Applications of parametric max-flow problem in computer vision A parametric max-flow problem: minimize for all l’s Du(l) – linear functions Vuv(xu , xv) – submodular ES algorithm [Eisner & Severence’76], [Gusfield’80] General case: has finite number of breakpoints: - 2K calls to maxflow (K is # of breakpoints) - K may be exponential! Monotonic case: Du(l) are all non-decreasing/all non-increasing - Nestedness property much more efficient implementation of ES - K=O(n) - Worst-case complexity can be improved to that of a single max-flow [Gallo, Grigoriadis, Tarjan’89] Applications Cosegmentation [Rother,Kolmogorov,Minka,Blake’CVPR06] Ratio minimization (*) PDE cuts (*) Co-segmentation (*) Learning Training ….. - Given 2 images, find segmentations so that histograms match - Key subproblem: 1 image + target histogram segmentation - Minimize PDE cuts [BKCD’ECCV06] Trust region graph cuts – TRGC [RKMB’06] - Approximate Ehist(z) as a linear function - Given current x, choose new approximation - Interpolate between current and new : - Compute minimum xl of for - Choose l that minimizes E(xl) Goal: compute “gradient flow” of contour C - gradient descent of some functional F(C) set of contours within small distance e from C C C C’ - best contour in the neighbourhood C’ space of all contours Searching for l: Each step – parametric max-flow problem: A: as in [RKMB’06] (binary search) B: Essentially, first breakpoint in [0,1] C: compute all solutions in [0,1] input image target histogram controls time step - In general: non-monotonic case - This work: smallest detectable move can be computed via monotonic max-flow algorithm (much faster!) strategy A strategy B strategy C