Title: Implicit Fairing of Arbitrary Meshes Using Diffusion and Curvature Flow
1Implicit Fairing of Arbitrary Meshes Using
Diffusion and Curvature Flow
- Mathieu Desbrun Mark Meyer Peter Schröder
Alan Barr - Caltech
2Smoothing Arbitrary Meshes
- Context
- Meshes often come from 3D scans
- Noisy
- Non uniform triangles
- No parameterization, just positions in space
- Smoothing removal of rough detail features
- while preserving the global shape
3Current Smoothing Results
- OK on semi-uniform meshes, Problems
with non uniform - yet slow on large meshes
meshes distortion!
4Contributions (I)
- Implicit fairing
- Implicit integration scheme for efficiency
- Exact volume preservation
- Constraints during smoothing
5Contributions (II)
- Two new operators
- Laplace operator that respects length scale
- Curvature normal operator, for better shape
denoising
6Previous Approaches (I)
- Early work (CAGD)
- Minimization of complex functionals
- Ex
- Taubin 95 (and follow-ons)
- Signal processing approach using L(M)
(low-pass filter)
7Previous Approaches (II)
- Iterative energy minimization
- Membrane energy
- Follows the Laplacian
- Iterations of an explicit Euler scheme
-
(Diffusion)
8Implicit Integration (I)
- Diffusion
- Strict stability requirements
- Small time steps for explicit integration
- Implicit schemes are strongly suggested
- BW98
9Implicit Integration (II)
Explicit Euler scheme
Implicit Euler scheme
vs.
10Introducing Implicit Fairing
- Small time steps for large meshes
11Introducing Implicit Fairing
- Implicit integration for larger time steps
- (PB) Conjugate Gradient for efficiency
- Equivalent to
12Volume Preservation
- Diffusion induces shrinkage!
- Enforce exact volume preservation
- Compute volume
- Rescale after integration step(s)
- Incorporate other invariants
- Volume, surface area, ...
- Build directly into PDE
13Constraint Enforcement
- Fixed points/Fixed regions
- Set the Laplacian to zero
- Soft constraints
- Locally adjust smoothing amount (l)
- Spray-paint the object directly
- Complex constraints BW98
14Where are we now? (I)
- Implicit fairing
- Implicit integration scheme for efficiency
- Exact volume preservation
- Constraints during smoothing
15Where are we now? (II)
- Two new operators
- Laplace operator that respects length scale
- Curvature normal operator, for better shape
denoising
16Regular Diffusion
- Problem with the umbrella operator
-
smoothed as much as - It assumes a regular parameterization!
-
17Regular Diffusion Effects
Initial mesh
Regular fairing
18Accurate Diffusion
- Length-scale weighted operator
- (Fujiwara operator)
-
- Creates scale-dependent operator
- But stability issues
- Explicit time step too small dt lt lmin2 / l
- Implicit integration a must!
19Regular/Accurate Diffusion
Scale-dependent fairing
Initial mesh
Regular fairing
20More Results With Diffusion
Initial mesh
21More Results With Diffusion
After upsampling and one fairing
22Shape Denoising
- Why diffuse the noise?
- Normal component
- Tangent component, distorting the shape!
- Alternatively curvature flow equation
(Laplace-Beltrami) - Problem how can we find k n reliably?
23Discrete Curvature
- Differential geometry definition
-
- After derivation
-
- Dependent on angles and lengths
- Zero curvature ensured when flat
24Curvature operator
Curvature visualization using false colors
25Curvature Flow Evaluation
Initial Mesh
Regular Diffusion
Curvature Flow
Improved Diffusion
26Results on 3D Scanned Data
Initial mesh
After one fairing
27Results on 3D Scanned Data
28Video
29Conclusion
- New tools for irregular meshes
- Implicit fairing
- Adds efficiency/stability
- Improved Laplace operator
- Limits distortion on arbitrary meshes
- New curvature normal operator
- Shape denoising with curvature flow
30Future Work
- More smoothing
- Other kinds of geometrical flow
- Edge preservation in computer vision
- Modeling
- Prescribed curvature normals
- Animation
- Applications of curvature operator
- Info? http//www.multires.caltech.edu/mathieu