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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order Clique Reduction in Binary Graph Cut Hiroshi Ishikawa Nagoya City University Department of Information and Biological Sciences

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Contribution of this work Reduce any higher-order binary MRF into first order Adds variables Can also be used for multi-label energy, with the Fusion Move technique

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Energy Minimization Close to Y Smooth Given Y Find X Assigns X v (= 0 or 1 ) to each pixel v All pixels Neighboring pixels

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Energy Minimization Good (Low Energy)Bad (High Energy) Better (Lower Energy)Worse (Higher Energy) ABCD

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Energy Minimization Good (Low Energy)Bad (High Energy) 12 Bad 40 Good

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Energy Minimization 10 As 0 Ds Better (Lower Energy)Worse (Higher Energy) 8 Bs 3 Cs 10 As 0 Ds 4 Bs 7 Cs ABCD

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Third Order (Clique up to 4 pixels) Higher-Order Energy First Order (Clique up to 2 pixels) General Order C : a set of cliques Clique

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June First-Order MRF Minimization Graph cuts Greig et al. 89 Boykov et al. CVPR98, PAMI2001( -exp.) Kolmogorov & Zabih. PAMI2004 Belief propagation Felzenszwalb & Huttenlocher. IJCV2006 Meltzer et al. ICCV2005 Tree-reweighted message passing Kolmogorov. PAMI2006

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order MRF Minimization Graph cuts Kolmogorov & Zabih. PAMI2004 Freedman & Drineas. CVPR2005 Woodford et al. CVPR2008 Kohli et al. PAMI08, Cremers&Grady ECCV06 Rother et al. CVPR2009 Komodakis & Paragios. CVPR2009 Belief propagation Lan et al. ECCV2006 Potetz. CVPR2008

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order MRF Minimization Graph cuts Kolmogorov & Zabih. PAMI2004 Freedman & Drineas. CVPR2005 Woodford et al. CVPR2008 Kohli et al. PAMI08, Cremers&Grady ECCV06 Rother et al. CVPR2009 Komodakis & Paragios. CVPR2009 Belief propagation Lan et al. ECCV2006 Potetz. CVPR2008

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Functions of Binary Variables Pseudo-Boolean function (PBF) Function of binary ( 0 or 1 ) variables Can always write uniquely as a polynomial One variable x : E 0 (1 x) + E 1 x Two variables x, y : E 00 (1 x) (1 y) + E 01 (1 x) y + E 10 x (1 y) + E 11 x y Three variables x, y, z : E 000 (1 x) (1 y) (1 z) + E 001 (1 x) (1 y) z + … + E 111 x y z n th order binary MRF = ( n +1) th degree PBF

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June nd -Order (Cubic) Case Kolmogorov & Zabih. PAMI2004 Freedman & Drineas. CVPR2005 Reduce cubic PBF into quadratic one using x y z B ={0,1}

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June nd -Order (Cubic) Case If a < 0 So, in a minimization problem, we can substitute by Thus

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order Case if a < 0

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order Case For a > 0 and d > 3, nothing similar is known our contribution Imagine such a formula:

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Higher-Order Case For a > 0 and d > 3, nothing similar is known our contribution Imagine such a formula: Notice LHS is symmetric i.e., if we swap the value of two variables, LHS is unchanged So RHS must be symmetric, too.

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Symmetric Polynomial Fact Any symmetric polynomial can be written as a polynomial in terms of elementary symmetric polynomials. If f (x, y, z, t) is quadratic symmetric, it can be written with a polynomial P(u,v) :

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Quartic (Degree 4) Case

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Quartic (Degree 4) Case

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Quartic (Degree 4) Case An exhaustive search for a, b, c, d, e yields

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Quintic (Degree 5) Case Similarly, and so on, until one can guess…

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June General Case where

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June General Case For each monomial, the number of new variable is: For instance, general quintic looks like: So the number is exponential in degree

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Multiple labels: Fusion Move Assume labels Labeling Y assigns a label Y v to each v Fusion Move Iteratively update Y : 1. Generate a proposed labeling P Lempitsky et al. ICCV2007

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Multiple labels: Fusion Move Assume labels Labeling Y assigns a label Y v to each v Fusion Move Iteratively update Y : 1. Generate a proposed labeling P 2. Merge Y and P The merge defines a binary problem: For each v, change Y v to P v or not Lempitsky et al. ICCV2007

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Multiple labels: Fusion Move Fusion Move Iteratively update Y : 1. Generate a proposed labeling P 2. Merge Y and P The merge defines a binary problem: For each v, change Y v to P v or not YPX

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Multiple labels: Fusion Move Fusion Move Iteratively update Y : 1. Generate a proposed labeling P 2. Merge Y and P The merge defines a binary problem: For each v, change Y v to P v or not YPX

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Fusion Move with QPBO QPBO (Roof duality) Minimizes submodular E globally. For non-submodular E, assigns each pixel 0, 1, or unlabeled With fusion move, by not changing unlabeled pixels to P, E doesnt increase Hammer et al. 1984, Boros et al. 1991, 2006 Kolmogorov & Rother PAMI2007, Rother et al. CVPR2007

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Experiment: Denoising by FoE FoE (Fields of Experts) Roth & Black CVPR2005 A higher-order prior for natural images C : a set of cliques C :

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Experiment: Denoising by FoE OriginalNoise-added1 st order3 rd order

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Experiment: Denoising by FoE Lan et al.PotetzThis work = 10 = PSNR (larger the better)Energy (smaller the better) Lan et al.PotetzThis work Lan et al. ECCV2006 ~8 hours Potetz. CVPR2008~30 mins This work~10 mins

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Experiment: Denoising by FoE Energy & PSNR Two proposal generation strategies PSNR E(×1000) time (sec.) blur & random expansion blur & random expansion

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Summary Reduce any higher-order binary MRF into first order Adds variables Number exponential in order For multi-label, can be used with Fusion Move with QPBO

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CVPR2009: IEEE Computer Society Conference on Computer Vision and Pattern Recognition Miami Beach, Florida. June Thank you! Code available at Acknowledgements Stefan Roth, Brian Potetz, and Vladimir Kolmogorov

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