Mumford-Shah Level-Set Ideas for Problems in Medical Imaging Paris Dauphine 2005

Transcription

1 1/26 Mumford-Shah Level-Set Ideas for Problems in Medical Imaging Paris Dauphine 2005 Wolfgang Ring, Marc Droske, Ronny Ramlau Institut für Mathematik, Universität Graz

2 General Mumford-Shah like 2/26 functionals

3 We consider the problem of determining an unknown function v : D R N with might be singular (discontinuous) across an unknown singularity set Γ D from given data y d Y. 3/26

4 We consider the problem of determining an unknown function v : D R N with might be singular (discontinuous) across an unknown singularity set Γ D from given data y d Y. 3/26 Suppose for the moment that the singularity set Γ is known. Then the function v D\Γ is an element in a Hilbert space X(Γ) which might be dependent on Γ.

5 We consider the problem of determining an unknown function v : D R N with might be singular (discontinuous) across an unknown singularity set Γ D from given data y d Y. 3/26 Suppose for the moment that the singularity set Γ is known. Then the function v D\Γ is an element in a Hilbert space X(Γ) which might be dependent on Γ. Note: X(Γ) is a space of functions with domain of definition given by D \ Γ. The norm on X(Γ) does not measure what happens on/across Γ.

6 The relation between v and the given (exact) data is supposed to be of the form y d = Kv (1) 4/26 where K = K(Γ) with K : X(Γ) Y is a continuous (possibly nonlinear) operator.

7 Simultaneous determination of functional and geometric data 5/26 The singularity set Γ and the function v X(Γ) are to be simultaneously determined as solution to min J MS (v, Γ) Γ G v X(Γ) J MS (v, Γ) = 1 2 Kv y d 2 Y + ν 2 v 2 X(Γ) + µ Γ 1 ds

8 Simultaneous determination of functional and geometric data 5/26 The singularity set Γ and the function v X(Γ) are to be simultaneously determined as solution to min J MS (v, Γ) Γ G v X(Γ) J MS (v, Γ) = 1 2 Kv y d 2 Y + ν 2 v 2 X(Γ) + µ Data fit Γ 1 ds Regularization without penalization on/across Γ Control of Γ

9 Elimination of the functional variable Shape optimization approach 6/26

10 Elimination of the functional variable Shape optimization approach Step 1: For fixed Γ G solve 6/26 Denote the solution v(γ). min J MS (v, Γ). v X(Γ)

11 Elimination of the functional variable Shape optimization approach Step 1: For fixed Γ G solve 6/26 Denote the solution v(γ). min J MS (v, Γ). v X(Γ) Step 2: Consider the shape optimization problem min Γ G J MS(v(Γ), Γ). Calculate a descent direction F shape sensitivity analysis. using techniques from

12 Step 3: Update the geometric variable Γ in the chosen descent direction using a level-set formulation for the propagation of the geometric variable. Solve 7/26 u t + F u = 0 for an appropriate time-step. Here Γ = {u = 0}.

13 Step 1: Solution of the optimality system w.r.t. v 8/26

14 Step 1: Solution of the optimality system w.r.t. v The optimal v(γ) is found by solving the optimality system v J MS (v(γ), Γ) = 0. 8/26

15 Step 1: Solution of the optimality system w.r.t. v The optimal v(γ) is found by solving the optimality system v J MS (v(γ), Γ) = 0. 8/26 For linear K: with v(γ) X(Γ). K (Kv(Γ) y d ) + νv(γ) = 0

16 Shape sensitivity analysis for reduced functionals 9/26

17 Shape sensitivity analysis for reduced functionals We consider the reduced functional 9/26 Ĵ(Γ) = J MS (v(γ), Γ) where v(γ) = argmin v J MS (v, Γ).

18 Shape sensitivity analysis for reduced functionals We consider the reduced functional 9/26 Ĵ(Γ) = J MS (v(γ), Γ) where v(γ) = argmin v J MS (v, Γ). We get dĵ(γ; F ) = vj ( v(γ), Γ ) v (Γ; F ) + Γ J(v(Γ), Γ; F ) (2)

19 Shape sensitivity analysis for reduced functionals We consider the reduced functional 9/26 Ĵ(Γ) = J MS (v(γ), Γ) where v(γ) = argmin v J MS (v, Γ). We get dĵ(γ; F ) = vj ( v(γ), Γ ) v (Γ; F ) + Γ J(v(Γ), Γ; F ) (2) Since v(γ) satisfies the Euler-Lagrange equations for J MS w.r.t. v, the first term in (2) vanishes.

