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Preconditioned Level Set Flows

Preconditioned Level Set Flows. Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster. Introduction.

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Preconditioned Level Set Flows

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  1. Preconditioned Level Set Flows Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster

  2. Introduction • We often look for geometric objects (with unknown topology) rather than for functions, e.g. in - image segmentation - computer graphics / surface restoration - topology optimization - inverse obstacle scattering- inclusion / cavity / crack detection - ….. Level Set Flows AIP 2007, Vancouver, June 07

  3. Introduction • Because of the desired flexibility you need, you have to use level set methods (or something similar) Osher & Sethian, JCP 1987, Sethian, Cambridge Univ. Press 1999, Osher & Fedkiw, Springer, 2002 Level Set Flows AIP 2007, Vancouver, June 07

  4. Level Set Methods • Because of the desired flexibility you need, you have to use level set methods (or something similar) Normal and mean curvature Level Set Flows AIP 2007, Vancouver, June 07

  5. Shape Optimization Approach • In a natural way such problems can be formulated as shape optimization problems where K is a class of admissible shapes (eventually including additional constraints). • Mumford-Shah / Chan-Vese functionals in segmentation • Least-squares / reciprocity gap in crack / inclusion dectection Level Set Flows AIP 2007, Vancouver, June 07

  6. Level Set Flows • Solution by level set flows: find velocities such that resulting evolution of shapes decreases objective functional JSantosa 96, Osher-Santosa 01, Dorn 01/02 • Geometric flow of the level sets of f can be translated into nonlinear differential equation for f („level set equation“) Level Set Flows AIP 2007, Vancouver, June 07

  7. Shape Calculus d 0 ( ) V J ­ V 0 ( ( ) ) ( ( ) ) J ­ J ­ V 7 ! t t n n = n d t • Formulas for the change of the objective functional are obtained by shape calculus • Shape sensitivity is a linear functional of the normal velocity on the boundary Level Set Flows AIP 2007, Vancouver, June 07

  8. Descent Directions • Descent directions are obtained by chosing the normal velocity as a representation of the negative shape sensitivity in a Hilbert space • Asymptotic expansion for small-time motion with that normal velocity mb 03/04 Level Set Flows AIP 2007, Vancouver, June 07

  9. Optimization Viewpoint • Descent method, time stept can be chosen by standard optimization rules (Armijo-Goldstein) • Descent method independent ofparametrization, can change topology by splitting • Level set method used to perform update • Change of scalar product is effective preconditioning ! Level Set Flows AIP 2007, Vancouver, June 07

  10. Differential Geometry Viewpoint • Interpretation from global differential geometry: - take manifold of shapes with appropriate metric to obtain Riemannian manifold - tangent spaces of such a manifold can be identified with normal velocities - Riemannian structure induces scalar product on tangent spaces • We usually take the other way: „scalar product on tangent spaces induces Riemannian structure“Michor-Mumford 05 Level Set Flows AIP 2007, Vancouver, June 07

  11. De Giorgi Viewpoint 1 2 ( ( ) ) ( ( ) ) d ­ V ­ J ­ V i + ¿ ¿ m n ! n n ; ; ; 2 V ¿ n • Generalized gradient flow structure in a metric space (concept of minimizing movements) DeGiorgi 1974, Ambrosio-Gigli-Savare 2005 • Flow obtained as the limit of variational problems (implicit Euler) • We can alternatively chose a metric and expand it to second order ! Level Set Flows AIP 2007, Vancouver, June 07

  12. Shape Metrics ( ) k k d b b ­ ­ ¡ = @ @ ­ ­ 1 2 ; 1 2 • Class of shape metrics obtained by using Hilbert space norms on signed distance functions • Expansion yields Sobolev spaces on normal velocities • Natural in level set form Level Set Flows AIP 2007, Vancouver, June 07

  13. Shape Metrics Z Z Z Z Z 0 ( ( ) ) ( ( ) ) d d d d d J J V ­ ­ W V V W + - ¾ c · x - c c - · ¾ ¾ ¾ = = = n n n n n @ @ @ @ ­ ­ ­ ­ ­ • Simple Example • Shape sensitivity • Flow in L2 metric Level Set Flows AIP 2007, Vancouver, June 07

  14. Shape Metrics V c - · = n • Resulting Velocity Level Set Flows AIP 2007, Vancouver, June 07

  15. Shape Metrics Z ¢ V h i d · - c V W r V r W = ¢ ¾ ¾ n = n n ¾ n ¾ n ; @ ­ • Possible alternative: H1 metric • Resulting Velocity • Zero-order flow instead of 2nd order parabolic – very weak step size condition !Sundaramoorthi, Yezzi,Mennucci 06 Level Set Flows AIP 2007, Vancouver, June 07

