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Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Total generalized variation: From regularization theory to applications in imaging Kristian Bredies Institute for

  1. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Total Generalized Variation Definition: �� � u div k v d x TGV k � v ∈ C k c (Ω , Sym k (I R d )) , α ( u ) = sup � Ω � div l v � ∞ ≤ α l , l = 0 , . . . , k − 1 � α = ( α 0 , . . . , α k − 1 ) > 0 weights Basic properties: [B./Kunisch/Pock ’10] TGV k α is proper, convex, lower semi-continuous TGV k α is translation and rotation invariant TGV k α + � · � 1 gives Banach space BGV k α (Ω) ker(TGV k α ) = P k − 1 (Ω) polynomials of degree less than k α measures piecewise P k − 1 only at the interfaces TGV k Introduction TGV Applications Summary K. Bredies 8 / 51

  2. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Total Generalized Variation Definition: �� � u div k v d x TGV k � v ∈ C k c (Ω , Sym k (I R d )) , α ( u ) = sup � Ω � div l v � ∞ ≤ α l , l = 0 , . . . , k − 1 � α = ( α 0 , . . . , α k − 1 ) > 0 weights Basic properties: [B./Kunisch/Pock ’10] TGV k α is proper, convex, lower semi-continuous TGV k α is translation and rotation invariant TGV k α + � · � 1 gives Banach space BGV k α (Ω) ker(TGV k α ) = P k − 1 (Ω) polynomials of degree less than k α measures piecewise P k − 1 only at the interfaces TGV k Introduction TGV Applications Summary K. Bredies 8 / 51

  3. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Total Generalized Variation Definition: �� � u div k v d x TGV k � v ∈ C k c (Ω , Sym k (I R d )) , α ( u ) = sup � Ω � div l v � ∞ ≤ α l , l = 0 , . . . , k − 1 � α = ( α 0 , . . . , α k − 1 ) > 0 weights Basic properties: [B./Kunisch/Pock ’10] TGV k α is proper, convex, lower semi-continuous TGV k α is translation and rotation invariant TGV k α + � · � 1 gives Banach space BGV k α (Ω) ker(TGV k α ) = P k − 1 (Ω) polynomials of degree less than k α measures piecewise P k − 1 only at the interfaces TGV k Introduction TGV Applications Summary K. Bredies 8 / 51

  4. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Total Generalized Variation Definition: �� � u div k v d x TGV k � v ∈ C k c (Ω , Sym k (I R d )) , α ( u ) = sup � Ω � div l v � ∞ ≤ α l , l = 0 , . . . , k − 1 � α = ( α 0 , . . . , α k − 1 ) > 0 weights Basic properties: [B./Kunisch/Pock ’10] TGV k α is proper, convex, lower semi-continuous TGV k α is translation and rotation invariant TGV k α + � · � 1 gives Banach space BGV k α (Ω) ker(TGV k α ) = P k − 1 (Ω) polynomials of degree less than k α measures piecewise P k − 1 only at the interfaces TGV k Introduction TGV Applications Summary K. Bredies 8 / 51

  5. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Denoising � u − f � 2 + TGV k Solve: min α ( u ) 2 u ∈ L 2 (Ω) Noisy image Introduction TGV Applications Summary K. Bredies 9 / 51

  6. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Denoising � u − f � 2 + TGV k Solve: min α ( u ) 2 u ∈ L 2 (Ω) TV regularization Introduction TGV Applications Summary K. Bredies 9 / 51

  7. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Denoising � u − f � 2 + TGV k Solve: min α ( u ) 2 u ∈ L 2 (Ω) TV-TV 2 infimal-convolution regularization Introduction TGV Applications Summary K. Bredies 9 / 51

  8. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Denoising � u − f � 2 + TGV k Solve: min α ( u ) 2 u ∈ L 2 (Ω) TGV 2 α regularization Introduction TGV Applications Summary K. Bredies 9 / 51

  9. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Denoising � u − f � 2 + TGV k Solve: min α ( u ) 2 u ∈ L 2 (Ω) TGV 3 α regularization Introduction TGV Applications Summary K. Bredies 9 / 51

  10. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Questions How to interpret TGV ? How is higher-order information incorporated? Introduction TGV Applications Summary K. Bredies 10 / 51

  11. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Questions How to interpret TGV ? How is higher-order information incorporated? Can TGV be used as regularization functional? Goal: Solve, for a large class of K , 1 2 � Ku − f � 2 2 + TGV k min α ( u ) u ∈ L p (Ω) Introduction TGV Applications Summary K. Bredies 10 / 51

  12. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Questions How to interpret TGV ? How is higher-order information incorporated? Can TGV be used as regularization functional? Goal: Solve, for a large class of K , 1 2 � Ku − f � 2 2 + TGV k min α ( u ) u ∈ L p (Ω) Are the theoretical results applicable in practice? Are there efficient minimization algorithms? Does the model lead to improvements in image reconstruction? Introduction TGV Applications Summary K. Bredies 10 / 51

