Multi-view reconstruction Vladimir Kolmogorov Yuri Boykov Carsten Rother University College London University of Western Ontario Ratio minimization - Q(·) assumed to be non-negative - can handle: - submodular / modular - modular / submodular (if numerator is negative for some x) - some other - including ratios of geometric functionals [B&K ICCV03, ICCV05] - generalizing to 3D previous formulations: [Cox et al’96], [Jermyn,Ishikawa’01] can be converted to a parametric max-flow problem - Minimize for different l’s. - Find l such that Related to isoperimetric problem (bias to circles) solved efficiently via Newton's (Dinkelbach’s) method Example 2: flux / length or length / area Example. 1 No shape bias ! One dominant solution is a global optimizer for ratio Theorem: other dominant configurations are optimal solutions for constrained ratio optimization problems for Example Divergence of photoconsistency gradients could be useful if unconstrained ratio minimizer is not a practically useful solution (e.g. too small) Visual-hull from photo-flux [Boykov&Lempitsky BMVC2006] Best for Applications of constrained ratio optimization (in 3D) Segmentation Surface fitting Multi-view reconstruction Optimizing ratio for increasingly larger lower bound on surface area Optimizing ratio for increasingly larger lower bound on surface area Divergence of photoconsistency gradients [Boykov&Lempitsky BMVC2006] Divergence of estimated surface normals [Lempitsky et.al. CVPR 2007]