Vladimir Kolmogorov Yuri Boykov Carsten Rother University of

Vladimir Kolmogorov Yuri Boykov Carsten Rother University of Western Ontario University College London 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 ICCV 03, ICCV 05] - generalizing to 3 D previous formulations: [Cox et al’ 96], [Jermyn, Ishikawa’ 01] - can be converted to a parametric max-flow problem for different l’s. - Minimize Related to isoperimetric problem (bias to circles) - Find l such that 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 Example for Divergence of photoconsistency gradients Visual-hull from photo-flux [Boykov&Lempitsky BMVC 2006] Best for • could be useful if unconstrained ratio minimizer is not a practically useful solution (e. g. too small) Applications of constrained ratio optimization (in 3 D) Multi-view reconstruction Divergence of photoconsistency gradients [Boykov&Lempitsky BMVC 2006] Optimizing ratio for increasingly larger lower bound on surface area Surface fitting Divergence of estimated surface normals [Lempitsky et. al. CVPR 2007] Optimizing ratio for increasingly larger lower bound on surface area Segmentation