1. Front propagation in inverse problems and imaging Martin Burger Institute for Computational and Applied Mathematics European Institute for Molecular Imaging (EIMI) Center for Nonlinear Science (CeNoS) Westfälische Wilhelms-Universität Münster

2. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 We often look for geometric objects (with unknown topology) rather than for functions, e.g. in - image segmentation - computer graphics / surface restoration - topology / shape optimization - inverse obstacle scattering - inclusion / cavity / crack detection - detection of hear infarctions - ….. Introduction

3. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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 Introduction

4. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Because of the desired flexibility you need, you have to use level set methods (or something similar) Normal and mean curvature Level Set Methods

5. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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

6. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Level Set Flows Solution by level set flows: find velocities such that resulting evolution of shapes decreases objective functional J Santosa 96, Osher-Santosa 01, Dorn 01/02 Geometric flow of the level sets of  can be translated into nonlinear differential equation for  („level set equation“)

7. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Calculus 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 d d t J ( ­ ( t )) = J 0 ( ­ ( t )) V n V n 7! J 0 ( ­ ) V n

8. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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

9. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Optimization Viewpoint Descent method, time step  can be chosen by standard optimization rules (Armijo-Goldstein) Descent method independent of parametrization, can change topology by splitting Level set method used to perform update Change of scalar product is effective preconditioning !

10. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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

11. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 De Giorgi Viewpoint 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 ! 1 2 ¿ d ( ­ ( V n ; ¿ ) ; ­ ) 2 + J ( ­ ( V n ; ¿ )) ! m i n V n

12. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics 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 d ( ­ 1 ; ­ 2 ) = k b @ ­ 1 b @ ­ 2 k ¡

13. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics Simple Example Shape sensitivity Flow in L 2 metric J ( ­ ) = -c Z ­ d x + Z @ ­ d ¾ J 0 ( ­ ) V n = Z @ ­ ( ·-c ) V n d ¾ Z @ ­ V n W n d ¾ = Z @ ­ ( c-· ) W n d ¾

14. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics Resulting Velocity V n = c-·

15. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics Possible alternative: H 1 metric Resulting Velocity Zero-order flow instead of 2nd order parabolic – very weak step size condition ! Sundaramoorthi, Yezzi, Mennucci 06 h V n ; W n i = Z @ ­ r ¾ V n ¢ r ¾ W n d ¾ ¢ ¾ V n = ·-c

16. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Image Segmentation Active contour model with L 2 and H 1 metric Sundaramoorthi

17. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics Other alternatives: H 1/2 and 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 H 1 scalar product in a surrounding domain Automatic extension, ideal for level set flows

18. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics H 1/2 used with success in topology optimization Allaire, Jouve et al

19. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Example: Obstacle Problem H -1/2 good choice for source reconstructions, obstacle problems Model Example

20. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Sensitivities First shape variation given by Second shape variation is coercive at stationary points H -1/2 is the right scalar product !

21. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Velocity choice Simple calculation shows that velocity for L 2 metric is given by By equivalent realization of the H -1/2 metric we obtain alternative choice Immediate extension velocity field V n = ¡ u § V n = ¡ @ u § @ n V = ¡ r u §

22. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Trajectories 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 !!) V = ¡ r u §

23. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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 !

24. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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 (?)

25. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics Shape deformations nicely controlled by Hausdorff-Metric Not differentiable due to max and sup, hence no reasonable expansion Smooth approximations possible Faugeras et al 06 d ( ­ 1 ; ­ 2 ) = max f sup x 2 @ ­ 1 d i s t ( x ; @ ­ 2 ) ; sup x 2 @ ­ 2 d i s t ( x ; @ ­ 1 )g

26. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Shape Metrics For fast methods, use variable metrics – based on second derivatives ! Inexact Newton methods: positive definite approximation of second derivative Hintermüller-Ring 04 For least-squares problems natural choice: Gauss-Newton (Levenberg-Marquardt,..) mb 04, Ascher et al 07 h V n ; W N i H ( ­ ) ¼ J 00 ( ­ )( V n ; W n )

27. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method Example: cavity detection

28. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method No noise Iterations 2,4,6,8

29. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method Residual and L 1 -error

30. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method Residual and L 1 -error, noisy data

31. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method 0.1 % noise Iterations 5,10,20,25

32. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 Levenberg-Marquardt Method 1% noise 2% noise 3% noise 4% noise

33. 4.6.2007Front Propagation ICIAM 07, Zürich, July 07 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 Download and Contact