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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

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total generalized variation: From regularization theory to applications in imaging

Kristian Bredies

Institute for Mathematics and Scientific Computing University of Graz

First French-German Mathematical Image Analysis Conference January 13, 2014

  • K. Bredies

1 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS 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

  • K. Bredies

2 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS 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

  • K. Bredies

3 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The total variation model in imaging

f Problem: Reconstruct image u from (blurred) noisy data noisy indirect measurements, etc. Widely used approach: min

u∈BV(Ω)

Ku − f 2

2

2 + α TV(u) Total variation TV: Convex energy Allows for discontinuities Enforces “sparse” gradient

  • K. Bredies

4 / 51 Introduction TGV Applications Summary

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SLIDE 5

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The total variation model in imaging

u Problem: Reconstruct image u from (blurred) noisy data noisy indirect measurements, etc. Widely used approach: min

u∈BV(Ω)

Ku − f 2

2

2 + α TV(u) Total variation TV: Convex energy Allows for discontinuities Enforces “sparse” gradient

  • K. Bredies

4 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The total variation model in imaging

u Problem: Reconstruct image u from (blurred) noisy data noisy indirect measurements, etc. Widely used approach: min

u∈BV(Ω)

Ku − f 2

2

2 + α TV(u) Total variation TV: Convex energy Allows for discontinuities Enforces “sparse” gradient Drawbacks: Unaware of higher-order smoothness Texture is not captured

  • K. Bredies

4 / 51 Introduction TGV Applications Summary

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SLIDE 7

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The total variation model in imaging

u Problem: Reconstruct image u from (blurred) noisy data noisy indirect measurements, etc. Widely used approach: min

u∈BV(Ω)

Ku − f 2

2

2 + α TV(u) Total variation TV: Convex energy Allows for discontinuities Enforces “sparse” gradient Drawbacks: Unaware of higher-order smoothness Texture is not captured

  • K. Bredies

4 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Convex higher-order image models

↓ Higher-order TV: Φ(u) =

  • Ω

d|∇2u| [Lysaker/Lundervold/Tai ’03] [Hinterberger/Scherzer ’04] Favors smooth solutions Edges are not preserved

  • K. Bredies

5 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Convex higher-order image models

↓ Higher-order TV: Φ(u) =

  • Ω

d|∇2u| [Lysaker/Lundervold/Tai ’03] [Hinterberger/Scherzer ’04] Favors smooth solutions Edges are not preserved TV-TV2 infimal convolution: Φ(u) = min

u=u1+u2

  • Ω

d|∇u1| + β

  • Ω

d|∇2u2| [Chambolle/Lions ’97] Models piecewise smooth images Staircase effect dominates solutions

  • K. Bredies

5 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Convex higher-order image models

↓ Higher-order TV: Φ(u) =

  • Ω

d|∇2u| [Lysaker/Lundervold/Tai ’03] [Hinterberger/Scherzer ’04] Favors smooth solutions Edges are not preserved TV-TV2 infimal convolution: Φ(u) = min

u=u1+u2

  • Ω

d|∇u1| + β

  • Ω

d|∇2u2| [Chambolle/Lions ’97] Models piecewise smooth images Staircase effect dominates solutions Different approach is needed

  • K. Bredies

5 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS 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

  • K. Bredies

6 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Joint work with Karl Kunisch and Thomas Pock Definition: Total Generalized Variation TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights
  • K. Bredies

7 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Joint work with Karl Kunisch and Thomas Pock Definition: Total Generalized Variation TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Idea: Incorporate information from ∇u, . . . , ∇ku Formal observation:

  • Ω

|∇ku| dx = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), v∞ ≤ 1

  • K. Bredies

7 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Joint work with Karl Kunisch and Thomas Pock Definition: Total Generalized Variation TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Idea: Incorporate information from ∇u, . . . , ∇ku Formal observation:

  • Ω

|∇ku| dx = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), v∞ ≤ 1

  • K. Bredies

7 / 51 Introduction TGV Applications Summary

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SLIDE 15

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Joint work with Karl Kunisch and Thomas Pock Definition: Total Generalized Variation TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Idea: Incorporate information from ∇u, . . . , ∇ku Formal observation: inf

Ek−l(pl)=0

  • Ω

|∇lu + pl| dx = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divk−l v∞ ≤ 1

  • K. Bredies

7 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Definition: TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Basic properties: [B./Kunisch/Pock ’10] TGVk