  13. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Interpretation: TGV of second order Minimum characterization: � � TGV 2 α ( u ) = w ∈ BD(Ω) α 1 min |∇ u − w | + α 0 |E ( w ) | Ω Ω R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } BD(Ω) = { w ∈ L 1 (Ω , I Vector fields of bounded deformation Intuitive interpretation: Locally: ∇ u smooth Locally: u jumps � w = ∇ u ≈ optimal � w = 0 ≈ optimal � TGV 2 � TGV 2 � loc |∇ 2 u | � α ∼ α 0 α ∼ α 1 loc |∇ u | Optimal balancing between ∇ u and ∇ 2 u � TGV 2 α (pw. smooth) < TGV 2 α (staircases) � preferred Radon norm � measure-based decomposition Introduction TGV Applications Summary K. Bredies 11 / 51

  14. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Interpretation: TGV of second order Minimum characterization: � � TGV 2 α ( u ) = w ∈ BD(Ω) α 1 min |∇ u − w | + α 0 |E ( w ) | Ω Ω R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } BD(Ω) = { w ∈ L 1 (Ω , I Vector fields of bounded deformation Intuitive interpretation: Locally: ∇ u smooth Locally: u jumps � w = ∇ u ≈ optimal � w = 0 ≈ optimal � TGV 2 � TGV 2 � loc |∇ 2 u | � α ∼ α 0 α ∼ α 1 loc |∇ u | Optimal balancing between ∇ u and ∇ 2 u � TGV 2 α (pw. smooth) < TGV 2 α (staircases) � preferred Radon norm � measure-based decomposition Introduction TGV Applications Summary K. Bredies 11 / 51

  15. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Interpretation: TGV of second order Minimum characterization: � � TGV 2 α ( u ) = w ∈ BD(Ω) α 1 min |∇ u − w | + α 0 |E ( w ) | Ω Ω R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } BD(Ω) = { w ∈ L 1 (Ω , I Vector fields of bounded deformation Intuitive interpretation: Locally: ∇ u smooth Locally: u jumps � w = ∇ u ≈ optimal � w = 0 ≈ optimal � TGV 2 � TGV 2 � loc |∇ 2 u | � α ∼ α 0 α ∼ α 1 loc |∇ u | Optimal balancing between ∇ u and ∇ 2 u � TGV 2 α (pw. smooth) < TGV 2 α (staircases) � preferred Radon norm � measure-based decomposition Introduction TGV Applications Summary K. Bredies 11 / 51

  16. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Interpretation: TGV of second order Minimum characterization: � � TGV 2 α ( u ) = w ∈ BD(Ω) α 1 min |∇ u − w | + α 0 |E ( w ) | Ω Ω R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } BD(Ω) = { w ∈ L 1 (Ω , I Vector fields of bounded deformation Intuitive interpretation: Locally: ∇ u smooth Locally: u jumps � w = ∇ u ≈ optimal � w = 0 ≈ optimal � TGV 2 � TGV 2 � loc |∇ 2 u | � α ∼ α 0 α ∼ α 1 loc |∇ u | Optimal balancing between ∇ u and ∇ 2 u � TGV 2 α (pw. smooth) < TGV 2 α (staircases) � preferred Radon norm � measure-based decomposition Introduction TGV Applications Summary K. Bredies 11 / 51

  17. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING TGV regularization Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Minimize: Inverse Problem: Tikhonov-functional solve Ku = f R d bounded domain � Ku − f � 2 Ω ⊂ I H min K : L p (Ω) → H 2 u ∈ L p (Ω) + TGV 2 α ( u ) linear and continuous Introduction TGV Applications Summary K. Bredies 12 / 51

  18. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING TGV regularization Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Minimize: Inverse Problem: Tikhonov-functional solve Ku = f R d bounded domain � Ku − f � 2 Ω ⊂ I H min K : L p (Ω) → H 2 u ∈ L p (Ω) + TGV 2 α ( u ) linear and continuous � Nonsmooth optimization problem Which conditions ensure existence of solutions? Are the solutions stable with respect to f ? Introduction TGV Applications Summary K. Bredies 12 / 51

  19. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING TGV regularization Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Minimize: Inverse Problem: Tikhonov-functional solve Ku = f R d bounded domain � Ku − f � 2 Ω ⊂ I H min K : L p (Ω) → H 2 u ∈ L p (Ω) + TGV 2 α ( u ) linear and continuous � Nonsmooth optimization problem Which conditions ensure existence of solutions? Are the solutions stable with respect to f ? � Show topological equivalence with BV(Ω) Introduction TGV Applications Summary K. Bredies 12 / 51

  20. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Vector fields of bounded deformation BD(Ω) = { w ∈ L 1 (Ω , I R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } � w � BD = � w � 1 + �E ( w ) � M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space R d | ker( E ) = { w : Ω → I w ( x ) = Ax + b , A T = − A } ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality : � w − Rw � 1 ≤ C �E ( w ) � M R : BD(Ω) → ker( E ) linear projection Introduction TGV Applications Summary K. Bredies 13 / 51

  21. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Vector fields of bounded deformation BD(Ω) = { w ∈ L 1 (Ω , I R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } � w � BD = � w � 1 + �E ( w ) � M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space R d | ker( E ) = { w : Ω → I w ( x ) = Ax + b , A T = − A } ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality : � w − Rw � 1 ≤ C �E ( w ) � M R : BD(Ω) → ker( E ) linear projection Introduction TGV Applications Summary K. Bredies 13 / 51

