1. Martin Burger Institut für Numerische und Angewandte Mathematik European Institute for Molecular Imaging CeNoS Convex and Nonconvex Relaxation Approaches

2. Martin Burger Convex and Nonconvex Relaxation 2 Graz, 27.9.2008 Topology Optimization – State of the Art Various approaches have been proposed for solving topology optimization problems in imaging and engineering applications Particular success has been achieved with - Level Set Methods - Topological Asymptotics - Ad-hoc approximations (e.g. SIMP / RAMP)

3. Martin Burger Convex and Nonconvex Relaxation 3 Graz, 27.9.2008 Topology Optimization – State of the Art In particular combinations of level set methods and topological asymptotics have become a powerful tool, which can be tuned to fast local convergence in many applications mb-Hackl-Ring 04, Allaire et al 05-08, Amstutz-Andrä 05, Wang et al 05-08, Hackl 07, He et al 06, Fulmanski et al 08 In some respects, there are still drawbacks

4. Martin Burger Convex and Nonconvex Relaxation 4 Graz, 27.9.2008 Topology Optimization – State of the Art Disadvantages -Volume constraints difficult (level set methods approximate signed distance functions, no continuous dependence of volume) - Linear constraints and simple relations for indicator function are destroyed (significant nonlinearity in level set methods) - Methods only converge local even for simple objectives (possible global convergence by topological derivatives, but difficult to check and with potentially high effort)

5. Martin Burger Convex and Nonconvex Relaxation 5 Graz, 27.9.2008 Indicator Function In some (few) cases it may be desireable to have different approaches based on (approximations) of the indicator function Non-convex constraint since indicator function only takes values zero and one  ( x ) = ½ 1 x 2 ­ 0 x = 2 ­

6. Martin Burger Convex and Nonconvex Relaxation 6 Graz, 27.9.2008 Example I: Imaging In segmentation often difficulties appear if local minimizers are computed only (objects not found, not characterized well …) Example: piecewise constant Mumford-Shah model (= Chan- Vese model) J is perimeter regularization (total variation)

7. Martin Burger Convex and Nonconvex Relaxation 7 Graz, 27.9.2008 Example I: Imaging Numerical evidence: local minimizers Chan-Vese algorithm seems to compute global minimizers (exact minimization w.r.t. constants alternated with gradient step) only with globally smeared Dirac delta, e.g.

8. Martin Burger Convex and Nonconvex Relaxation 8 Graz, 27.9.2008 Example II: Local Stress Constraints Structural Topology Optimization with local stress constraints Subject to Similar for von Mises stress

9. Martin Burger Convex and Nonconvex Relaxation 9 Graz, 27.9.2008 Example II: Local Stress Constraints Bilinear constraint creates enormous trouble In particular no constraint qualification (e.g. Slater) ! Linear reformulation (Stolpe-Svanberg 02)

10. Martin Burger Convex and Nonconvex Relaxation 10 Graz, 27.9.2008 Relaxation It is attractive to introduce a relaxation of this constraint, i.e. look for a function u such that instead of How to maintain the connection with the original optimization problem ? u ( x ) 2 [ 0 ; 1 ] Â ( x ) = ½ 1 x 2 ­ 0 x = 2 ­

11. Martin Burger Convex and Nonconvex Relaxation 11 Graz, 27.9.2008 Relaxation Basically there are two options - Convex relaxation: obtain some reformulation of original optimization problem and try to - Nonconvex relaxation: penalize deviation from desired values e.g. P ( u ) = Z u ( 1 ¡ u ) H ( u ) ! m i nsu b j ec tt o 0 · u · 1 H ( u ) + ² ¡ 1 P ( u ) ! m i nsu b j ec tt o 0 · u · 1

12. Martin Burger Convex and Nonconvex Relaxation 12 Graz, 27.9.2008 Relaxation + and – - Convex relaxation: needs special structures and special investigations in order to be applied. If it can be applied, it is possible to compute global minima - Nonconvex relaxation: can be applied in a universal way. Does in general not help with global minimization, at least continuation in the penalization parameter is possible

13. Martin Burger Convex and Nonconvex Relaxation 13 Graz, 27.9.2008 Nonconvex Relaxation Replace minimization of H subject to 0-1 constraints by Note: interesting only if H penalizes oscillations, otherwise complexity explodes (infinite-dimensional combinatorial optimization) Typically via perimeter constraint H ( u ) + ² ¡ 1 P ( u ) ! m i nsu b j ec tt o 0 · u · 1 H ( u ) = H 0 ( u ) + ® J ( u )

