A Second Order Non-smooth Variational Model for Restoring Manifold-valued Images

Transcription

1 A Second Order Non-smooth Variational Model for Restoring Manifold-valued Images Ronny Bergmann Mathematical Image Processing and Data Analysis Department of Mathematics University of Kaiserslautern July 10, th Rhein-Main Arbeitskreis Mathematics of Computation, Darmstadt joint work with M. Bačák (Télécom ParisTech), G. Steidl (U Kaiserslautern), A. Weinmann (TU Munich)

2 Contents Introduction Second Order Differences on Manifolds Proximum & CPPA Examples 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 2

3 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 3

4 Restoring Images: Denoising and Inpainting given noisy image f : V R, V G= {1,...,N} {1,...,M} reconstruct original image u 0 : G R pixel from G\V have to be inpainted approach: Minimize variational model E(u) := F(f, u) + α R(u), α>0 data term regularization term first order models (total variation, TV) isotropic model with discrete gradient u : R iso (u) := u = u i+1,j u i,j 2 + u i,j+1 u i,j 2 i,j i,j anisotropic model: R aniso (u) := i,j ( ui+1,j u i,j + u i,j+1 u i,j ) R. Bergmann A Second Order TV-type Model on Manifolds 4

5 Denoising Gray-valued Images A drawback of first order models original noisy first order model R. Bergmann A Second Order TV-type Model on Manifolds 5

6 Denoising Gray-valued Images A drawback of first order models and a remedy. original noisy second order model R. Bergmann A Second Order TV-type Model on Manifolds 5

7 Riemannian Manifolds Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Here: Images with pixel values in a Riemannian manifold M Goal: Second Order Model for images f: V M Applications with manifold-valued images S 1 S 2 InSAR, HSI(HSV) color space, phase space directions, chromaticity-brightness colorspace SO(3) orientations, electron backscattered diffraction P(s) DT-MRI, covariance matrices R. Bergmann A Second Order TV-type Model on Manifolds 6

8 Symmetric Positive Definite Matrices: x Ax > 0. each pixel is a 3 3 symmetric positive definite matrix, ellipsoids: x Ax = 1 R. Bergmann A Second Order TV-type Model on Manifolds 7

9 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 8

10 First and Second Order Differences Let s restrict ourselves to signals for now... On R n line γ(t) =x + t(y x) Riemannian manifold M geodesic path γ x,z (t) G = {1,...,N}, V G distance x y 2 first order model f i u i α u i u i+1 2 i V i G\{N} geodesic distance d: M M R 0 first order model d(f i, u i ) 2 + α d(u i, u i+1 ) i V i G\{N} R. Bergmann A Second Order TV-type Model on Manifolds 9

11 First and Second Order Differences Let s restrict ourselves to signals for now... On R n line γ(t) =x + t(y x) Riemannian manifold M geodesic path γ x,z (t) G = {1,...,N}, V G distance x y 2 first order model f i u i α u i u i+1 2 i V i G\{N} second oder difference x 2y + z 2 y geodesic distance d: M M R 0 first order model d(f i, u i ) 2 + α d(u i, u i+1 ) i V i G\{N} How to model that on M? y x z x z R. Bergmann A Second Order TV-type Model on Manifolds 9

12 First and Second Order Differences Let s restrict ourselves to signals for now... On R n line γ(t) =x + t(y x) Riemannian manifold M geodesic path γ x,z (t) G = {1,...,N}, V G distance x y 2 first order model f i u i α u i u i+1 2 i V i G\{N} second oder difference (x + z) y 2 y geodesic distance d: M M R 0 first order model d(f i, u i ) 2 + α d(u i, u i+1 ) i V i G\{N} idea: mid point formulation y c x z x c(x, z) z c(x, z) R. Bergmann A Second Order TV-type Model on Manifolds 9

13 First and Second Order Differences Let s restrict ourselves to signals for now... On R n line γ(t) =x + t(y x) Riemannian manifold M geodesic path γ x,z (t) G = {1,...,N}, V G distance x y 2 first order model f i u i α u i u i+1 2 i V i G\{N} second oder difference 2 c(x, z) y 2, c(x, y): mid point y geodesic distance d: M M R 0 first order model d(f i, u i ) 2 + α d(u i, u i+1 ) i V i G\{N} idea: mid point formulation y c x z x c(x, z) z c(x, z) R. Bergmann A Second Order TV-type Model on Manifolds 9