  22. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Vector fields of bounded deformation BD(Ω) = { w ∈ L 1 (Ω , I R d ) | E ( w ) ∈ M (Ω , Sym 2 (I R d )) } � w � BD = � w � 1 + �E ( w ) � M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space R d | ker( E ) = { w : Ω → I w ( x ) = Ax + b , A T = − A } ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality : � w − Rw � 1 ≤ C �E ( w ) � M R : BD(Ω) → ker( E ) linear projection Introduction TGV Applications Summary K. Bredies 13 / 51

  23. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Topological equivalence R d sufficiently smooth ⇒ Theorem: Ω ⊂ I c � u � BV ≤ � u � 1 + TGV 2 ∀ u ∈ BGV 2 α ( u ) ≤ C � u � BV α (Ω) Introduction TGV Applications Summary K. Bredies 14 / 51

  24. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Topological equivalence R d sufficiently smooth ⇒ Theorem: Ω ⊂ I c � u � BV ≤ � u � 1 + TGV 2 ∀ u ∈ BGV 2 α ( u ) ≤ C � u � BV α (Ω) Proof: � � 1 We have � u � BV ≤ C 1 �∇ u − Rw � M + � u � 1 ∀ w ∈ BD(Ω) 2 Sobolev-Korn inequality + minimum characterization: �∇ u − Rw � M ≤ �∇ u − w � M + � w − Rw � 1 � � ≤ C 3 α 1 �∇ u − w � M + α 0 �E ( w ) � M inf w ∈ BD(Ω) �∇ u − Rw � M ≤ C 3 TGV 2 ⇒ α ( u ) � u � 1 + TGV 2 � � 3 With the help of 1 : ⇒ � u � BV ≤ C 4 α ( u ) 4 Finally: TGV 2 α ( u ) ≤ α 1 TV( u ) � u � 1 + TGV 2 ⇒ α ( u ) ≤ C 5 � u � BV Introduction TGV Applications Summary K. Bredies 14 / 51

  25. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Topological equivalence R d sufficiently smooth ⇒ Theorem: Ω ⊂ I c � u � BV ≤ � u � 1 + TGV 2 ∀ u ∈ BGV 2 α ( u ) ≤ C � u � BV α (Ω) Proof: � � 1 We have � u � BV ≤ C 1 �∇ u − Rw � M + � u � 1 ∀ w ∈ BD(Ω) 2 Sobolev-Korn inequality + minimum characterization: �∇ u − Rw � M ≤ �∇ u − w � M + � w − Rw � 1 � � ≤ C 3 α 1 �∇ u − w � M + α 0 �E ( w ) � M inf w ∈ BD(Ω) �∇ u − Rw � M ≤ C 3 TGV 2 ⇒ α ( u ) � u � 1 + TGV 2 � � 3 With the help of 1 : ⇒ � u � BV ≤ C 4 α ( u ) 4 Finally: TGV 2 α ( u ) ≤ α 1 TV( u ) � u � 1 + TGV 2 ⇒ α ( u ) ≤ C 5 � u � BV Introduction TGV Applications Summary K. Bredies 14 / 51

  26. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Topological equivalence R d sufficiently smooth ⇒ Theorem: Ω ⊂ I c � u � BV ≤ � u � 1 + TGV 2 ∀ u ∈ BGV 2 α ( u ) ≤ C � u � BV α (Ω) Proof: � � 1 We have � u � BV ≤ C 1 �∇ u − Rw � M + � u � 1 ∀ w ∈ BD(Ω) 2 Sobolev-Korn inequality + minimum characterization: �∇ u − Rw � M ≤ �∇ u − w � M + � w − Rw � 1 � � ≤ C 3 α 1 �∇ u − w � M + α 0 �E ( w ) � M inf w ∈ BD(Ω) �∇ u − Rw � M ≤ C 3 TGV 2 ⇒ α ( u ) � u � 1 + TGV 2 � � 3 With the help of 1 : ⇒ � u � BV ≤ C 4 α ( u ) 4 Finally: TGV 2 α ( u ) ≤ α 1 TV( u ) � u � 1 + TGV 2 ⇒ α ( u ) ≤ C 5 � u � BV Introduction TGV Applications Summary K. Bredies 14 / 51

  27. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Topological equivalence R d sufficiently smooth ⇒ Theorem: Ω ⊂ I c � u � BV ≤ � u � 1 + TGV 2 ∀ u ∈ BGV 2 α ( u ) ≤ C � u � BV α (Ω) Proof: � � 1 We have � u � BV ≤ C 1 �∇ u − Rw � M + � u � 1 ∀ w ∈ BD(Ω) 2 Sobolev-Korn inequality + minimum characterization: �∇ u − Rw � M ≤ �∇ u − w � M + � w − Rw � 1 � � ≤ C 3 α 1 �∇ u − w � M + α 0 �E ( w ) � M inf w ∈ BD(Ω) �∇ u − Rw � M ≤ C 3 TGV 2 ⇒ α ( u ) � u � 1 + TGV 2 � � 3 With the help of 1 : ⇒ � u � BV ≤ C 4 α ( u ) 4 Finally: TGV 2 α ( u ) ≤ α 1 TV( u ) � u � 1 + TGV 2 ⇒ α ( u ) ≤ C 5 � u � BV Introduction TGV Applications Summary K. Bredies 14 / 51

