Title: Variational Formulations and PDEs ononto Implicit Surfaces Good Bye Triangulated Surfaces
1Variational Formulations and PDEs on/onto
Implicit SurfacesGood Bye Triangulated Surfaces?
- Guillermo Sapiro
- University of Minnesota
- With M. Bertalmio, L. T. Cheng, S. Osher, and F.
Memoli
2The problem
- Solve PDEs and variational problems for data
defined from a generic surface onto a generic
surface - Surfaces from the data bases at Stanford and
GATECH
3Variational problems and PDEs on surfaces?
- Mean curvature motion (Ilmanen, etc)
- Mathematical physics
- Computer graphics
- Image processing
- Regularization of inverse problems (e.g.,
EEGMRI, e.g., Faugeras et al.) - 3D surface mapping
4Examples
- Images from G. Turk , J. Dorsey, P. Thompson
5Classical approach
- Work with triangulated/meshed surfaces
- Discretization on non-uniform grids
- Projections onto triangulated surfaces
- Limit to functions (e.g., Kimmel)
- Very limited framework
- Work on surfaces mapped to the plane
- Loose the geometry, ads complexity
- No work on target surfaces reported
6Our approach
- Define the variational problems and PDEs
following the theory of Harmonic Maps - Well defined framework
- Represent the surfaces in implicit form
- Classical numerics on Cartesian grids
- No projections
- Motivated by Osher-Sethian level-sets and Osher
variational levels-sets (though here the surface
is not moving)
7Harmonic Maps Theory (Tang-Sapiro-Caselles)
- Find a map between manifolds (M,g) and (N,h)
minimizing - Gradient descent (p2)
See also Perona, Chan-Shen, Sochen et al., Hoppe
et al, Zorin et al., etc
8ExampleIsotropic direction smoothing
9ExampleColor Image Enhancement
10The embedding
- IM -gt N
- M is a generic surface (domain)
- With Bertalmio, Cheng, Osher
- N is a generic surface (target)
- With Memoli, Osher
11Embedding the domain surface
- Example IM-gtR
- A map from a generic domain surface onto the real
line
12Embedding the domain surface (cont.)
Figure from G. Turk
13Embedding the domain surface (cont.)
14Embedding the domain surface (cont.)
- The gradient descent flow Heat flow on intrinsic
surfaces - All the computations are done in the Cartesian
grid!
15Example
16Example
17Example
18Example
19L1 Denoising on Implicit Surfaces
20Example Curvature Smoothing
21Example
22Example L1 denoising with constraints
23Example Debluring
24Unit vector/color denoising on implicit surfaces
- I is a map from the 3D surface to the 3D unit
sphere
25Example Chroma denoising
26Example General vector denoising
Original
L1
L2
27Pattern formation on implicit 3D surfaces
- Follows Turing, Kass-Witkin, Turk
28Examples
29Example
30Example
31Example
32Vector field visualization
33Vector field visualization
34Embedding the target manifold
35Embedding the target manifold (cont.)
36Concluding remarks
- No more need for triangulated surfaces for
variational problems and PDEs - Results locally independent of embedding function
- Extended to open domain and target surfaces
- Open problems More theory