14 A Second Order TV-type Model Mid points between x, z M: C x,z := { ( c M: c = γ T ) x,z 2 for any geodesic γ x,z, T := L(γ x,z ) } The Absolute Second Order Difference: d 2 (x, y, z) := min c C x,z d(c, y), x, y, z M. Second Order TV-type Model for M-valued signals f E(u) := d(f i, u i ) 2 + α d(u i, u i+1 )+β d 2 (u i 1, u i, u i+1 ) i V i G\{N} i G\{1,N} R. Bergmann A Second Order TV-type Model on Manifolds 10

15 Example: The Circle S 1 Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Cyclic data: p, q, r S 1 := { s R 2 : s 2 = 1 } x, y, z [, ) d 2 (x, y, z): Several unwrappings q p r z x y z x y x 2y + z R. Bergmann A Second Order TV-type Model on Manifolds 11

16 Example: The Circle S 1 Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Cyclic data: p, q, r S 1 := { s R 2 : s 2 = 1 } x, y, z [, ) d 2 (x, y, z): Several unwrappings q p r z z x y z x y x y z x y x 2y + z R. Bergmann A Second Order TV-type Model on Manifolds 11

17 Example: The Circle S 1 Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Cyclic data: p, q, r S 1 := { s R 2 : s 2 = 1 } x, y, z [, ) d 2 (x, y, z): Several unwrappings q p r z z z x y z x y x y z x y x y z x y x 2y + z R. Bergmann A Second Order TV-type Model on Manifolds 11

18 Example: The Circle S 1 Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Cyclic data: p, q, r S 1 := { s R 2 : s 2 = 1 } x, y, z [, ) d 2 (x, y, z): Minimize over all unwrappings q p r z z x y z x y x y z x y z x y z x y d 2 (x, y, z) := min (x + µ) 2(y + µ)+(z + µ) = (x 2y + z) 2 µ R R. Bergmann A Second Order TV-type Model on Manifolds 11

19 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 12

20 Proximal Mappings Proximal algorithms are standard tool for solving non-smooth, constrained minimization problems. To compute a minimizer of E(u) we need proximal mappings For a proper, closed, convex function ϕ: R n (, + ] and λ>0 the proximal mapping is defined by [Moreau, 1965; Rockafellar, 1976] 1 prox λϕ (g) :=arg min u R n 2 u g λϕ(u). For a function ϕ: M n (, + ] and λ>0 the proximal mapping is defined by [Ferreira, Oliveira, 2002] 1 n prox λϕ (g) :=arg min d(u i, g i ) 2 + λϕ(u). u M n 2 i=1 Note: For a minimizer u of ϕ it holds prox λϕ (u )=u. R. Bergmann A Second Order TV-type Model on Manifolds 13

21 The Cyclic Proximal Point Algorithm To find arg min x ϕ(x) use Picard iteration: Proximal Point Algorithm fast evaluation of prox λϕ needed x (k) = prox λϕ (x (k 1) ), k > 0 For ϕ = c ϕ l use Cyclic Proximal Point Algorithm (CPPA) [Bertsekas, 2011] l=1 l+1 (k+ x c ) = prox λk ϕ l (x (k+ l c ) ), i = 0,...,c 1, k > 0 For real-valued data: Converges to a minimizer if {λ k } l 2 (Z)\l 1 (Z). R. Bergmann A Second Order TV-type Model on Manifolds 14

22 Minimizing the Second Order Model To minimize E(u) := i V d(f i, u i ) 2 + α d(u i, u i+1 )+β i G\{N} take each summand as one ϕ l in the CPPA. For the involved proximal maps we have i G\{1,N} d 2 (u i 1, u i, u i+1 ) ϕ(u i )=d(f i, u i ) 2, analytical solution [Ferreira, Oliveira, 2002] ϕ(u i, u i+1 )=d(u i, u i+1 ), analytical solution [Weinmann, Storath, Demaret, 2014] ϕ(u i 1, u i, u i+1 )=d 2 (u i 1, u i, u i+1 ) analytical solution for S 1 [B., Laus, Steidl, Weinmann, 2014] numerical solution otherwise [Bačák, B., Steidl, Weinmann, 2015] R. Bergmann A Second Order TV-type Model on Manifolds 15