  28. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Existence and stability Corollary: � u − P 1 u � d / ( d − 1) ≤ C TGV 2 Coercivity: α ( u ) P 1 → Π 1 linear projection on the affine functions Π 1 Theorem: 1 < p ≤ d / ( d − 1) Optimization problem     K : L p (Ω) → H 1     2 � Ku − f � 2 min   linear and continuous, ⇒ u ∈ L p (Ω) + TGV 2 α ( u ) H Hilbert space         K injective on Π 1 possesses a solution Proof: Direct method + coercivity of TGV 2 α Introduction TGV Applications Summary K. Bredies 15 / 51

  29. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Existence and stability Corollary: � u − P 1 u � d / ( d − 1) ≤ C TGV 2 Coercivity: α ( u ) P 1 → Π 1 linear projection on the affine functions Π 1 Theorem: 1 < p ≤ d / ( d − 1) Optimization problem     K : L p (Ω) → H 1     2 � Ku − f � 2 min   linear and continuous, ⇒ u ∈ L p (Ω) + TGV 2 α ( u ) H Hilbert space         K injective on Π 1 possesses a solution Proof: Direct method + coercivity of TGV 2 α u n ⇀ u in L p (Ω) (subseq.) � Stability: f n → f in H ⇒ TGV 2 α ( u n ) → TGV 2 α ( u ) Introduction TGV Applications Summary K. Bredies 15 / 51

  30. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING General orders Next step: Generalization with respect to k � Examine BD(Ω , Sym k (I R d )) = { w ∈ L 1 (Ω , Sym k (I R d )) | E ( w ) ∈ M (Ω , Sym k +1 (I R d )) } � w � BD = � w � 1 + �E ( w ) � M Symmetric tensor fields of bounded deformation Introduction TGV Applications Summary K. Bredies 16 / 51

  31. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) [B. ’11] Theorem: 1 u ∈ D (Ω , Sym k (I R d )) ∗ � ∇ k +1 ⊗ u = 0 in Ω ⇒ distribution with E ( u ) = 0 2 ker( E ) is a subspace of Π k (Ω , Sym k (I R d )) Introduction TGV Applications Summary K. Bredies 17 / 51

  32. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) [B. ’11] Theorem: 1 u ∈ D (Ω , Sym k (I R d )) ∗ � ∇ k +1 ⊗ u = 0 in Ω ⇒ distribution with E ( u ) = 0 2 ker( E ) is a subspace of Π k (Ω , Sym k (I R d )) Furthermore: There is a T such that ∇ k +1 ⊗ u = T E ( u ) for smooth u The fundamental solution for div E reads as: k � k + 1 � Γ η � ( − 1) l E l � div l ( E l +1 η ) � k = l + 1 l =0 E m fundamental solution for ∆ m Introduction TGV Applications Summary K. Bredies 17 / 51

  33. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) R d be a Lipschitz domain Theorem: Let Ω ⊂ I 1 Trace mapping: γ : BD(Ω , Sym k (I R d )) → L 1 ( ∂ Ω , Sym k (I R d )) continuous with respect to strict convergence 2 Gauß-Green theorem: � � � � ( γ u ⊗ ν ) · v d H d − 1 − u · div v d x = v · d E ( u ) Ω ∂ Ω Ω R d )), v ∈ C 1 (Ω , Sym k +1 (I for u ∈ BD(Ω , Sym k (I R d )) R d , Sym k (I R d )) with 3 Zero extension: Eu ∈ BD(I E ( Eu ) = E ( u ) − � ( γ u ⊗ ν ) H d − 1 � ∂ Ω Introduction TGV Applications Summary K. Bredies 18 / 51

  34. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) R d be a Lipschitz domain Theorem: Let Ω ⊂ I 1 Trace mapping: γ : BD(Ω , Sym k (I R d )) → L 1 ( ∂ Ω , Sym k (I R d )) continuous with respect to strict convergence 2 Gauß-Green theorem: � � � � ( γ u ⊗ ν ) · v d H d − 1 − u · div v d x = v · d E ( u ) Ω ∂ Ω Ω R d )), v ∈ C 1 (Ω , Sym k +1 (I for u ∈ BD(Ω , Sym k (I R d )) R d , Sym k (I R d )) with 3 Zero extension: Eu ∈ BD(I E ( Eu ) = E ( u ) − � ( γ u ⊗ ν ) H d − 1 � ∂ Ω Introduction TGV Applications Summary K. Bredies 18 / 51

  35. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) R d be a Lipschitz domain Theorem: Let Ω ⊂ I 1 Trace mapping: γ : BD(Ω , Sym k (I R d )) → L 1 ( ∂ Ω , Sym k (I R d )) continuous with respect to strict convergence 2 Gauß-Green theorem: � � � � ( γ u ⊗ ν ) · v d H d − 1 − u · div v d x = v · d E ( u ) Ω ∂ Ω Ω R d )), v ∈ C 1 (Ω , Sym k +1 (I for u ∈ BD(Ω , Sym k (I R d )) R d , Sym k (I R d )) with 3 Zero extension: Eu ∈ BD(I E ( Eu ) = E ( u ) − � ( γ u ⊗ ν ) H d − 1 � ∂ Ω Introduction TGV Applications Summary K. Bredies 18 / 51

