Vladimir Kolmogorov Yuri Boykov Carsten Rother

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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 BK ICCV03, ICCV05
• - generalizing to 3D previous
  formulations
  Cox et al96, Jermyn,Ishikawa01

can be converted to a parametric max-flow problem

• - Minimize
  for different ls.
• - Find l such that

Related to isoperimetric problem (bias to circles)
solved efficiently via Newton's (Dinkelbachs)
method
Example 2 flux / length or
length / area
Example. 1
No shape bias !
One dominant solution is a global optimizer for
ratio
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 BoykovLempitsky
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
BoykovLempitsky BMVC2006
Divergence of estimated surface normals
Lempitsky et.al. CVPR 2007