23 A Proximal Mapping on S 1 Introduction Second Order Differences on Manifolds Proximum & CPPA Examples Theorem [B., Laus, Steidl, Weinmann, 2014] The minimizer(s) of are given using { 1 prox λd2 (g) =arg min u (S 1 ) j=1 } d(u j, g j ) 2 + λd 2 (u 1, u 2, u 3 ) { s := sgn(g 1 2g 2 +g 3 ) 2, w := (1, 2, 1), and m := min λ, 1 } 6 ( g, w ) 2 as 1 if ( g, w ) 2 <by prox λd2 (g) =(g smw) 2 2 if ( g, w ) 2 = by prox λd2 (g) =(g ± smw) 2 R. Bergmann A Second Order TV-type Model on Manifolds 16

24 Proximal Mapping of a Second Order Difference on M To Compute { 1 prox λd2 (g) =arg min u M i=1 } d(u i, g i ) 2 + λd 2 (u 1, u 2, u 3 ) (sub)gradient descent method requires gradient of d 2 (x, y, z) =d(c(x, z), y) M 3d 2 =( M d 2 (, y, z), M d 2 (x,, z), M d 2 (x, y, )). For y c(x, z) and an ONB {ξ 1,...,ξ n } of T x M M d 2 (x,, z)(y) = log y c(x, z) T y M log y c(x, z) y M d 2 (, y, z) by chain rule n logc(x) y M d 2 (, y, z)(x) =, D x c[ξ k ] log c(x) y c(x) k=1 c(x), ξ k T x M. R. Bergmann A Second Order TV-type Model on Manifolds 17

25 Proximal Mapping of a Second Order Difference on M To Compute { 1 prox λd2 (g) =arg min u M i=1 } d(u i, g i ) 2 + λd 2 (u 1, u 2, u 3 ) (sub)gradient descent method requires gradient of d 2 (x, y, z) =d(c(x, z), y) M 3d 2 =( M d 2 (, y, z), M d 2 (x,, z), M d 2 (x, y, )). For y c(x, z) and an ONB {ξ 1,...,ξ n } of T x M M d 2 (x,, z)(y) = log y c(x, z) T y M log y c(x, z) y M d 2 (, y, z) by chain rule n logc(x) y M d 2 (, y, z)(x) =, D x c[ξ k ] log c(x) y c(x) k=1 c(x), ξ k T x M. R. Bergmann A Second Order TV-type Model on Manifolds 17

26 Geodesic Variation How do small changes in x affect the geodesic? Introduction Second Order Differences on Manifolds Proximum & CPPA Examples σ x,ξk (s), k = 1,...,n, s ( ε, ε): curve starting in σ x,ξk (0) =x with direction σ x,ξ k (0) =ξ k ζ k (s): direction in σ x,ξk (s) towards z. geodesic variation of the geodesic γ x,z Γ k (s, t) := exp σx,ξk (s) (tζ k (s)), s ( ε, ε), t [0, T], T = d(x, z) R. Bergmann A Second Order TV-type Model on Manifolds 18

27 Jacobi Fields Indicate directions of change along the geodesic and rephrase computation of the gradient to... The Jacobi field J k, k = 1,...,n, along γ x,z is defined by J k (t) := s Γ k(s, t) s=0, t [0, T] Lemma D x c[ξ k ]=J k ( T 2 ) Lemma A Jacobi field J k fulfills the System of ODEs D 2 t 2 J k + R(J k, γ x,z ) γ x,z = 0, J k (0) =ξ k, J k (T) =0 with the Riemannian curvature tensor R: TM TM TM TM. R. Bergmann A Second Order TV-type Model on Manifolds 19

28 Riemannian Symmetric Spaces...for certain manifolds there s still hope. For every x Mexists an isometry s x on M, i.e. s x (x) =x, D x s[ξ] = ξ, for all ξ T x M covariant derivative Examples S n Hyperbolic Spaces P(s) Grasmannians R = 0 Riemannian symmetric spaces are the most beautiful and most important Riemannian manifolds. (J.-H. Eschenburg, 1997) R. Bergmann A Second Order TV-type Model on Manifolds 20