  36. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) [B. ’11] Theorem: 1 Embedding result: BD(Ω , Sym k (I R d )) ֒ → L d / ( d − 1) (Ω , Sym k (I R d )) 2 Sobolev-Korn inequality: � u − Ru � d / ( d − 1) ≤ C �E ( u ) � M Introduction TGV Applications Summary K. Bredies 19 / 51

  37. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) [B. ’11] Theorem: 1 Embedding result: BD(Ω , Sym k (I R d )) ֒ → L d / ( d − 1) (Ω , Sym k (I R d )) 2 Sobolev-Korn inequality: � u − Ru � d / ( d − 1) ≤ C �E ( u ) � M Consequently: Minimum characterization: k � � TGV k α ( u ) = min α k − l |E ( u l − 1 ) − u l | u l ∈ BD(Ω , Sym l (I R d )) , Ω l =1 u 0 = u , u k =0 � Ku − f � 2 + TGV k H Existence of solutions: min α ( u ) 2 u ∈ L p (Ω) if K is injective on Π k − 1 Introduction TGV Applications Summary K. Bredies 19 / 51

  38. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING The spaces BD(Ω , Sym k (I R d )) [B. ’11] Theorem: 1 Embedding result: BD(Ω , Sym k (I R d )) ֒ → L d / ( d − 1) (Ω , Sym k (I R d )) 2 Sobolev-Korn inequality: � u − Ru � d / ( d − 1) ≤ C �E ( u ) � M Consequently: Minimum characterization: k � � TGV k α ( u ) = min α k − l |E ( u l − 1 ) − u l | u l ∈ BD(Ω , Sym l (I R d )) , Ω l =1 u 0 = u , u k =0 � Ku − f � 2 + TGV k H Existence of solutions: min α ( u ) 2 u ∈ L p (Ω) if K is injective on Π k − 1 � “ TV applicable ⇒ TGV applicable” Introduction TGV Applications Summary K. Bredies 19 / 51

  39. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Regularization properties Joint work with Martin Holler Theorem: [B./Holler ’13] u n solution ∼ parameters α n , data f n , � f n − f † � ≤ δ n Multiparameter choice/source condition: 1 α n i → 0 and δ 2 n /α n i → 0 2 lim n →∞ α n i /α n i − 1 > 0 3 Ku † = f † for u † ∈ BV(Ω) Then: Convergence: u n ⇀ ∗ u ∗ � subsequentially TGV k , l α ∗ ( u n ) → TGV k α ∗ ( u ∗ ) u ∗ minimizing-TGV k α ∗ solution Introduction TGV Applications Summary K. Bredies 20 / 51

  40. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Deconvolution Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: 1 2 � u ∗ k − f � 2 min 2 u ∈ L 2 (Ω) + TGV 2 α ( u ) Noisy data f Introduction TGV Applications Summary K. Bredies 21 / 51

  41. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Deconvolution Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: 1 2 � u ∗ k − f � 2 min 2 u ∈ L 2 (Ω) + TGV 2 α ( u ) TV-regularized solution Introduction TGV Applications Summary K. Bredies 21 / 51

  42. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Deconvolution Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: 1 2 � u ∗ k − f � 2 min 2 u ∈ L 2 (Ω) + TGV 2 α ( u ) TGV 2 α -regularized solution Introduction TGV Applications Summary K. Bredies 21 / 51

  43. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Optimization methods Exemplarily: Second-order scalar case TGV 2 , 0 α = TGV 2 α u ∈ L p (Ω) F ( u ) + TGV 2 min α ( u ) Approach: 1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem y ∈ Y � Ax , y � + G ( x ) − F ∗ ( y ) min x ∈ X max 3 Use primal-dual algorithm of [Chambolle/Pock ’11] Introduction TGV Applications Summary K. Bredies 22 / 51

  44. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Optimization methods Exemplarily: Second-order scalar case TGV 2 , 0 α = TGV 2 α u ∈ L p (Ω) F ( u ) + TGV 2 min α ( u ) Approach: 1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem y ∈ Y � Ax , y � + G ( x ) − F ∗ ( y ) min x ∈ X max 3 Use primal-dual algorithm of [Chambolle/Pock ’11] Introduction TGV Applications Summary K. Bredies 22 / 51

  45. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Optimization methods Exemplarily: Second-order scalar case TGV 2 , 0 α = TGV 2 α u ∈ L p (Ω) F ( u ) + TGV 2 min α ( u ) Approach: 1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem y ∈ Y � Ax , y � + G ( x ) − F ∗ ( y ) min x ∈ X max 3 Use primal-dual algorithm of [Chambolle/Pock ’11] y n +1 =( I + σ∂ F ∗ ) − 1 ( y n + σ Ax n )    x n +1 =( I + τ∂ G ) − 1 ( x n − τ A ∗ y n +1 ) x n +1 =2 x n +1 − x n   Introduction TGV Applications Summary K. Bredies 22 / 51