20 Descent directions and the choice of an appropriate metric 10/26

21 Shape differential and steepest descent direction 11/26

22 Shape differential and steepest descent direction Suppose F Z, (Z... space of perturbations). The shape differential δj(γ) Z of J at Ω is an element in Z satisfying 11/26 dj(γ; F ) = δj(γ), F Z,Z. (δj... covariant repr. of the shape derivative)

23 Shape differential and steepest descent direction Suppose F Z, (Z... space of perturbations). The shape differential δj(γ) Z of J at Ω is an element in Z satisfying 11/26 dj(γ; F ) = δj(γ), F Z,Z. (δj... covariant repr. of the shape derivative) A steepest descent direction F sd is an element in Z satisfying dj(γ; F sd ) = min dj(γ; F ). F Z F Z 1 (F sd... contravariant repr. of the shape derivative)

24 Choice of the Metric For the update of the geometry, the descent direction must be determined. 12/26

25 Choice of the Metric For the update of the geometry, the descent direction must be determined. 12/26 The descent direction depends on the choice of the space Z and on the respective norm.

26 Choice of the Metric For the update of the geometry, the descent direction must be determined. 12/26 The descent direction depends on the choice of the space Z and on the respective norm. Different norms lead to different behaviors of the algorithm.

27 Choice of the Metric For the update of the geometry, the descent direction must be determined. 12/26 The descent direction depends on the choice of the space Z and on the respective norm. Different norms lead to different behaviors of the algorithm. Restrictive condition on the choice of Z: δ Z.

28 A preconditioned gradient method 13/26

29 A preconditioned gradient method Suppose that there exists a G = G(Γ) H 1 (Γ) such that dj MS (Γ; F ) = G, F Γ H 1 (Γ),H 1 0 (Γ). 13/26

30 A preconditioned gradient method Suppose that there exists a G = G(Γ) H 1 (Γ) such that dj MS (Γ; F ) = G, F Γ H 1 (Γ),H 1 0 (Γ). 13/26 To find the steepest descent direction w.r.t. the H norm, we have to solve the constrained optimization problem min G, F Γ F H0 1(Γ) H 1 (Γ),H1 0 (Γ) F H 1 0 (Γ) 1

31 Introducing the Lagrange functional ( ( L(F, λ) = G, F Γ + λ Γ F 2 + F 2) ds 1) Γ 14/26 we get the optimality condition F = 1 2λ ( Γ + id) 1 G (3)

32 Introducing the Lagrange functional ( ( L(F, λ) = G, F Γ + λ Γ F 2 + F 2) ds 1) Γ 14/26 we get the optimality condition F = 1 2λ ( Γ + id) 1 G (3) The gradient direction G is preconditioned by an inverse elliptic operator. The multiplier λ is chosen such that the H0 1 -norm of F is one.

33 A piecewise constant 15/26 Mumford-Shah approach for x-ray tomography (with Ronny Ramlau)

34 Objectives: 16/26

35 Objectives: Simultaneously reconstruct the density distribution and the boundaries of regions with approximately homogeneous densities from Radon-transform data. 16/26 Segment directly from the data, estimate objects Reduce ill-posedness by reducing the dimension of the unknown inversion of noisy data Comparable in quality and speed to established methods

36 Objectives: Simultaneously reconstruct the density distribution and the boundaries of regions with approximately homogeneous densities from Radon-transform data. 16/ Segment directly from the data, estimate objects Reduce ill-posedness by reducing the dimension of the unknown inversion of noisy data Comparable in quality and speed to established methods

37 Objectives: Simultaneously reconstruct the density distribution and the boundaries of regions with approximately homogeneous densities from Radon-transform data. Segment directly from the data, estimate objects 16/ Reduce ill-posedness by reducing the dimension of the unknown inversion of noisy data Comparable in quality and speed to established methods

38 The functional and the resulting geometric flow 17/26 We fix the space of admissible densities as: X(Γ) = P C(D \ Γ) = { n(γ) i=1 f i χ Ω Γ i } : f i R Ω Γ i... connected component of D \ Γ. L 2 (D) (4)