29 A Specific Frame Along γ x,z Decompose the Jacobi field in Riemannian symmetric spaces With parallel transport of an orthonormal frame {Θ 1 =Θ 1 (t),...,θ n =Θ n (t)}, i.e. n J k (t) = a k,i (t)θ i (t). The system of ODEs D 2 t 2 J k + R(J k, γ x,z ) γ x,z = 0, J k (0) =ξ k, J k (T) =0 simplifies to a a linear system of ODEs with constant coefficients ( R(Θi a (t)+ga(t) =0, G :=, γ ) γ x,z x,z, Θ ) n j. γ x,z i,j=1 Choose {ξ 1,...,ξ n } at t = 0 that diagonalize G with eigenvalues κ i Decoupled ODEs i=1 R. Bergmann A Second Order TV-type Model on Manifolds 21

30 The Derivative D x c[ξ k ] in Symmetric Spaces Finally we are able to evaluate the Jacobi field at the mid point T/2 Corollary Let κ i denote the eigen values of G. With T = d(x, z) we have for k = 1,...,n ) ( κk T sinh 2 sinh( ( κ k T) Θ k( T 2 ), κ k < 0, ( ) D x c[ξ k ]=J T k 2) = κk T sin 2 sin( κ k T) Θ k( T 2 ), κ k > 0, 1 2 Θ k( T 2 ), κ k = 0. Gradient descent to approximate the Proximal Mapping of the second order difference on a manifold. R. Bergmann A Second Order TV-type Model on Manifolds 22

31 Examples of Gradients d 2 (, y, z)(x) the sphere S n choose ξ 1 = γ x,z (0) =log x z and extend to ONB on T x M D x c[ξ 1 ]= 1 2 Θ 1( T 2 ), D xc[ξ k ]= sin T 2 sin T Θ k( T 2 ), k = 2,...,n. R. Bergmann A Second Order TV-type Model on Manifolds 23

32 Examples of Gradients d 2 (, y, z)(x) the sphere S n choose ξ 1 = γ x,z (0) =log x z and extend to ONB on T x M D x c[ξ 1 ]= 1 2 Θ 1( T 2 ), D xc[ξ k ]= sin T 2 sin T Θ k( T 2 ), k = 2,...,n. symmetric positive definite matrices P(s), n = s(s+1) 2, Decompose v = log x z = s i=1 λ iv i v i T x P(s) ={x} Sym(s) and choose ξ ij := { 1 2 (v iv j + v j v i ), i = j, 1 2 (v i v j + v j v i ) i j,, 1 i j s as ONB of T x P(s). From κ i,j = 1 4 (λ j λ j ) 2 we obtain 1 2 Θ ij( T 2 ) λ i = λ j, D x c[ξ ij ]= sinh( T 4 λ i λ j ) sinh( T 2 λ i λ j ) Θ ij( T 2 ) λ i λ j. R. Bergmann A Second Order TV-type Model on Manifolds 23

33 Minimizing the Second Order Model: An efficient Splitting To minimize E(u) := i V d(f i, u i ) 2 + α d(u i, u i+1 )+β i G\{N} i G\{1,N} d 2 (u i 1, u i, u i+1 ) take each summand as ϕ l in the CPPA: c = 3N 3 proximal maps. Improved by ϕ(u i )=d(f i, u i ) 2, evaluate in parallel ϕ(u i, u i+1 )=d(u i, u i+1 ), even-odd-splitting ϕ(u i 1, u i, u i+1 )=d 2 (u i 1, u i, u i+1 ), splitting w. r. t. mod3 reduced to c = 6 parallel/vector-valued proximal maps for signals c = 15 for images. R. Bergmann A Second Order TV-type Model on Manifolds 24

34 Convergence of the CPPA on Manifolds Hadamard manifold: complete Riemannian manifold M with 2d(c(x, z), c(y, z)) d(x, y), for all x, y, z M triangles are thinner than in R n smaller angle sum than in R n Theorem (Exact CPPA) [Bačák, B., Steidl, Weinmann, 2015] Let M be a Hadamard manifold and let ϕ = c l=1 ϕ l,ϕ l : M R, have a global minimizer. Assume there exist p H, C > 0 such that for each l = 1,...,L and all x, y Mwe have ϕ l (x) ϕ l (y) Cd(x, y)(1 + d(x, p)). Then the sequence {x (k) } k N defined by the CPPA x (k+1) := prox λk ϕ c prox λk ϕ c 1... prox λk ϕ 1 (x (k) ) with {λ k } l 2 (Z)\l 1 (Z) converges for every x (0) to a minimizer of ϕ. R. Bergmann A Second Order TV-type Model on Manifolds 25