  46. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Saddle-point formulation Finite difference approximations ∇ h , E h , div h = − ( ∇ h ) ∗ etc. Supremum definition of TGV 2 : F ( u ) + � u , (div h ) 2 v � min u max v − I {� v � ∞ ≤ α 0 } ( v ) − I {� ω � ∞ ≤ α 1 } ( div h v ) Introduction TGV Applications Summary K. Bredies 23 / 51

  47. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Saddle-point formulation Finite difference approximations ∇ h , E h , div h = − ( ∇ h ) ∗ etc. Supremum definition of TGV 2 : F ( u ) + � u , (div h ) 2 v � min u max v − I {� v � ∞ ≤ α 0 } ( v ) − I {� ω � ∞ ≤ α 1 } ( div h v ) Introduce constraint ω = div h v and Lagrange multiplier w Introduction TGV Applications Summary K. Bredies 23 / 51

  48. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Saddle-point formulation Finite difference approximations ∇ h , E h , div h = − ( ∇ h ) ∗ etc. Supremum definition of TGV 2 : F ( u ) + � u , div h ω � + � w , div h v − ω � min u , w max v ,ω − I {� v � ∞ ≤ α 0 } ( v ) − I {� ω � ∞ ≤ α 1 } ( ω ) Introduce constraint ω = div h v and Lagrange multiplier w Introduction TGV Applications Summary K. Bredies 23 / 51

  49. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Saddle-point formulation Finite difference approximations ∇ h , E h , div h = − ( ∇ h ) ∗ etc. Supremum definition of TGV 2 : F ( u ) + � u , div h ω � + � w , div h v − ω � min u , w max v ,ω − I {� v � ∞ ≤ α 0 } ( v ) − I {� ω � ∞ ≤ α 1 } ( ω ) Introduce constraint ω = div h v and Lagrange multiplier w � 0 � u � � v � −∇ h � x = , y = , A = , −E h w ω − I F ∗ ( v , ω ) = I {� v � ∞ ≤ α 0 } ( v )+ I {� ω � ∞ ≤ α 1 } ( ω ) G ( u , w ) = F ( u ) , Introduction TGV Applications Summary K. Bredies 23 / 51

  50. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Primal-dual algorithm 1 [B. ’12] Iteration: ω n +1 = P {� · � ∞ ≤ α 1 } ω n + σ ( ∇ h ¯ u n − ¯ � w n ) � � dual update v n +1 = P {� · � ∞ ≤ α 0 } v n + σ E h (¯ w n ) � � u n +1 = ( I + τ∂ F ) − 1 ( u n + τ div h ω n +1 ) � primal update w n +1 = w n + τ (div h v n +1 + ω n +1 ) u n +1 = 2 u n +1 − u n , w n +1 = 2 w n +1 − w n � ¯ ¯ extragradient Introduction TGV Applications Summary K. Bredies 24 / 51

  51. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Primal-dual algorithm 1 [B. ’12] Iteration: ω n +1 = P {� · � ∞ ≤ α 1 } ω n + σ ( ∇ h ¯ u n − ¯ � w n ) � � dual update v n +1 = P {� · � ∞ ≤ α 0 } v n + σ E h (¯ w n ) � � u n +1 = ( I + τ∂ F ) − 1 ( u n + τ div h ω n +1 ) � primal update w n +1 = w n + τ (div h v n +1 + ω n +1 ) u n +1 = 2 u n +1 − u n , w n +1 = 2 w n +1 − w n � ¯ ¯ extragradient P {� · � ∞ ≤ α 0 } , P {� · � ∞ ≤ α 1 } amount to pointwise operations ( I + τ∂ F ) − 1 resolvent mapping � assumed to be known Converges for appropriate choice of σ, τ > 0 [Chambolle/Pock ’11] Introduction TGV Applications Summary K. Bredies 24 / 51

  52. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Applications Examples for Algorithm 1: ( I + τ∂ F ) − 1 ( u ) F ( u ) 1 u + τ f 2 � u − f � 2 H 1 + τ � u − f � 1 f + S τ ( u − f ) S τ soft-shrinkage operator 1 2 � Ku − f � 2 ( I + τ K ∗ K ) − 1 ( u + τ K ∗ f ) H solve, e.g., with CGNE Primary application: Denoising problems Introduction TGV Applications Summary K. Bredies 25 / 51

  53. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Applications Examples for Algorithm 1: ( I + τ∂ F ) − 1 ( u ) F ( u ) 1 u + τ f 2 � u − f � 2 H 1 + τ � u − f � 1 f + S τ ( u − f ) S τ soft-shrinkage operator 1 2 � Ku − f � 2 ( I + τ K ∗ K ) − 1 ( u + τ K ∗ f ) H solve, e.g., with CGNE Primary application: Denoising problems Introduction TGV Applications Summary K. Bredies 25 / 51