α is proper, convex, lower semi-continuous

TGVk

α is translation and rotation invariant

TGVk

α + · 1 gives Banach space BGVk α(Ω)

ker(TGVk

α) = Pk−1(Ω) polynomials of degree less than k

TGVk

α measures piecewise Pk−1 only at the interfaces

  • K. Bredies

8 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Definition: TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Basic properties: [B./Kunisch/Pock ’10] TGVk

α is proper, convex, lower semi-continuous

TGVk

α is translation and rotation invariant

TGVk

α + · 1 gives Banach space BGVk α(Ω)

ker(TGVk

α) = Pk−1(Ω) polynomials of degree less than k

TGVk

α measures piecewise Pk−1 only at the interfaces

  • K. Bredies

8 / 51 Introduction TGV Applications Summary

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SLIDE 18

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Definition: TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Basic properties: [B./Kunisch/Pock ’10] TGVk

α is proper, convex, lower semi-continuous

TGVk

α is translation and rotation invariant

TGVk

α + · 1 gives Banach space BGVk α(Ω)

ker(TGVk

α) = Pk−1(Ω) polynomials of degree less than k

TGVk

α measures piecewise Pk−1 only at the interfaces

  • K. Bredies

8 / 51 Introduction TGV Applications Summary

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SLIDE 19

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Definition: TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Basic properties: [B./Kunisch/Pock ’10] TGVk

α is proper, convex, lower semi-continuous

TGVk

α is translation and rotation invariant

TGVk

α + · 1 gives Banach space BGVk α(Ω)

ker(TGVk

α) = Pk−1(Ω) polynomials of degree less than k

TGVk

α measures piecewise Pk−1 only at the interfaces

  • K. Bredies

8 / 51 Introduction TGV Applications Summary

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SLIDE 20

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Total Generalized Variation

Definition: TGVk

α(u) = sup

  • Ω

u divk v dx

  • v ∈ Ck

c (Ω, Symk(I

Rd)), divl v∞ ≤ αl, l = 0, . . . , k − 1

  • α = (α0, . . . , αk−1) > 0 weights

Basic properties: [B./Kunisch/Pock ’10] TGVk

α is proper, convex, lower semi-continuous

TGVk

α is translation and rotation invariant

TGVk

α + · 1 gives Banach space BGVk α(Ω)

ker(TGVk

α) = Pk−1(Ω) polynomials of degree less than k

TGVk

α measures piecewise Pk−1 only at the interfaces

  • K. Bredies

8 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Denoising

Solve: min

u∈L2(Ω)

u − f 2 2 + TGVk

α(u)

Noisy image

  • K. Bredies

9 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Denoising

Solve: min

u∈L2(Ω)

u − f 2 2 + TGVk

α(u)

TV regularization

  • K. Bredies

9 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Denoising

Solve: min

u∈L2(Ω)

u − f 2 2 + TGVk

α(u)

TV-TV2 infimal-convolution regularization

  • K. Bredies

9 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Denoising

Solve: min

u∈L2(Ω)

u − f 2 2 + TGVk

α(u)

TGV2

α regularization

  • K. Bredies

9 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Denoising

Solve: min

u∈L2(Ω)

u − f 2 2 + TGVk

α(u)

TGV3

α regularization

  • K. Bredies

9 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Questions

How to interpret TGV? How is higher-order information incorporated?

  • K. Bredies

10 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS 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, min

u∈Lp(Ω) 1 2Ku − f 2 2 + TGVk α(u)

  • K. Bredies

10 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS 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, min

u∈Lp(Ω) 1 2Ku − f 2 2 + TGVk α(u)

Are the theoretical results applicable in practice? Are there efficient minimization algorithms? Does the model lead to improvements in image reconstruction?

  • K. Bredies

10 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Interpretation: TGV of second order

Minimum characterization: TGV2

α(u) =

min

w∈BD(Ω) α1

  • Ω

|∇u − w| + α0

  • Ω

|E(w)| BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} Vector fields of bounded deformation Intuitive interpretation: Locally: ∇u smooth w = ∇u ≈ optimal TGV2

α ∼ α0

  • loc |∇2u|

Locally: u jumps w = 0 ≈ optimal TGV2

α ∼ α1

  • loc |∇u|

Optimal balancing between ∇u and ∇2u TGV2

α(pw. smooth) < TGV2 α(staircases) preferred

Radon norm measure-based decomposition

  • K. Bredies

11 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Interpretation: TGV of second order

