2. www.cs.technion.ac.il/~ron Numerical Geometry of Images: Shape Reconstruction Ron Kimmel Geometric Image Processing Lab Computer Science Department Technion-Israel Institute of Technology

3. Shape reconstruction techniques Shape from:Inputs Shading Stereo Auto-stereograms Photometric stereo Structured/coded light IR-pulses (3DV-Systems)

4. Stereo

5. Shape from Stereo

6. Shape from autostereograms for i = (stripesize+1):xsize+stripesize, for j = 1:ysize, stereo(j,i) = stereo(j,i-floor(stripesize*(1-z(j,i)))); end %for j end %for i

7. Image Formation qF. Guichard 93 Mondrian world: Lambertian surface patches

8. Image formation Lambetian model V l N

9. Lambertian surfaces qImages formation

10. Shape from photometric stereo Given Lambertian shading model, multiple images each with different light source. At each point we have 3 unknowns that can be computed from at least 3 images. Next, the problem is surface integration from its gradient vector field

11. Shape from photometric stereo Next, the problem is surface reconstruction from Define the integral measure For which the Euler-Lagrange is a Poisson equation or in more compact notations

12. Shape from photometric stereo Efficient numerical algorithm for solving a Poisson equation depends on the boundary conditions. For periodic b.c. Fourier transform is an option

13. Shape from photometric stereo Efficient numerical algorithm for solving a Poisson equation depends on the boundary conditions. For point constraints, see Kimmel-Yavneh Algebraic Multi-Grid approach.

14. Multigrid Poisson solvers Solve a Poisson equation

15. Shape from shading qHorn: characteristic strip expansion qBruckstein: equal height contours qRouy-Turin: Minimal cost approach qKimmel-Sethian: Fast Marching

16. Summary: Shape reconstruction qShape from shading qExplicit methods (Horn/Bruckstein)

17. Open Questions qA working automatic shape from stereo algorithm does not exist (but we are getting there). www.cs.technion.ac.il/~ron