MAP Estimation of Semi-Metric MRFs via Hierarchical Graph Cuts M. Pawan Kumar Daphne Koller MAP Estimation b(k) Semi-Metric Potentials Bounds Aim: To obtain accurate, efficient maximum a posteriori (MAP) estimation for Markov random fields (MRF) with semi-metric pairwise potentials lk a(i) ab(i,k) = wab d(i,k) For =1 (Metric) li ab(i,k) d(i,i) = 0, d(i,j) = d(j,i) > 0 Linear Program: O(log H) va vb d(i,j) - d(j,k) ≤  d(i,k) Graph Cuts: 2 dmax/dmin minf Q(f) Variables V, Labels L f : {a,b, …} {1, …, H} Our Method: O(log H) Q(f) = ∑ a(f(a)) + ∑ ab(f(a),f(b)) f(a)-f(b) f(a)-f(b) r-HST Metrics r-HST Metric Labeling Efficient Divide-and-Conquer Approach Combine fi using -Expansion A A Initialize f0 = f1 At each iteration Choose an fi ft(a) = ft-1(a) OR ft(a) = fi(a) B B C C l1 l2 l3 l4 Optimal move using graph cuts l1 l2 l3 l4 l5 l6 Distance dT  path length f1 = minf Q(f) f2 = minf Q(f) f3 = minf Q(f) Repeat B ≤ A/r C ≤ A/r f(a)  {1,2} f(a)  {3,4} f(a)  {5,6} Overview Analysis Bound of 1 for unary potentials, 2r/(r-1) for pairwise potentials Mathematical Induction Unary potential bound follows from -Expansion d  1dT1 + 2dT2 + …. A A minf Q(f;dT1) fT1 minf Q(f;dT2) fT2 . B B C C . va vb va vb va vb l1 l2 l3 l4 Combine fT1, fT2 …. Bound = 1 Bound = 1 Bound = 2dmax/dmin = 2r/(r-1) Use -Expansion True for children Learning a Mixture of rHSTs (Hierarchical Clustering) Refinement (Hard EM) ∑tdTt(i,k) min maxi,k d(i,k) l1 l3 l4 Initial labeling f Cluster Cj Root 1 cluster Derandomization Boosting-style descent yik: contribution of (i,k) to current labeling Choose random π yik = Residual For li in cluster Cj Find first lk in π s.t. d(i,k) ≤ T min ∑yik dT(i,k) l2 l3 l1 l4 Permutation π yik = ∑wab[f(a)=i][f(b)=k] Update yik. Repeat. min ∑yik dT(i,k) l3 Bounds Decrease T by r For =1, O(log H) New labeling f’ Cluster Cj+1 Repeat l4 l1 For 1, O((log H)2) Approximate E and M Fakcharoenphol et al., 2000 Synthetic Experiments 100 randomly generated 4-connected grid graphs of size 100x100 Q Exp Swap TRW BP RSwp RExp Our +EM T-L1 48645 48721 47506 50942 48045 47998 47850 47823 T-L2 52094 51938 51318 60269 51842 51641 51587 51413 rHST 50221 51055 48132 52841 - 48146 Met 48112 48487 47355 48136 47538 47382 SMet 47613 47579 46612 47402 46651 46638 Time Exp Swap TRW BP RSwp RExp Our +EM T-L1 0.4 0.6 104.3 15.8 2.0 5.8 10.2 25.7 T-L2 0.9 179.0 45.6 10.7 30.7 12.8 64.1 rHST 0.3 0.5 713.7 150.4 - 1.9 5.0 Met 703.8 129.7 10.6 32.7 SMet 70.9.4 141.8 12.2 57.5 Image Denoising Clean up an image with noise and missing data Exp TRW BP Our + EM Q 86163 73383 526969 81820 Time 26.1 529.6 115.8 294.7 465.6 Exp TRW BP Our + EM Q 75641 68226 105845 72828 72332 Time 5.1 174.3 32.9 70.6 204.5 Stereo Reconstruction Find correspondence between two epipolar corrected images of a scene Exp TRW BP Our + EM Q 78776 62777 126824 65116 65008 Time 12.1 263.3 50.4 152.8 361.8 Exp TRW BP Our + EM Q 15322 13257 56280 14135 Time 4.5 169.1 29.6 72.1 203.1 Scene Registration Find correspondence between two scenes with common elements (building, fire) Exp TRW BP Our + EM Q 82036 81118 84396 81315 81258 Time 1.7 1371.1 218.0 104.9 373.6 Exp TRW BP Our + EM Q 68572 67616 70239 67682 67676 Time 1.3 1058.2 160.0 73.6 240.5