Minimum characterization: TGV2

α(u) =

min

w∈BD(Ω) α1

  • Ω

|∇u − w| + α0

  • Ω

|E(w)| BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} Vector fields of bounded deformation Intuitive interpretation: Locally: ∇u smooth w = ∇u ≈ optimal TGV2

α ∼ α0

  • loc |∇2u|

Locally: u jumps w = 0 ≈ optimal TGV2

α ∼ α1

  • loc |∇u|

Optimal balancing between ∇u and ∇2u TGV2

α(pw. smooth) < TGV2 α(staircases) preferred

Radon norm measure-based decomposition

  • K. Bredies

11 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Interpretation: TGV of second order

Minimum characterization: TGV2

α(u) =

min

w∈BD(Ω) α1

  • Ω

|∇u − w| + α0

  • Ω

|E(w)| BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} Vector fields of bounded deformation Intuitive interpretation: Locally: ∇u smooth w = ∇u ≈ optimal TGV2

α ∼ α0

  • loc |∇2u|

Locally: u jumps w = 0 ≈ optimal TGV2

α ∼ α1

  • loc |∇u|

Optimal balancing between ∇u and ∇2u TGV2

α(pw. smooth) < TGV2 α(staircases) preferred

Radon norm measure-based decomposition

  • K. Bredies

11 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Interpretation: TGV of second order

Minimum characterization: TGV2

α(u) =

min

w∈BD(Ω) α1

  • Ω

|∇u − w| + α0

  • Ω

|E(w)| BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} Vector fields of bounded deformation Intuitive interpretation: Locally: ∇u smooth w = ∇u ≈ optimal TGV2

α ∼ α0

  • loc |∇2u|

Locally: u jumps w = 0 ≈ optimal TGV2

α ∼ α1

  • loc |∇u|

Optimal balancing between ∇u and ∇2u TGV2

α(pw. smooth) < TGV2 α(staircases) preferred

Radon norm measure-based decomposition

  • K. Bredies

11 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

TGV regularization

Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Inverse Problem: solve Ku = f Ω ⊂ I Rd bounded domain K : Lp(Ω) → H linear and continuous Minimize: Tikhonov-functional min

u∈Lp(Ω)

Ku − f 2

H

2 + TGV2

α(u)

  • K. Bredies

12 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

TGV regularization

Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Inverse Problem: solve Ku = f Ω ⊂ I Rd bounded domain K : Lp(Ω) → H linear and continuous Minimize: Tikhonov-functional min

u∈Lp(Ω)

Ku − f 2

H

2 + TGV2

α(u)

Nonsmooth optimization problem Which conditions ensure existence of solutions? Are the solutions stable with respect to f ?

  • K. Bredies

12 / 51 Introduction TGV Applications Summary

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SLIDE 35

Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

TGV regularization

Joint work with Tuomo Valkonen Exemplarily: Solution of linear ill-posed inverse problems Total Generalized Variation of second order Inverse Problem: solve Ku = f Ω ⊂ I Rd bounded domain K : Lp(Ω) → H linear and continuous Minimize: Tikhonov-functional min

u∈Lp(Ω)

Ku − f 2

H

2 + TGV2

α(u)

Nonsmooth optimization problem Which conditions ensure existence of solutions? Are the solutions stable with respect to f ? Show topological equivalence with BV(Ω)

  • K. Bredies

12 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Vector fields of bounded deformation

BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} wBD = w1 + E(w)M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space ker(E) = {w : Ω → I Rd | w(x) = Ax + b, AT = −A} ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality: w − Rw1 ≤ CE(w)M R : BD(Ω) → ker(E) linear projection

  • K. Bredies

13 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Vector fields of bounded deformation

BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} wBD = w1 + E(w)M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space ker(E) = {w : Ω → I Rd | w(x) = Ax + b, AT = −A} ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality: w − Rw1 ≤ CE(w)M R : BD(Ω) → ker(E) linear projection

  • K. Bredies

13 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Vector fields of bounded deformation

BD(Ω) = {w ∈ L1(Ω, I Rd) | E(w) ∈ M(Ω, Sym2(I Rd))} wBD = w1 + E(w)M Well-known in the theory of mathematical plasticity Some properties: BD(Ω) is a Banach space ker(E) = {w : Ω → I Rd | w(x) = Ax + b, AT = −A} ⊂ BD(Ω) Space of infinitesimal rigid displacements Sobolev-Korn inequality: w − Rw1 ≤ CE(w)M R : BD(Ω) → ker(E) linear projection