14. Martin Burger Convex and Nonconvex Relaxation 14 Graz, 27.9.2008 Nonconvex Relaxation Further approximation on perimeter term possible For appropriate scaling of the penalty term P this approach yields Gamma-convergence to the original variational problem as epsilon tends to zero (cf. Modica-Mortola 87) Resulting problems have simple structure (quadratic regularization), but are parameter-dependent H 0 ( u ) + ² ¡ 1 P ( u ) + ® 2 ² Z j r u j 2

15. Martin Burger Convex and Nonconvex Relaxation 15 Graz, 27.9.2008 Nonconvex Relaxation Parameter-dependence to be exploited in two instances - Discretization: adaptivity needed to resolve arising interfaces at width of order epsilon - Continuation: Convex problems for large epsilon, decrease epsilon to obtain optimal topologies

16. Martin Burger Convex and Nonconvex Relaxation 16 Graz, 27.9.2008 Nonconvex Relaxation Structural topology with local stress constraints:

17. Martin Burger Convex and Nonconvex Relaxation 17 Graz, 27.9.2008 Nonconvex Relaxation Structural topology with local stress constraints: Quadratic objective functional, linear inequality and equality constraints (huge number) Numerical solution (mb-Stainko 06, Stainko 06): FE-Discretization in NGSolve Minimization with interior point code IPOPT Parameter-robust multigrid preconditioning and iterative solution of linear systems in each iteration step of IPOPT

18. Martin Burger Convex and Nonconvex Relaxation 18 Graz, 27.9.2008 Nonconvex Relaxation Long beam, load on bottom, constrained von Mises stress ² = 1 ² = 1 4 ² = 1 64 ² = 1 16 ² = 1 256

19. Martin Burger Convex and Nonconvex Relaxation 19 Graz, 27.9.2008 Nonconvex Relaxation Short beam, load on top, constrained total stress ² = 1 ² = 1 4 ² = 1 2 ² = 1 8

20. Martin Burger Convex and Nonconvex Relaxation 20 Graz, 27.9.2008 Nonconvex Relaxation Short beam, load on top, constrained total stress ² = 1 16 ² = 1 64 ² = 1 128 ² = 1 32

21. Martin Burger Convex and Nonconvex Relaxation 21 Graz, 27.9.2008 Nonconvex Relaxation Observations: - Starting with small epsilon basically equivalent to level set based local optimization (phase-field method, ask Charlie Elliott) - Additional freedom of continuation in epsilon. Sufficiently small decrease usually yields global (topological) optima. - As epsilon gets small adaptive refinement is necessary to resolve diffuse interface. Adaptation is potential danger for global optimization. Again ok with some care. - Good solver for convave minimization is needed

22. Martin Burger Convex and Nonconvex Relaxation 22 Graz, 27.9.2008 Convex Relaxation Convex relaxation approach can be considered at least for the following structure, including a state variable v and the design variable u Minimization with respect to u and v can be done subsequently, we start with u J = total variation (perimeter for 0-1 functions)

23. Martin Burger Convex and Nonconvex Relaxation 23 Graz, 27.9.2008 Convex Relaxation Minimization with respect to u is of the form Obvious relaxation of the form

24. Martin Burger Convex and Nonconvex Relaxation 24 Graz, 27.9.2008 Convex Relaxation Minimization with respect to u is of the form - Is the indicator function solution of the relaxed problem ? - If yes, how can we compute such special solutions ? - If yes, are solutions stable with respect to g ? (Needed due to data noise in imaging and in order to understand alternating minimization)

25. Martin Burger Convex and Nonconvex Relaxation 25 Graz, 27.9.2008 Convex Relaxation Layer-cake representation: Co-area formula:

26. Martin Burger Convex and Nonconvex Relaxation 26 Graz, 27.9.2008 Convex Relaxation Hence the functional can be decomposed into level sets For solution of original problem and solution of relaxed problem:

27. Martin Burger Convex and Nonconvex Relaxation 27 Graz, 27.9.2008 Convex Relaxation This implies (for almost every t ) Indicator function is also solution of relaxed problem Almost every level set of relaxed solution is a solution of the original problem Chan-Esedoglu 04, Chan-Esedoglu-Nikolova 04