39 The functional and the resulting geometric flow 17/26 We fix the space of admissible densities as: X(Γ) = P C(D \ Γ) = { n(γ) i=1 f i χ Ω Γ i } : f i R Ω Γ i... connected component of D \ Γ. L 2 (D) The Radon Mumford-Shah functional is given as J(f, Γ) = Rf g d 2 L 2 (R S 1 ) (4) + α Γ. (5)

40 Solving the Euler-Lagrange System (Step 1): f J(f(Γ), Γ) h = Rf(Γ) g d, Rh L2 (R S 1 ) = 0 h. (6) 18/26

41 Solving the Euler-Lagrange System (Step 1): f J(f(Γ), Γ) h = Rf(Γ) g d, Rh L2 (R S 1 ) = 0 h. (6) 18/26 This is equivalent to Af(Γ) = g (7) where A = (a ij ) and g = (g i ) t. a ij = 2 g i = x Ω Γ i y Ω Γ j R g d dx = Ω Γ i y x n i (x), n j (x) ds(y) ds(x). (8) x Ω Γ i g(ω x, ω) dω dx. (9) ω S 1

42 The shape derivative (Step 2): dj(γ; F ) = 4 k x Γ k p d(k) s p f p l q d(l) s q f q y Γ l y x y x, nk (y) ds(y) F (x) ds(x) 2 s p f p g d (ω x, ω) dω F (x) ds(x) k x Γ k p d(k) ω S 1 + κ(x) F (x) ds(x). k x Γ k f p...densitiy values, s p = ±1, κ...curvature. 19/26

43 Numerical observations 20/26

44 Numerical observations Mild degree of illposedness 20/ reconstruction of the densitiy: alpha = 0, nu = 0 reconstruction of the densitiy: alpha = 0, nu = Benefit of H 1 - preconditioning: Reduction iteration numbers of Works for real data simulations

45 Numerical observations Mild degree of illposedness Benefit of H 1 - preconditioning: Reduction of iteration numbers 20/26 Works for real data simulations ν = 10 2 ν = 10 1 ν = 10 0 ν = 10 1 ν = 10 2 ν = 10 3 ν = 0 α = α =

46 Numerical observations 20/ Mild degree of illposedness Benefit of H 1 - preconditioning: Reduction of iteration numbers Works for real data simulations

47 A Mumford-Shah like 21/26 approach for simultaneous registration and segmentation of multimodal data sets. (with Marc Droske)

48 Objectives: 22/26

49 Objectives: Simultaneously segment two images and match similar structures onto each other. 22/ Work with noisy multimodal images Transfer features which are strong in one image onto the other, where the feature is weak Work with real datasets

50 Objectives: Simultaneously segment two images and match similar structures onto each other. 22/26 Work with noisy multimodal images Transfer features which are strong in one image onto the other, where the feature is weak Work with real datasets

51 Objectives: Simultaneously segment two images and match similar structures onto each other. 22/26 Work with noisy multimodal images Transfer features which are strong in one image onto the other, where the feature is weak Work with real datasets

52 The functional: E MS (Γ, Φ, R, T ) = 1 R R 0 2 dx + µ R 2 dx 2 D 2 D\Γ + 1 T T 0 2 dx+ µ T 2 dx+αh N 1 (Γ)+νE reg (Φ) 2 D 2 D\Γ Φ 23/26 R, T... reference and template images, Φ... transformation between the images, Γ... edge-set in the reference image, Γ Φ = Φ(Γ)... edge-set in the template image, E reg... regularization term penalizing deviation from a rigid body motion.

53 Algorithm: Minimize w.r.t. T and R for fixed Γ, Φ. Consider the reduced functional 24/26 Ê(Γ, Φ) = E(Γ, Φ, R(Γ), T (Γ, Φ)). Calculate descent directions with respect to Γ and Φ. Use composite finite elements and multigrid solvers. Update Γ and φ. Use the level-set equation for the update of Γ.

54 An application from dentistry: 25/26

55 An application from dentistry: 25/26

56 An application from dentistry: 25/26

57 Further applications Optical flow estimation 26/26 Inversion of SPECT data Occlusions etc.