
ACAP way
2D 3D First order Cauchy-Riemann Generalized CR Second order Harmonic ACAP
Conformal maps – 3D?
SLIDE 16
16
Method
Optimize jacobian matrix on 3 canonical planes
Idea
SLIDE 17
17
Method
Formulation
Optimize jacobian sub-matrices
SLIDE 18
18
Method
Formulation
A compact form
SLIDE 19
19
Method
Formulation
A more compact form
SLIDE 20
20
Method
- Problem
– Operator D supposes there's no rotation
- Solution
– Estimate rotation contained in – Cancel rotation part with
Rotation invariance
SLIDE 21
21
Method
- Goal
– Minimize the energy functional
- Steps
1) Surface conformal map 2) Volume harmonic map 3) Local rotation estimation 4) Volume ACAP map
Algorithm
SLIDE 22
22
Results
Timing Harmonic : 48 secs ACAP : 154 secs Tetrahedra 5569k
SLIDE 23
23
Results
Remarks
- Approximately 3x longer to compute
- Results are always better than harmonic
SLIDE 24
24
Results
- Balance uniform scale and orthogonality
– Use a parameter – : Orthogonality only – : Uniform scale only
Variations
SLIDE 25
25
Conclusion
- Element inversion
– Cannot be garanteed with linear energies – Especially in concave parts
- Fixed boundaries
– Limit the movement of the interior – Conformal surface maps are not optimal
Limitations
SLIDE 26
26
Conclusion
- Volume-aware surface parameterization
Future work
SLIDE 27
27
Conclusions
Future work
- Volume-aware surface parameterization
SLIDE 28
28
Conclusions
Future work
- Smart singularity placement
SLIDE 29
29
Conclusions
- Generalized Cauchy-Riemann
- Fast
- Easy to implement
- Generalize to N dimensions
- To do
– Free boundaries – Smart placement of singularities – Optimal codomain geometry
SLIDE 30
30