28. Martin Burger Convex and Nonconvex Relaxation 28 Graz, 27.9.2008 Convex Relaxation Solve relaxed problem and take level sets to solve original problem Since relaxed problem is convex, we can guarantee to find global optimum

29. Martin Burger Convex and Nonconvex Relaxation 29 Graz, 27.9.2008 Stability In general, level sets are not stable with respect to (weak) BV convergence Use to write

30. Martin Burger Convex and Nonconvex Relaxation 30 Graz, 27.9.2008 Stability Hence, solution of original problem also solves a quadratic problem with 0-1 constraint Stability for quadratic problem can be used to obtain weak stability for original and relaxed problem with respect to g Standard proof: implies for minimizers Set-valued weak* convergence to minimizers of original problem g k ! g i n L 2 ~ u k b oun d e d i n L 1 \ BV

31. Martin Burger Convex and Nonconvex Relaxation 31 Graz, 27.9.2008 Quantitative Stability Relaxation can also be used to obtain quantitative stability estimates Let Equivalent minimization Z gu d x + R ( u ) ! m i n u 2 BV ( ­ ) R ( u ) = ½ J ( u ) i f 0 · u · 1 + 1e l se

32. Martin Burger Convex and Nonconvex Relaxation 32 Graz, 27.9.2008 Quantitative Stability Optimality condition Difference of optimality conditions for different data yields g + p = 0 p 2 @ R ( u ) h p 1 p 2 ; u 1 u 2 i = h g 1 g 2 ; u 1 u 2 i ¡¡¡¡¡

33. Martin Burger Convex and Nonconvex Relaxation 33 Graz, 27.9.2008 Quantitative Stability With upper and lower bound on u one can estimate generalized Bregman distance Direct estimate for subgradients Detailed interpretations of subgradients for total variation can yield information about closeness of contours ¡ ¡ ¡ k p 1 p 2 k L q · k g 1 g 2 k L q d R ( u 1 ; u 2 ) · k g 1 g 2 k L 1

34. Martin Burger Convex and Nonconvex Relaxation 34 Graz, 27.9.2008 Applications Several applications can be put in the general form by appropriate choice of v

35. Martin Burger Convex and Nonconvex Relaxation 35 Graz, 27.9.2008 Applications - Minimal compliance in structural optimization: v is displacement, G is energy density - Optimal design of composite membranes / photonic crystal Minimal eigenvalue for Helmholtz-equation Not directly applicable to eigenvalue maximization and bandgaps

36. Martin Burger Convex and Nonconvex Relaxation 36 Graz, 27.9.2008 Applications - Chan-Vese segmentation model Try to find a partition of the domain in two typical mean gray values Chan-Vese 99

37. Martin Burger Convex and Nonconvex Relaxation 37 Graz, 27.9.2008 Applications - Chan-Vese segmentation model After rescaling, Chan-Vese algorithm with smoothed delta can be interpreted as a (almost) projected gradient descent on the relaxed functional P damps updates out of feasible set u k + 1 = u k P ( u k ) £ ¸ ( c 1 ¡ f ) 2 ¡ ¸ ( c 2 ¡ f ) 2 + J ( u k ) ¤ ¡

38. Martin Burger Convex and Nonconvex Relaxation 38 Graz, 27.9.2008 Applications Chan-Vese segmentation model Data courtesy of Institute for Physiology, WWU

39. Martin Burger Convex and Nonconvex Relaxation 39 Graz, 27.9.2008 Applications Similar for other priors in regions - Region-based Mumford-Shah: Piecewise smooth in subregions - Histogram-based: maximally different histograms in the subregions Esedolgu et al 07

40. Martin Burger Convex and Nonconvex Relaxation 40 Graz, 27.9.2008 Applications Adaptive Priors Example: parametrized anisotropic perimeter MR-T1 Image Chan-Vese Segmentation

41. Martin Burger Convex and Nonconvex Relaxation 41 Graz, 27.9.2008 Adaptive Priors Problem: perimeter constraint leads to cut-off of small elongated structures (sulci in the brain)

42. Martin Burger Convex and Nonconvex Relaxation 42 Graz, 27.9.2008 Adaptive Priors for Brain Imaging Sulci are important for applications to EEG/ MEG inversion Brain activity is modeled by dipoles at the sulci boundaries pointing in normal direction Baillet et al, 2001

43. Martin Burger Convex and Nonconvex Relaxation 43 Graz, 27.9.2008 EEG/MEG Inversion Dipole activity creates electric and magnetic field on / outside the human skull. Measured by EEG / MEG Simulate quasistatic Simulate quasistatic Maxwell equations

