1. Shape reconstruction and inverse problems
Lecture 9
© Alexander & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

3. Measurement
Projection
Shape space
Measurement space

4. Reconstruction ?
Find
Given
Shape space
Measurement space 4

5. Reconstruction
Shape space
Measurement space 5

6. Given a (possibly noisy) measurement of unknown shape
Inverse problems
Given a (possibly noisy) measurement of unknown shape
Reconstruct the shape by minimizing the distance between given measurement and measurement obtained from shape 6

7. Inverse problems 7

8. Many shapes have the same measurement!
Ill-posedness
Many shapes have the same measurement!
Shape space
Measurement space 8

9. Prior knowledge We know that the measurements come from
deformations of the same object! 9

10. Regularization Deformations of the dog shape Shape space
Measurement space 10

11. Regularization Prior Shape space Measurement space 11
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 11

12. Inverse problems with intrinsic prior
Error
Regularization
Prior is given on the intrinsic geometry of the shape (intrinsic prior)
Error = distance between measurements
Regularization = intrinsic distance from prior shape
Prior shape is a deformation of the shape we need to reconstruct
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 12

13. Solution of inverse problems with intrinsic prior
Optimization variable: shape , represented as a set of
coordinates
Possible initialization: prior shape
Gradients of and w.r.t. are required
Does it sound familiar?
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 13

14. ?
Extrinsic dissimilarity
Intrinsic dissimilarity

15. Joint similarity as inverse problem
Measurement space = shape space
Identity projection operator
= intrinsic distance on shape space
= extrinsic distance on measurement space
Prior = the shape itself 15

16. Computation of the regularization term
Assume shape = deformation of prior with the same connectivity
Trivial correspondence
Compute L2 distortion of geodesic distances
and gradient
is a fixed (precomputed) matrix of geodesic distances on
depends on the variables (must be updated on every iteration)
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

17. Computation of using Dijkstra’s algorithm
Same approach as in joint similarity
Compute and fix the path of the geodesic
is a matrix of Euclidean distances between adjacent vertices
is a linear operator integrating the path length along fixed path
At each iteration, only changes
Computation of is straightforward
A. Bronstein, M. Bronstein, R. Kimmel, ICCV 2007

18. Inconsistency of Dijkstra’s algorithm
Dijkstra/Analytic
1.1
1.05
FMM/Analytic 1
Number of points
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009

19. Computation of using Fast Marching
Standard FMM
FMM with derivative
propagation
Distance update
Distance update
Distance derivative update
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009

20. Shape-from-X
Silhouette
Sparse points
Image
Shading
Stereo 20

21. Denoising Measurement space = shape space Identity projection operator
= intrinsic distance on shape space
= extrinsic distance on measurement space
Noisy measurement 21

22. Denoising Unknown shape Prior Measurement Reconstruction
(without prior)
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 22

23. Denoising Unknown shape Prior Measurement Reconstruction (with prior)
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 23

24. Bundle adjustment Clean measurement Shape Noisy measurement 24
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 24

25. Bundle adjustment Measurement space of 2D point clouds
Projection operator
(assuming known correspondence)
Noisy measurement 25

26. Bundle adjustment Unknown shape Prior Measurement Reconstruction
(without prior)
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 26

27. Bundle adjustment Unknown shape Prior Measurement Reconstruction
(with prior)
Devir, Rosman, Bronstein, Bronstein, Kimmel, 2009 27

28. Shape-to-image matching
Measured image
Reconstruction
Salzmann, Pilet, Ilic, Fua, 2007 28