1. MAP Estimation of Semi-Metric MRFs via Hierarchical Graph Cuts M. Pawan Kumar Daphne Koller Aim: To obtain accurate, efficient maximum a posteriori (MAP) estimation for Markov random fields (MRF) with semi-metric pairwise potentials MAP Estimation vava vbvb  a (i)  ab (i,k) Semi-Metric Potentials  b (k)  ab (i,k) = w ab d(i,k) f(a)-f(b) d(i,i) = 0, d(i,j) = d(j,i) > 0 d(i,j) - d(j,k) ≤  d(i,k) f : {a,b, …} {1, …, H} Bounds For  =1 (Metric) Linear Program: O(log H) Graph Cuts: 2 d max /d min Our Method: O(log H) lili lklk r-HST Metrics min f Q(f) Q(f) = ∑  a (f(a)) + ∑  ab (f(a),f(b)) Variables V, Labels L l1l1 l2l2 l3l3 l4l4 A A BB CC B ≤ A/r C ≤ A/r Overview Distance d T  path length d   1 d T1 +  2 d T2 + …. min f Q(f;d T1 ) f T1 min f Q(f;d T2 ) f T2.. Combine f T1, f T2 …. r-HST Metric Labeling Efficient Divide-and-Conquer Approach Analysis l1l1 l2l2 l3l3 l4l4 Use  -Expansion f 1 = min f Q(f) f(a)  {1,2} l5l5 l6l6 f 2 = min f Q(f) f(a)  {3,4} f 3 = min f Q(f) f(a)  {5,6} Combine f i using  -Expansion Initialize f 0 = f 1 Repeat At each iteration Choose an f i f t (a) = f t-1 (a) OR f t (a) = f i (a) Optimal move using graph cuts Image Denoising ExpTRWBPOur+ EM Q75641682261058457282872332 Time5.1174.332.970.6204.5 Stereo Reconstruction Scene Registration ExpTRWBPOur+ EM Q861637338352696981820 Time26.1529.6115.8294.7465.6 ExpTRWBPOur+ EM Q78776627771268246511665008 Time12.1263.350.4152.8361.8 ExpTRWBPOur+ EM Q15322132575628014135 Time4.5169.129.672.1203.1 ExpTRWBPOur+ EM Q8203681118843968131581258 Time1.71371.1218.0104.9373.6 ExpTRWBPOur+ EM Q6857267616702396768267676 Time1.31058.2160.073.6240.5 Learning a Mixture of rHSTs (Hierarchical Clustering ) min max i,k ∑  t d T t (i,k) d(i,k) Refinement (Hard EM) l1l1 l3l3 l4l4 l2l2 l3l3 l1l1 l4l4 l1l1 l3l3 l4l4 Permutation π Cluster C j Cluster C j+1 Root  1 cluster Choose random π For l i in cluster C j Find first l k in π s.t. d(i,k) ≤ T Decrease T by r Repeat Fakcharoenphol et al., 2000 Derandomization Boosting-style descent y ik = Residual min ∑y ik d T (i,k) Update y ik. Repeat. Bounds For  =1, O(log H) For  1, O((  log H) 2 ) Initial labeling f y ik : contribution of (i,k) to current labeling min ∑y ik d T (i,k) New labeling f’ Approximate E and M y ik = ∑w ab [f(a)=i][f(b)=k] Bound of 1 for unary potentials, 2r/(r-1) for pairwise potentials l1l1 l2l2 l3l3 l4l4 A A BB CC Mathematical Induction True for children vava vbvb Unary potential bound follows from  -Expansion Bound = 1 vava vbvb vava vbvb Bound = 2d max /d min = 2r/(r-1) Clean up an image with noise and missing data Find correspondence between two epipolar corrected images of a scene Find correspondence between two scenes with common elements (building, fire) QExpSwapTRWBPRSwpRExpOur+EM T-L14864548721475065094248045479984785047823 T-L25209451938513186026951842516415158751413 rHST50221510554813252841--48146 Met48112484874735548136--4753847382 SMet47613475794661247402--4665146638 Synthetic Experiments 100 randomly generated 4-connected grid graphs of size 100x100 TimeExpSwapTRWBPRSwpRExpOur+EM T-L10.40.6104.315.82.05.810.225.7 T-L20.40.9179.045.610.730.712.864.1 rHST0.30.5713.7150.4--1.95.0 Met0.30.5703.8129.7--10.632.7 SMet0.40.570.9.4141.8--12.257.5