Covering Trees and Lower-bounds on Quadratic Assignment Julian Yarkony, Charless Fowlkes, Alexander Ihler University of California, Irvine Overview & Goals Constructing a Covering Tree • Develop new algorithms for determining corresponding points between images • Create a single tree which covers all edges of graph • Duplicate nodes as required • Covering Trees • A new variational optimization of the treereweighted (TRW) bound • Achieves the same lowerbound as TRW • BAR (Bottleneck Assignment Rounding) • use the min-marginals from covering tree to construct an assignment Quadratic Assignment Problem Formulation • P parts, L locations • Xi 2 1…L specifies the location of part i • Minimize assignment cost • Minimal representation; no updates of edge parameters Parameters of the covering tree µCT distribute unary potentials from original problem across copies of the node : pairwise assignment cost i→Xi, j→Xj • Two parameter optimization updates: • Subgradient update : 1 -to-1 assignment constraint TRW and Dual Decomposition: • Select a set of spanning trees • Decompose problem µ into {µT} s. t. åT µT = µ • Fixed point update • Optimizing state X in the covering tree • May not be consistent across all copies • May not be a 1 -1 assignment • “Assignment Rounding” • find a valid assignment using CT estimates • Construct bound • lower bounds cost when part i in location k Original Graph Collection of trees Focus on large baselines (grey region) • Compare CT+BAR to CT alone • BAR improves estimate quality Data Set “House” data set • 110 views, 30 marked points for correspondence • Unary costs: shape context features • Pairwise costs: deviation in distance and angle • Only consider pairwise potentials between neighboring parts (2, 3, 4 neighbors per part) Comparing accuracy & timing • Fix the number of neighbors in graph • Similar accuracy, CT+BAR faster • 2 nbr Graph Matching vs. 4 nbr CT+BAR • Similar timing, CT+BAR more accurate : indicator of copy t taking on state k Update any set S of nodes simultaneously with step-size We show fixed point update is monotone for · 1/|S| Correspondence accuracy vs. view separation • Solving Bottleneck assignment problem • Search over all possible bottleneck values • Restrict to states less than bottleneck value • Check feasibility of value with bipartite matching : set of copies of node i : min-marginal of copy t Experimental Results Bottleneck Assignment Rounding • Find the best assignment consistent with bounds Optimizing the Covering Tree Bound : cost of assigning part i to location Xi {jyarkony, fowlkes, ihler}@ics. uci. edu • We compare our algorithm to "Graph Matching" proposed by Toressani et al. which also uses a dualdecomposition approach. • Gives a lower bound on the optimum • TRW algorithm optimizes the lower bound over µ Convergence of various optimization algorithms on 10 by 10 Ising grids with repulsive potentials References V. Kolmogorov. Convergent tree-reweighted message passing for energy minimization. IEEE Trans. Pattern Anal. Machine Intell. , 28(10): 1568– 1583, 2006. L. Torresani, V. Kolmogorov, and C. Rother. Feature correspondence via graph matching: Models and global optimization. In ECCV, pages 596– 609, 2008. M. J. Wainwright, T. Jaakkola, and A. S. Willsky. MAP estimation via agreement on (hyper)trees: message-passing and linear programming approaches. IEEE Trans. Inform. Theory, 51(11): 3697– 3717, 2005.