  54. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Applications Examples for Algorithm 1: ( I + τ∂ F ) − 1 ( u ) F ( u ) 1 u + τ f 2 � u − f � 2 H 1 + τ � u − f � 1 f + S τ ( u − f ) S τ soft-shrinkage operator 1 2 � Ku − f � 2 ( I + τ K ∗ K ) − 1 ( u + τ K ∗ f ) H solve, e.g., with CGNE Primary application: Denoising problems Introduction TGV Applications Summary K. Bredies 25 / 51

  55. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Primal-dual algorithm 2 Alternative: � Ku , p � + ˜ Additional dual variable: F ( u ) = max F ( u ) − G ( p ) p Needs only resolvents w.r.t. ∂ G , ∂ ˜ F , not ( I + τ∂ F ) − 1 Introduction TGV Applications Summary K. Bredies 26 / 51

  56. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Primal-dual algorithm 2 Alternative: � Ku , p � + ˜ Additional dual variable: F ( u ) = max F ( u ) − G ( p ) p Needs only resolvents w.r.t. ∂ G , ∂ ˜ F , not ( I + τ∂ F ) − 1 Iteration: [B. ’12] p n +1 = ( I + σ∂ G ) − 1 ( p n + σ K ¯ u n )    ω n +1 = P {� · � ∞ ≤ α 1 } ω n + σ ( ∇ h ¯ u n − ¯ � w n ) � v n +1 = P {� · � ∞ ≤ α 0 } v n + σ E h (¯  � w n ) �  u n + τ (div h ω n +1 − K ∗ p n +1 ) u n +1 = ( I + τ∂ ˜ � F ) − 1 � � w n +1 = w n + τ (div h v n +1 + ω n +1 ) u n +1 = 2 u n +1 − u n , w n +1 = 2 w n +1 − w n � ¯ ¯ Introduction TGV Applications Summary K. Bredies 26 / 51

  57. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Applications Examples for Algorithm 2: ( I + σ∂ G ) − 1 ( p ) F ( u ) G ( p ) � p � 2 1 p − σ f 2 � Ku − f � 2 + � f , p � H 2 1 + σ � Ku − f � 1 I {� p � ∞ ≤ 1 } ( p ) + � f , p � P {� p � ∞ ≤ 1 } ( p − σ f ) Here: ˜ F ( u ) = 0 ⇒ ( I + τ∂ ˜ F ) − 1 ( u ) = u Primary application: TGV 2 -regularized solution of Ku = f Introduction TGV Applications Summary K. Bredies 27 / 51

  58. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Applications Examples for Algorithm 2: ( I + σ∂ G ) − 1 ( p ) F ( u ) G ( p ) � p � 2 1 p − σ f 2 � Ku − f � 2 + � f , p � H 2 1 + σ � Ku − f � 1 I {� p � ∞ ≤ 1 } ( p ) + � f , p � P {� p � ∞ ≤ 1 } ( p − σ f ) Here: ˜ F ( u ) = 0 ⇒ ( I + τ∂ ˜ F ) − 1 ( u ) = u Primary application: TGV 2 -regularized solution of Ku = f Introduction TGV Applications Summary K. Bredies 27 / 51

  59. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Outline 1 Introduction 2 Total Generalized Variation Existence and stability for second order Regularization theory for general orders Optimization algorithms 3 Applications Compressive imaging JPEG(2000) decompression and zooming Quantitative susceptibility mapping Dual energy CT denoising 4 Summary Introduction TGV Applications Summary K. Bredies 28 / 51

  60. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Compressive imaging Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: Au = f R ( u ) min “Single-pixel camera” A basis analysis operator Original image R “sparsifying” penalty Introduction TGV Applications Summary K. Bredies 29 / 51

  61. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Compressive imaging Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: Au = f R ( u ) min “Single-pixel camera” A basis analysis operator 1600 measurements R “sparsifying” penalty Compressed sensing: R ( u ) = � u � 1 Captures solution with minimal “ L 0 -norm” with high probability Introduction TGV Applications Summary K. Bredies 29 / 51

  62. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Compressive imaging Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: Au = f R ( u ) min “Single-pixel camera” A basis analysis operator 1600 measurements R “sparsifying” penalty Compressed sensing: In particular: R ( u ) = � u � 1 R ( u ) = TV( u ) Captures solution with “Gradient sparsity” minimal “ L 0 -norm” with high probability Introduction TGV Applications Summary K. Bredies 29 / 51

  63. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Compressive imaging Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: Au = f R ( u ) min “Single-pixel camera” A basis analysis operator 1600 measurements R “sparsifying” penalty Compressed sensing: In particular: R ( u ) = � u � 1 R ( u ) = TV( u ) Captures solution with “Gradient sparsity” minimal “ L 0 -norm” with � Use TGV as high probability penalty Introduction TGV Applications Summary K. Bredies 29 / 51

  64. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Example TV-based reconstruction: Test data: From “Rice Single-Pixel Camera Project” http://dsp.rice.edu/cscamera Reconstruction from 768 samples 384 samples varying number of samples Algorithm: Primal-dual method 2 256 samples 192 samples Introduction TGV Applications Summary K. Bredies 30 / 51

  65. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Example TGV-based reconstruction: Test data: From “Rice Single-Pixel Camera Project” http://dsp.rice.edu/cscamera Reconstruction from 768 samples 384 samples varying number of samples Algorithm: Primal-dual method 2 256 samples 192 samples Introduction TGV Applications Summary K. Bredies 30 / 51

