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Fast Trust Region for Segmentation

Lena Gorelick Joint work with Frank Schmidt and Yuri Boykov Rochester Institute of Technology, Center of Imaging Science January 2013. Fast Trust Region for Segmentation. TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A A A A A A.

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Fast Trust Region for Segmentation

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  1. Lena Gorelick Joint work with Frank Schmidt and Yuri Boykov Rochester Institute of Technology, Center of Imaging Science January 2013 Fast Trust Region for Segmentation TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAA

  2. Image segmentation Basics S

  3. Standard Segmentation Energy Target Appearance Resulting Appearance Fg Probability Distribution Bg Intensity

  4. Minimize Distance to Target Appearance Model Non-linear harder to optimizeregional term

  5. Non-linear Energies with High- Order Terms • complex appearance models • shape non-linear regional term

  6. Related Work • Can be optimized with gradient descent • First order (linear) approximation models • We use more accurate non-linear approximation models based on trust region Ben Ayed et al. Image Processing 2008, Foulonneau et al., PAMI 2006 Foulonneau et al., IJCV 2009

  7. Our contributions • General class of non-linear regional functionals • Optimization algorithm based on trust region framework – Fast Trust Region

  8. Outline • Non-linear Regional Functionals • Overview of Trust Region Framework • Trust region sub-problem • Lagrangian Formulation for the sub-problem • Fast Trust Region method • Results

  9. Regional FunctionalExamples Volume Constraint

  10. Regional FunctionalExamples Bin Count Constraint

  11. Regional FunctionalExamples • Histogram Constraint

  12. Regional FunctionalExamples • Histogram Constraint

  13. Regional FunctionalExamples • Histogram Constraint

  14. Regional FunctionalExamples • Histogram Constraint

  15. Shape Prior • Volume Constraint is a very crude shape prior • Can be encoded using a set of shape moments mpq p+q is the order

  16. Shape Prior • Volume Constraint is a very crude shape prior

  17. Shape Prior using Shape Moments mpq

  18. Shape Prior using Shape moments • Shape Prior Constraint Dist( , )

  19. Optimization of Energies with Regional Functional

  20. Gradient Descent with Level Sets • Gradient Descent • First Order Taylor Approximation for R(S) • First Order approximation for B(S) (“curvature flow”) • Only robust with tiny steps • Slow • Sensitive to initialization Ben Ayed et al. CVPR 2010, Freedman et al. tPAMI 2004 http://en.wikipedia.org/wiki/File:Level_set_method.jpg

  21. Energy Specific vs. General • Speedup via energy- specific methods • Bhattacharyya Distance • Volume Constraint • In contrast: • Fast optimization algorithm for general high-order energies • Based on more accurate non-linear approximation models Ben Ayed et al. CVPR 2010, Werner, CVPR2008 Woodford, ICCV2009

  22. General Trust Region ApproachAn overview • The goal is to optimize Trust region Trust Region Sub-Problem • First Order Taylor for R(S) • Keep quadratic B(S)

  23. General Trust Region ApproachAn overview • The goal is to optimize Trust Region Sub-Problem

  24. How to Solve Trust Region Sub-Problem • Constrained optimizationminimize • Unconstrained Lagrangian Formulationminimize • Can be optimized globally using graph-cut Can be approximated with unary terms Boykov et al. ECCV 2006

  25. Spectrum of Solutions for different λ or d • Newton step • “Gradient Descent” • Exact Line Search (ECCV12)

  26. General Trust Region • Repeat • Solve Trust Region Sub-problemaround S0 with radius d • Update solution S0 • Update Trust Region Size d • Until Convergence

  27. Fast Trust Region • General Trust Region • Control of the distance constraint d • Lagrangian Formulation • Control of the Lagrange multiplier λ λ

  28. Comparison • Simulated Gradient Descent • Exact Line-Search (ECCV 12) • Newton step • Fast Trust Region (CVPR 13)

  29. Volume Constraint for Vertebrae segmentation Initializations Log-Lik. + length + volumeFast Trust Region Log-Lik. + length

  30. Shape Prior with Geometric moments for liver segmentation Fast Trust Region Log-LikelihoodsNo Shape Prior Second order geometric moments computed for the user provided initial ellipse

  31. Appearance model with KL Divergence Constraint Init Fast Trust Region “Gradient Descent” Exact Line Search Appearance model is obtained from the ground truth

  32. Appearance Model with Bhattacharyya Distance Constraint “ “ “Gradient Descent” Fast Trust Region Exact Line Search Appearance model is obtained from the ground truth

  33. Shape prior with Tchebyshev moments for spine segmentation Log-Lik. + length + Shape PriorFast Trust Region Second order Tchebyshev moments computed for the user scribble

  34. Appearance Model with Bhattacharrya Distance Constraint Ground Truth BHA. + length Fast Trust Region Appearance model is obtained from the ground truth

  35. Future Directions • Multi-label Fast Trust Region • Binary shape prior: • affine-invariant Legendre/Tchebyshev moments • Learning class specific distribution of moments • Multi-label shape prior • moments of multi-label atlas map • Experimental evaluation and comparison between level-sets and FTR.

  36. Thank you

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