Pseudo-Bound Optimization for Binary Energies Meng Tang 1 Ismail Ben Ayed 2 Yuri Boykov 1 1 University of Western Ontario, Canada 2 GE Healthcare Canada Binary Energy Minimization [1] Y. Boykov and M. Jolly, Interactive Graph Cuts for Optimal Boundary & Region Segmentation of Objects in N-D Images, in ICCV [2] C. Rother, V. Kolmogorov and A. Blake, GrabCut: Interactive Foreground Extraction using Iterated Graph Cuts, in SIGGRAPH [3] V. Kolmogorov, Y. Boykov, and C. Rother, Applications of parametric maxflow in computer vision, in ICCV  Non-submodular pairwise energy  High-order energy S * = arg S min E(S) Binary Energy: Difficult-to-optimize energies: Standard Bound Optimization solution space S StSt S t+1 At(S)At(S) E(S)E(S) E(St)E(St) E(S t+1 )  Never increases original energy E(S)  Converges to local minima  Bound may not approximate well New Idea: Pseudo-Bound Opt. solution space S StSt S t+1 At(S)At(S) E(S)E(S) E(St)E(St)  Make larger move S t → S t+1  Pseudo-bounds better approximate E(S)  Efficiently optimize pseudo-bounds using parametric max-flow algorithm From Model-fitting to Entropy Minimization mixed optimization Note: H(P|Q)  H(P) for any two distributions (equality when Q=P ) cross-entropyentropy binary optimization BCD (block-coordinate-descent) of I.Minimize over segmentation S for fixed  0,  1 [1] II.Minimize over θ 0, θ 1 for fixed labeling S E(S)E(S) E ( S|θ 0,θ 1 ) t StSt S t+1 (I) (II) E ( S|θ 0,θ 1 ) t+1 E ( S|θ 0,θ 1 ) = t is a bound of original energy E(S). BCD, as is in GrabCut [2], is bound Optimization for E(S). Parametric Pseudo-Bound Cuts (pPBC) Ƒ t (S, λ) = + λ R t (S) Pairwise SubmodularUnary Bounding relaxation S λ = min s Ƒ t (S, λ) S t+1 = argmin E(S λ ) Parametric Maxflow [3] Update iterate A t (S) Bound SλSλ n-links s t a cut t-link Edge capacities depend linearly on λ. examples of pseudo-bounds For entropy-based energy: For non-submodular pairwise: Ƒ t (S,λ) + λ(|S|-|S t |) = E ( S|θ 0,θ 1 ) t |S||S| |St||St| f(|S|) |V0||V0| Volume prior: m pq s p s q, for m pq >0 Application --- High-order EnergyApplication --- Non-submodular PairwiseConclusion and Future Work BCD pPBC E=1.4026×10 6 E=1.4198×10 6 Initializations 1 2 Interactive Segmentation Unsupervised Segmentation Matching Target Distribution Input imageGround truthResults Input LSA-AUX [4] t=0.1s LSA-TR [4] t=1.9s pPBC-T t=3.1s Segmentation with Curvature Regularization Binary Image Deconvolution Average energy with 10 random noisy images. Input imagesResults  Novel generalization of majorize-minimize principle  Efficient opt. of pseudo-bounds via parametric max-flow  Showed that BCD (in GrabCut ) is a bound Optimization  Devise specific pseudo-bounds for several energies  Achieve state-of-the-art results for many binary energies  Future work: thorough comparison, multi-label energies Comparison of Auxiliary Cuts, FTR, pPBC and its limited version (λ ≤ 0). noisy image LSA-AUX [4] LSA-TR [4] pPBC-T Top (bottom) row: nose sigma = 0.1 (0.2) [4] L. Gorelick, Y. Boykov, O. Veksler, I. Ayed and A. Delong, Submodularization for Binary Pairwise Energies, in CVPR high-order