35 Convergence with Inexact Proximal Mappings our second order model functional E(u) fulfills these requirements of ϕ but the second order proximal maps are only approximately given Theorem (Inexact CPPA) [Bačák, B., Steidl, Weinmann, 2015] Let M be a Hadamard manifold, let ϕ be given as in the last Theorem. Let {x (k) } k N be the sequence generated by the inexact cyclic PPA with and d (x (k+ l c ) l 1 ) (k+, prox λk ϕl (x c ) ) < ε k c, ε k <. k=0 Then the sequence {x (k) } k N converges to a minimizer of ϕ. R. Bergmann A Second Order TV-type Model on Manifolds 26

36 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 27

37 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal function f: [0, 1] S 1 sampled to obtain data f o =(f o,i ) 500 i=1 f could be a wrapped version of the gray plot, hence jumps >at and 16 are due to the representation system R. Bergmann A Second Order TV-type Model on Manifolds 28

38 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal function f: [0, 1] S 1 sampled to obtain data f o =(f o,i ) 500 i=1 adding wrapped Gaussian noise, σ = 0.2 noisy data f n =(f 0 + η) 2 R. Bergmann A Second Order TV-type Model on Manifolds 28

39 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal comparison of f o & f n with R. Bergmann A Second Order TV-type Model on Manifolds 28

40 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal comparison of f o & f n with f 1 denoising TV 1 (α = 3 4, β = 0) but: stair casing R. Bergmann A Second Order TV-type Model on Manifolds 28

41 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal comparison of f o & f n with f 2 denoising TV 2 (α = 0, β = 3 2 ) but: no plateaus R. Bergmann A Second Order TV-type Model on Manifolds 28

42 A One-Dimensional Example of Cyclic Data Denoising a phase valued signal comparison of f o & f n with f 3 denoising TV 1 & TV 2 (α = 1 2, β = 1) smallest mean squared error R. Bergmann A Second Order TV-type Model on Manifolds 28

43 Combined Inpainting and Denoising original image noisy image, σ = 0.3 R. Bergmann A Second Order TV-type Model on Manifolds 29

44 Combined Inpainting and Denoising original image (lost area in black) noisy image, σ = 0.3 R. Bergmann A Second Order TV-type Model on Manifolds 29

45 Combined Inpainting and Denoising original image (lost area in black) noisy image, σ = noiseless model, α= 1 8. R. Bergmann A Second Order TV-type Model on Manifolds 29

46 Combined Inpainting and Denoising original image (lost area in black) noisy image, σ = noiseless model, α= 1 (& diags). 8 noisy model, α= 1 (& diags). 8 R. Bergmann A Second Order TV-type Model on Manifolds 29

47 Combined Inpainting and Denoising original image (lost area in black) noisy image, σ = noiseless model, α= 1 8 (& diags), β = γ = 1 4. noisy model, α= 1 8 (& diags), β = γ = 1 4. R. Bergmann A Second Order TV-type Model on Manifolds 29

48 Bernoulli s Lemniscate on the Sphere S 2 a 2 ( ), γ(t) := cos(t), cos 1 sin(t) t [0, 2], a = sin 2 (t) Generate a sphere-valued signal by putting it into the tangential plane of the north pole γ S (t) =log p (γ(t)), p =(0, 0, 1) noisy lemniscate of Bernoulli on S 2, Gaussian noise, σ = 30, on T ps 2. R. Bergmann A Second Order TV-type Model on Manifolds 30

49 Bernoulli s Lemniscate on the Sphere S 2 a 2 ( ), γ(t) := cos(t), cos 1 sin(t) t [0, 2], a = sin 2 (t) Generate a sphere-valued signal by putting it into the tangential plane of the north pole γ S (t) =log p (γ(t)), p =(0, 0, 1) reconstruction with TV 1, α = 0.21, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 30

50 Bernoulli s Lemniscate on the Sphere S 2 a 2 ( ), γ(t) := cos(t), cos 1 sin(t) t [0, 2], a = sin 2 (t) Generate a sphere-valued signal by putting it into the tangential plane of the north pole γ S (t) =log p (γ(t)), p =(0, 0, 1) reconstruction with TV 2, α = 0, β = 10, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 30

51 Bernoulli s Lemniscate on the Sphere S 2 a 2 ( ), γ(t) := cos(t), cos 1 sin(t) t [0, 2], a = sin 2 (t) Generate a sphere-valued signal by putting it into the tangential plane of the north pole γ S (t) =log p (γ(t)), p =(0, 0, 1) reconstruction with TV 1 & TV 2, α = 0.16, β = 12.4, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 30