  • K. Bredies

13 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Topological equivalence

Theorem: Ω ⊂ I Rd sufficiently smooth ⇒ cuBV ≤ u1 + TGV2

α(u) ≤ CuBV

∀u ∈ BGV2

α(Ω)

  • K. Bredies

14 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Topological equivalence

Theorem: Ω ⊂ I Rd sufficiently smooth ⇒ cuBV ≤ u1 + TGV2

α(u) ≤ CuBV

∀u ∈ BGV2

α(Ω)

Proof:

1 We have uBV ≤ C1

  • ∇u − RwM + u1
  • ∀w ∈ BD(Ω)

2 Sobolev-Korn inequality + minimum characterization:

∇u − RwM ≤ ∇u − wM + w − Rw1 ≤ C3

  • α1∇u − wM + α0E(w)M
  • ⇒

infw∈BD(Ω) ∇u − RwM ≤ C3 TGV2

α(u) 3 With the help of 1 :

⇒ uBV ≤ C4

  • u1 + TGV2

α(u)

  • 4 Finally: TGV2

α(u) ≤ α1 TV(u)

⇒ u1 + TGV2

α(u) ≤ C5uBV

  • K. Bredies

14 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Topological equivalence

Theorem: Ω ⊂ I Rd sufficiently smooth ⇒ cuBV ≤ u1 + TGV2

α(u) ≤ CuBV

∀u ∈ BGV2

α(Ω)

Proof:

1 We have uBV ≤ C1

  • ∇u − RwM + u1
  • ∀w ∈ BD(Ω)

2 Sobolev-Korn inequality + minimum characterization:

∇u − RwM ≤ ∇u − wM + w − Rw1 ≤ C3

  • α1∇u − wM + α0E(w)M
  • ⇒

infw∈BD(Ω) ∇u − RwM ≤ C3 TGV2

α(u) 3 With the help of 1 :

⇒ uBV ≤ C4

  • u1 + TGV2

α(u)

  • 4 Finally: TGV2

α(u) ≤ α1 TV(u)

⇒ u1 + TGV2

α(u) ≤ C5uBV

  • K. Bredies

14 / 51 Introduction TGV Applications Summary

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Topological equivalence

Theorem: Ω ⊂ I Rd sufficiently smooth ⇒ cuBV ≤ u1 + TGV2

α(u) ≤ CuBV

∀u ∈ BGV2

α(Ω)

Proof:

1 We have uBV ≤ C1

  • ∇u − RwM + u1
  • ∀w ∈ BD(Ω)

2 Sobolev-Korn inequality + minimum characterization:

∇u − RwM ≤ ∇u − wM + w − Rw1 ≤ C3

  • α1∇u − wM + α0E(w)M
  • ⇒

infw∈BD(Ω) ∇u − RwM ≤ C3 TGV2

α(u) 3 With the help of 1 :

⇒ uBV ≤ C4

  • u1 + TGV2

α(u)

  • 4 Finally: TGV2

α(u) ≤ α1 TV(u)

⇒ u1 + TGV2

α(u) ≤ C5uBV

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Topological equivalence

Theorem: Ω ⊂ I Rd sufficiently smooth ⇒ cuBV ≤ u1 + TGV2

α(u) ≤ CuBV

∀u ∈ BGV2

α(Ω)

Proof:

1 We have uBV ≤ C1

  • ∇u − RwM + u1
  • ∀w ∈ BD(Ω)

2 Sobolev-Korn inequality + minimum characterization:

∇u − RwM ≤ ∇u − wM + w − Rw1 ≤ C3

  • α1∇u − wM + α0E(w)M
  • ⇒

infw∈BD(Ω) ∇u − RwM ≤ C3 TGV2

α(u) 3 With the help of 1 :

⇒ uBV ≤ C4

  • u1 + TGV2

α(u)

  • 4 Finally: TGV2

α(u) ≤ α1 TV(u)

⇒ u1 + TGV2

α(u) ≤ C5uBV

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Existence and stability

Corollary: Coercivity: u − P1ud/(d−1) ≤ C TGV2

α(u)

P1 → Π1 linear projection on the affine functions Π1 Theorem: 1 < p ≤ d/(d − 1) K : Lp(Ω) → H linear and continuous, H Hilbert space K injective on Π1          ⇒          Optimization problem min

u∈Lp(Ω)

1 2Ku − f 2 + TGV2

α(u)

possesses a solution Proof: Direct method + coercivity of TGV2

α

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Existence and stability

Corollary: Coercivity: u − P1ud/(d−1) ≤ C TGV2

α(u)

P1 → Π1 linear projection on the affine functions Π1 Theorem: 1 < p ≤ d/(d − 1) K : Lp(Ω) → H linear and continuous, H Hilbert space K injective on Π1          ⇒          Optimization problem min

u∈Lp(Ω)

1 2Ku − f 2 + TGV2

α(u)

possesses a solution Proof: Direct method + coercivity of TGV2

α

Stability: f n → f in H ⇒

  • un ⇀ u in Lp(Ω) (subseq.)