  16. Image Segmentation • Active contour model with L2 and H1 metric Sundaramoorthi Level Set Flows AIP 2007, Vancouver, June 07

  17. Shape Metrics • Other alternatives: H1/2and H-1/2 metric • Sounds complicated, but can be realized as Dirichlet or Neumann traces of Sobolev functions on domain ! Can take harmonic extensions and their H1 scalar product in a surrounding domain • Automatic extension, ideal for level set flows Level Set Flows AIP 2007, Vancouver, June 07

  18. Shape Metrics • H1/2used in topology optimization Allaire, Jouve et al 06/07 Level Set Flows AIP 2007, Vancouver, June 07

  19. Example: Obstacle Problem • H-1/2 good choice for source reconstructions, obstacle problems • Model Example Level Set Flows AIP 2007, Vancouver, June 07

  20. Shape Sensitivities • First shape variationgiven by • Second shape variation is coercive at stationary points • H-1/2 is the right scalar product ! Level Set Flows AIP 2007, Vancouver, June 07

  21. Velocity choice @ u § V ¡ u V = § ¡ n = n @ n V r ¡ u = § • Simple calculation shows that velocity for L2 metric is given by • By equivalent realization of the H-1/2 metric we obtain alternative choice • Immediate extension velocity field Level Set Flows AIP 2007, Vancouver, June 07

  22. Trajectories V r ¡ u = § • Due to existence of a velocity field we can analyze the shape evolution via trajectories • Note: from elliptic regularity theory, velocity field is Hölder continuous (existence), but not Lipschitz (no Picard-Lindelöf !!) Level Set Flows AIP 2007, Vancouver, June 07

  23. Trajectories • But the velocity field is almost Lipschitz • This is enough for uniqueness (Osgood‘s theorem) • No stability estimate for ODE ! • Thus, topology could change ! Level Set Flows AIP 2007, Vancouver, June 07

  24. Complete Analysis • Main result: level set flow is well-defined, independent of level set representation, and converges to a minimizer of the functional • Well-definedness by approximation argument and trajectory analysis • Independence by existence of uniqueness of trajectories • Convergence by energy estimates mb-Matevosyan 07 (?) Level Set Flows AIP 2007, Vancouver, June 07

  25. Shape Metrics ( ) f ( ) d d @ ­ ­ ­ i t m a x s u p s x = 1 2 2 @ ­ 2 ; ; ; x 1 ( ) g d @ ­ i t s u p s x 1 @ ­ 2 ; x 2 • Shape deformations nicely controlled by Hausdorff-Metric • Not differentiable due to max and sup, hence no reasonable expansion • Smooth approximations possibleFaugeras et al 06 Level Set Flows AIP 2007, Vancouver, June 07

  26. Shape Metrics 0 0 h i ( ) ( ) V W J ­ V W ¼ ( ) N H ­ n n n ; ; • For fast methods, use variable metrics – based on second derivatives ! • Inexact Newton methods: positive definite approximation of second derivativeHintermüller-Ring 04 • For least-squares problems natural choice: Gauss-Newton (Levenberg-Marquardt, ..)mb 04, Ascher et al 07 Level Set Flows AIP 2007, Vancouver, June 07

  27. Levenberg-Marquardt Method Example: cavity detection Level Set Flows AIP 2007, Vancouver, June 07

  28. Levenberg-Marquardt Method • No noise • Iterations2,4,6,8 Level Set Flows AIP 2007, Vancouver, June 07

  29. Levenberg-Marquardt Method Residual and L1-error Level Set Flows AIP 2007, Vancouver, June 07

  30. Levenberg-Marquardt Method Residual and L1-error, noisy data Level Set Flows AIP 2007, Vancouver, June 07

  31. Levenberg-Marquardt Method 0.1 % noise Iterations 5,10,20,25 Level Set Flows AIP 2007, Vancouver, June 07

  32. Levenberg-Marquardt Method 1% noise 2% noise 3% noise 4% noise Level Set Flows AIP 2007, Vancouver, June 07

  33. Download and Contact Papers and talks at www.math.uni-muenster.de/u/burger or by email martin.burger@uni-muenster.de Based on joint work with: Norayr Matevosyan, Stan Osher Thanks for input and suggestions to: B.Hackl, W.Ring, M.Hintermüller, U.Ascher Austrian Science Foundation FWF, SFB F 013 Level Set Flows AIP 2007, Vancouver, June 07

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