52 S 2 -valued data: An Image of Directions original data R. Bergmann A Second Order TV-type Model on Manifolds 31

53 S 2 -valued data: An Image of Directions noisy data R. Bergmann A Second Order TV-type Model on Manifolds 31

54 S 2 -valued data: An Image of Directions TV 1 model, α = 0.035, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 31

55 S 2 -valued data: An Image of Directions TV 2 model, β = 8.6, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 31

56 Inpainting of P(3) valued images Illustrate symmetric positive definite 3 3 matrices as ellipsoids original data R. Bergmann A Second Order TV-type Model on Manifolds 32

57 Inpainting of P(3) valued images Illustrate symmetric positive definite 3 3 matrices as ellipsoids original data lost (a lot of) data R. Bergmann A Second Order TV-type Model on Manifolds 32

58 Inpainting of P(3) valued images Illustrate symmetric positive definite 3 3 matrices as ellipsoids original data inpainted with α = β = 0.05, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 32

59 Inpainting of P(3) valued images Illustrate symmetric positive definite 3 3 matrices as ellipsoids original data inpainted with α = 0.1, MAE = R. Bergmann A Second Order TV-type Model on Manifolds 32

60 Diffusion Tensor Magnetic Resonance Imaging Dataset P(3) of the human head, traversal plane z = 28 original data, denoised, α = 0.01, β = 0.05 (Data available from The Camino Project, cmic.cs.ucl.ac.uk/camino) R. Bergmann A Second Order TV-type Model on Manifolds 33

61 Diffusion Tensor Magnetic Resonance Imaging Dataset P(3) of the human head, traversal plane z = 28 original data, denoised, α = 0.01, β = 0.05 (Data available from The Camino Project, cmic.cs.ucl.ac.uk/camino) a detail R. Bergmann A Second Order TV-type Model on Manifolds 33

62 1 Introduction 2 Second Order Differences on Manifolds 3 Proximal Mappings and the Cyclic Proximal Point Algorithm 4 Examples 5 Conclusion R. Bergmann A Second Order TV-type Model on Manifolds 34

63 Conclusion We have for manifold valued images f: V M a model for second order differences a first and second order TV-type functional E(u) applicable to denoising and inpainting cyclic proximal point algorithm to minimize the functional E(u) computed the proximal mappings using Jacobi fields proof of convergence Future work an isotropic second order TV other couplings other algorithms, e.g. a stochastic Douglas-Rachford algorithm R. Bergmann A Second Order TV-type Model on Manifolds 35

64 Literature [1] M. Bačák, R. Bergmann, G. Steidl, and A. Weinmann. A second order non-smooth variational model for restoring manifold-valued images. Preprint, ArXiv # , [2] R. Bergmann, R. H. Chan, R. Hielscher, J. Persch, and G. Steidl. Restoration of Manifold-Valued Images by Half-Quadratic Minimization. Preprint, ArXiv # , [3] R. Bergmann, F. Laus, G. Steidl, and A. Weinmann. Second order differences of cyclic data and applications in variational denoising. SIAM J. Imaging Sci., [4] R. Bergmann and A. Weinmann. A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Preprint, ArXiv # , [5] R. Bergmann and A. Weinmann. Inpainting of cyclic data using first and second order differences. EMMCVPR 2015, R. Bergmann A Second Order TV-type Model on Manifolds 36

65 Literature [1] M. Bačák, R. Bergmann, G. Steidl, and A. Weinmann. A second order non-smooth variational model for restoring manifold-valued images. Preprint, ArXiv # , [2] R. Bergmann, R. H. Chan, R. Hielscher, J. Persch, and G. Steidl. Restoration of Manifold-Valued Images by Half-Quadratic Minimization. Preprint, ArXiv # , [3] R. Bergmann, F. Laus, G. Steidl, and A. Weinmann. Second order differences of cyclic data and applications in variational denoising. SIAM J. Imaging Sci., [4] R. Bergmann and A. Weinmann. A second order TV-type approach for inpainting and denoising higher dimensional combined cyclic and vector space data. Preprint, ArXiv # , [5] R. Bergmann and A. Weinmann. Inpainting of cyclic data using first and second order differences. EMMCVPR 2015, Thank you for your attention. R. Bergmann A Second Order TV-type Model on Manifolds 36