TGV2

α(un) → TGV2 α(u)

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

General orders

Next step: Generalization with respect to k Examine BD(Ω, Symk(I Rd)) = {w ∈ L1(Ω, Symk(I Rd)) | E(w) ∈ M(Ω, Symk+1(I Rd))} wBD = w1 + E(w)M Symmetric tensor fields of bounded deformation

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: [B. ’11]

1 u ∈ D(Ω, Symk(I

Rd))∗ distribution with E(u) = 0

  • ⇒

∇k+1 ⊗ u = 0 in Ω

2 ker(E) is a subspace of Πk(Ω, Symk(I

Rd))

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: [B. ’11]

1 u ∈ D(Ω, Symk(I

Rd))∗ distribution with E(u) = 0

  • ⇒

∇k+1 ⊗ u = 0 in Ω

2 ker(E) is a subspace of Πk(Ω, Symk(I

Rd)) Furthermore: There is a T such that ∇k+1 ⊗ u = TE(u) for smooth u The fundamental solution for div E reads as: Γη

k = k

  • l=0

(−1)l k + 1 l + 1

  • El

divl(El+1η)

  • Em fundamental solution for ∆m
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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: Let Ω ⊂ I Rd be a Lipschitz domain

1 Trace mapping:

γ : BD(Ω, Symk(I Rd)) → L1(∂Ω, Symk(I Rd)) continuous with respect to strict convergence

2 Gauß-Green theorem:

  • Ω

u · div v dx =

  • ∂Ω

(γu ⊗ ν) · v dHd−1 −

  • Ω

v · dE(u) for u ∈ BD(Ω, Symk(I Rd)), v ∈ C1(Ω, Symk+1(I Rd))

3 Zero extension: Eu ∈ BD(I

Rd, Symk(I Rd)) with E(Eu) = E(u) − (γu ⊗ ν)Hd−1 ∂Ω

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: Let Ω ⊂ I Rd be a Lipschitz domain

1 Trace mapping:

γ : BD(Ω, Symk(I Rd)) → L1(∂Ω, Symk(I Rd)) continuous with respect to strict convergence

2 Gauß-Green theorem:

  • Ω

u · div v dx =

  • ∂Ω

(γu ⊗ ν) · v dHd−1 −

  • Ω

v · dE(u) for u ∈ BD(Ω, Symk(I Rd)), v ∈ C1(Ω, Symk+1(I Rd))

3 Zero extension: Eu ∈ BD(I

Rd, Symk(I Rd)) with E(Eu) = E(u) − (γu ⊗ ν)Hd−1 ∂Ω

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: Let Ω ⊂ I Rd be a Lipschitz domain

1 Trace mapping:

γ : BD(Ω, Symk(I Rd)) → L1(∂Ω, Symk(I Rd)) continuous with respect to strict convergence

2 Gauß-Green theorem:

  • Ω

u · div v dx =

  • ∂Ω

(γu ⊗ ν) · v dHd−1 −

  • Ω

v · dE(u) for u ∈ BD(Ω, Symk(I Rd)), v ∈ C1(Ω, Symk+1(I Rd))

3 Zero extension: Eu ∈ BD(I

Rd, Symk(I Rd)) with E(Eu) = E(u) − (γu ⊗ ν)Hd−1 ∂Ω

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: [B. ’11]

1 Embedding result:

BD(Ω, Symk(I Rd)) ֒ → Ld/(d−1)(Ω, Symk(I Rd))

2 Sobolev-Korn inequality: u−Rud/(d−1) ≤CE(u)M

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: [B. ’11]

1 Embedding result:

BD(Ω, Symk(I Rd)) ֒ → Ld/(d−1)(Ω, Symk(I Rd))

