Numerical Geometry of Images: Shape Reconstruction Ron Kimmel Geometric Image Processing Lab Computer Science Department Technion-Israel Institute of Technology

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

Stereo

Shape from Stereo

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

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

Image formation Lambetian model V l N

Lambertian surfaces qImages formation

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

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

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

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.

Multigrid Poisson solvers Solve a Poisson equation

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

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

Open Questions qA working automatic shape from stereo algorithm does not exist (but we are getting there).