  66. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Image decompression Joint work with Martin Holler JPEG compression scheme: Lossy procedure High compression � disturbing artifacts (ringing, blocking) Original image (8 bpp) Introduction TGV Applications Summary K. Bredies 31 / 51

  67. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Image decompression Joint work with Martin Holler JPEG compression scheme: Lossy procedure High compression � disturbing artifacts (ringing, blocking) JPEG image (0.062 bpp) Introduction TGV Applications Summary K. Bredies 31 / 51

  68. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Image decompression Joint work with Martin Holler JPEG compression scheme: Lossy procedure High compression � disturbing artifacts (ringing, blocking) Goal: Remove artifacts Respect given information JPEG image (0.062 bpp) Introduction TGV Applications Summary K. Bredies 31 / 51

  69. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Application: Image decompression Joint work with Martin Holler JPEG compression scheme: Lossy procedure High compression � disturbing artifacts (ringing, blocking) Goal: Remove artifacts Respect given information � Use TGV image model JPEG image (0.062 bpp) Introduction TGV Applications Summary K. Bredies 31 / 51

  70. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression model JPEG compression scheme: 8x8 blocks DCT Lossless Quantization compression Source Quantization Compressed Image data table Image data Problem: Many images give same JPEG object � convex set C Standard decompression � particular choice � artifacts Introduction TGV Applications Summary K. Bredies 32 / 51

  71. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression model JPEG compression scheme: 8x8 blocks DCT Lossless Quantization compression Source Quantization Compressed Image data table Image data Problem: Many images give same JPEG object � convex set C Standard decompression � particular choice � artifacts Idea: Optimize over all possible choices Introduction TGV Applications Summary K. Bredies 32 / 51

  72. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression model JPEG compression scheme: 8x8 blocks DCT Lossless Quantization compression Source Quantization Compressed Image data table Image data Problem: Many images give same JPEG object TGV Model: � convex set C u ∈ L 2 (Ω) TGV 2 min α ( u ) Standard decompression + I C ( u ) � particular choice � artifacts Idea: Optimize over all possible choices Introduction TGV Applications Summary K. Bredies 32 / 51

  73. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression: Example Decompression of grayscale images: 0.062 bpp standard decompression Introduction TGV Applications Summary K. Bredies 33 / 51

  74. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression: Example Decompression of grayscale images: 0.062 bpp JPEG-TGV decompression Introduction TGV Applications Summary K. Bredies 33 / 51

  75. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression: Example Decompression of color images: 0.051 bpp standard decompression Introduction TGV Applications Summary K. Bredies 34 / 51

  76. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG decompression: Example Decompression of color images: 0.051 bpp JPEG-TGV decompression Introduction TGV Applications Summary K. Bredies 34 / 51

  77. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Towards real-life application Software: Handles all flavors of JPEG (grayscale/color, chroma subsampling, etc.) Fast OpenMP + GPU (CUDA) implementation Interactive applet available Introduction TGV Applications Summary K. Bredies 35 / 51

  78. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Extension to JPEG 2000 JPEG 2000 compression scheme: 00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 10000000011 W 00000010 =20 00010000 =16 01011100 =92 00000011 =30 Wavelet- Uncompressed image Bit-level coding JPEG2000 file Transformed image Introduction TGV Applications Summary K. Bredies 36 / 51

  79. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Extension to JPEG 2000 JPEG 2000 compression scheme: 00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 10000000011 W 00000010 =20 00010000 =16 01011100 =92 00000011 =30 Wavelet- Uncompressed image Bit-level coding JPEG2000 file Transformed image Source image set: Same structure Introduction TGV Applications Summary K. Bredies 36 / 51

  80. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Extension to JPEG 2000 JPEG 2000 compression scheme: 00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 10000000011 W 00000010 =20 00010000 =16 01011100 =92 00000011 =30 Wavelet- Uncompressed image Bit-level coding JPEG2000 file Transformed image Source image set: Same structure � TGV-based decompression can also be applied Introduction TGV Applications Summary K. Bredies 36 / 51

  81. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG 2000: Example Decompression of color images: 0.019 bpp standard decompression Introduction TGV Applications Summary K. Bredies 37 / 51

  82. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING JPEG 2000: Example Decompression of color images: 0.019 bpp JPEG2000-TGV decompression Introduction TGV Applications Summary K. Bredies 37 / 51

  83. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Free extra: Wavelet zooming JPEG 2000: Approximation coeffi- cients (+ precision) Some wavelet coeffi- cients (+ precision) Wavelet zooming: Approximation coeffi- cients (full precision) No wavelet coefficients � same framework can be used Introduction TGV Applications Summary K. Bredies 38 / 51

  84. Mathematical Optimization and INSTITUTE FOR MATHEMATICS Applications in Biomedical Sciences AND SCIENTIFIC COMPUTING Wavelet zooming: Example Test data: Barbara’s headscarf constant interp. cubic interp. Zooming: 64 × 64 → 256 × 256 Haar wavelet+TGV CDF wavelet+TGV Introduction TGV Applications Summary K. Bredies 39 / 51

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