44. Martin Burger Convex and Nonconvex Relaxation 44 Graz, 27.9.2008 EEG/MEG Inversion Inversion tries to find dipole location from measured EEG / MEG data Highly undeterdetermined, needs strong prior knowledge on possible dipole source locations and orientations Obtained from segmentation and classification of MR images

45. Martin Burger Convex and Nonconvex Relaxation 45 Graz, 27.9.2008 Brain segmentation Usual prior in terms of signed distance function to the surface „Isotropic perimeter“ is curve integral of Favours rounded structures (circle-like) Can be understood from optimality conditions and subgradients (related to mean curvature). Alternatively from isoperimetric problems, minimizing perimeter at fixed volume (Wulff shape)

46. Martin Burger Convex and Nonconvex Relaxation 46 Graz, 27.9.2008 Brain segmentation In order to allow elongated structures we use „Anisotropic perimeter“, curve integral of Favours elongated structures (ellipses) Elongation more and more pronounced with  tending to 0 and 1

47. Martin Burger Convex and Nonconvex Relaxation 47 Graz, 27.9.2008 Brain segmentation In the above definition, main axes of the ellipses lie in coordinate directions. We still need to introduce a rotation (angle  )

48. Martin Burger Convex and Nonconvex Relaxation 48 Graz, 27.9.2008 Brain segmentation Isoperimetric problems at fixed  and  Can be seen from corresponding anisotropic mean curvature flow (descent flow for regularization functional)

49. Martin Burger Convex and Nonconvex Relaxation 49 Graz, 27.9.2008 Brain segmentation Segmentation result at fixed  and 

50. Martin Burger Convex and Nonconvex Relaxation 50 Graz, 27.9.2008 Brain segmentation Local definition of  and in particular  is needed Iterate segmentation with update in each pixel

51. Martin Burger Convex and Nonconvex Relaxation 51 Graz, 27.9.2008 Brain segmentation Extension to three dimensions, two angles needed

52. Martin Burger Convex and Nonconvex Relaxation 52 Graz, 27.9.2008 Brain segmentation Adaptive prior in segmentation and classification of normal directions

53. Martin Burger Convex and Nonconvex Relaxation 53 Graz, 27.9.2008 Generalization Generalizations to more general structure

54. Martin Burger Convex and Nonconvex Relaxation 54 Graz, 27.9.2008 Generalization Simple idea: add new variable w = u and penalize the constraint (Moreau-Yosida Regularization of the first part)

55. Martin Burger Convex and Nonconvex Relaxation 55 Graz, 27.9.2008 Generalization Use again u 2 = u before relaxing Relaxation exact, still convex problem for u at fixed v and w convex for w at fixed v and u. Overall functional nonconvex

56. Martin Burger Convex and Nonconvex Relaxation 56 Graz, 27.9.2008 Generalization Test example: simplest inverse obstacle problem M ½ ­ or M ½ @ ­

57. Martin Burger Convex and Nonconvex Relaxation 57 Graz, 27.9.2008 Generalization Evolution of alternating minimization approach, no data noise

58. Martin Burger Convex and Nonconvex Relaxation 58 Graz, 27.9.2008 Generalization Evolution of alternating minimization approach, 3% data noise

59. Martin Burger Convex and Nonconvex Relaxation 59 Graz, 27.9.2008 Generalization Test example: simplest inverse obstacle problem M ½ ­ or M ½ @ ­

60. Martin Burger Convex and Nonconvex Relaxation 60 Graz, 27.9.2008 Mathematical Imaging@WWU Christoph Brune Alex Sawatzky Frank Wübbeling Thomas Kösters Martin Benning Marzena Franek Bärbel Schlake Christina Stöcker Mary Wolfram Thomas Grosser Jahn Müller

61. Martin Burger Convex and Nonconvex Relaxation 61 Graz, 27.9.2008 Based on further collaborations with Michael Hintermüller (Graz) Roman Stainko (Linz / DTU Lyngby) Denis Neiter (Ecole Polytechnique, Internship at WWU) Carsten Wolters (WWU, University Hospital) Funding: Regularization with Singular Energies (DFG), SFB 656 (DFG), Cartoon-Reconstruction and Segmentation in Nanoscopy (BMBF), European Institute for Molecular Imaging (WWU + SIEMENS Medical Solutions)