Linear, Discrete, and Quadratic Constraints in Single-image 3D Reconstruction

In this thesis, we investigate the formulation, optimization and ambiguities in single-image 3D surface reconstruction from geometric and photometric constraints. We examine linear, discrete and quadratic constraints for shape from planar curves, shape from texture, and shape from shading. The problem of recovering 3D shape from the projection of planar curves on a surface is strongly motivated by perception studies. Applications include single-view modeling and uncalibrated structured light. When the curves intersect, the problem leads to a linear system for which a direct least-squares method is sensitive to noise. We derive a more stable solution and show examples where the same method produces plausible surfaces from the projection of parallel (non-intersecting) planar cross sections. The problem of reconstructing a smooth surface under constraints that have discrete ambiguities arise in areas such as shape from texture, shape from shading, photometric stereo and shape from defocus. While the problem is computationally hard, heuristics based on semidefinite programming may reveal the shape of the surface. Finally, we examine the shape from shading problem without boundary conditions as a polynomial system. This formulation allows, in generic cases, a complete solution for ideal polyhedral objects. For the general case we propose a semidefinite programming relaxation procedure, and an exact line search iterative procedure with a new smoothness term that favors folds at edges. We use this numerical technique to inspect shading ambiguities.Ph