2 Sobolev-Korn inequality: u−Rud/(d−1) ≤CE(u)M

Consequently: Minimum characterization: TGVk

α(u) =

min

ul∈BD(Ω,Syml(I Rd)), u0=u, uk=0 k

  • l=1

αk−l

  • Ω

|E(ul−1) − ul| Existence of solutions: min

u∈Lp(Ω)

Ku − f 2

H

2 + TGVk

α(u)

if K is injective on Πk−1

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

The spaces BD(Ω, Symk(I Rd))

Theorem: [B. ’11]

1 Embedding result:

BD(Ω, Symk(I Rd)) ֒ → Ld/(d−1)(Ω, Symk(I Rd))

2 Sobolev-Korn inequality: u−Rud/(d−1) ≤CE(u)M

Consequently: Minimum characterization: TGVk

α(u) =

min

ul∈BD(Ω,Syml(I Rd)), u0=u, uk=0 k

  • l=1

αk−l

  • Ω

|E(ul−1) − ul| Existence of solutions: min

u∈Lp(Ω)

Ku − f 2

H

2 + TGVk

α(u)

if K is injective on Πk−1 “TV applicable ⇒ TGV applicable”

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Regularization properties

Joint work with Martin Holler Theorem: [B./Holler ’13] un solution ∼ parameters αn, data f n, f n − f † ≤ δn Multiparameter choice/source condition:

1 αn i → 0 and δ2 n/αn i → 0 2 limn→∞ αn i /αn i−1 > 0 3 Ku† = f † for u† ∈ BV(Ω)

Then: Convergence: un ⇀∗ u∗ TGVk,l

α∗(un) → TGVk α∗(u∗)

  • subsequentially

u∗ minimizing-TGVk

α∗ solution

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Deconvolution

Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: min

u∈L2(Ω)

1 2u ∗ k − f 2

2

+ TGV2

α(u)

Noisy data f

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Deconvolution

Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: min

u∈L2(Ω)

1 2u ∗ k − f 2

2

+ TGV2

α(u)

TV-regularized solution

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Deconvolution

Problem: Solve u ∗ k = f k convolution kernel Tikhonov functional: min

u∈L2(Ω)

1 2u ∗ k − f 2

2

+ TGV2

α(u)

TGV2

α-regularized solution

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Optimization methods

Exemplarily: Second-order scalar case TGV2,0

α = TGV2 α

min

u∈Lp(Ω) F(u) + TGV2 α(u)

Approach:

1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem

min

x∈X max y∈Y Ax, y + G(x) − F∗(y) 3 Use primal-dual algorithm of [Chambolle/Pock ’11]

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Optimization methods

Exemplarily: Second-order scalar case TGV2,0

α = TGV2 α

min

u∈Lp(Ω) F(u) + TGV2 α(u)

Approach:

1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem

min

x∈X max y∈Y Ax, y + G(x) − F∗(y) 3 Use primal-dual algorithm of [Chambolle/Pock ’11]

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Optimization methods

Exemplarily: Second-order scalar case TGV2,0

α = TGV2 α

min

u∈Lp(Ω) F(u) + TGV2 α(u)

Approach:

1 Discretize with finite differences 2 Reformulate as convex-concave saddle-point problem

min

x∈X max y∈Y Ax, y + G(x) − F∗(y) 3 Use primal-dual algorithm of [Chambolle/Pock ’11]

     y n+1 =(I + σ∂F∗)−1(y n + σAxn) xn+1 =(I + τ∂G)−1(xn − τA∗y n+1) xn+1 =2xn+1 − xn

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Saddle-point formulation

Finite difference approximations ∇h, Eh, divh = −(∇h)∗ etc. Supremum definition of TGV2: min

u max v

F(u) + u, (divh)2v −I{v∞≤α0}(v) − I{ω∞≤α1}( divh v)

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Saddle-point formulation

Finite difference approximations ∇h, Eh, divh = −(∇h)∗ etc. Supremum definition of TGV2: min

u max v

F(u) + u, (divh)2v −I{v∞≤α0}(v) − I{ω∞≤α1}( divh v) Introduce constraint ω = divh v and Lagrange multiplier w

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Saddle-point formulation

Finite difference approximations ∇h, Eh, divh = −(∇h)∗ etc. Supremum definition of TGV2: min

u,w max v,ω

F(u) + u, divh ω + w, divh v − ω −I{v∞≤α0}(v) − I{ω∞≤α1}(ω) Introduce constraint ω = divh v and Lagrange multiplier w

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Saddle-point formulation

Finite difference approximations ∇h, Eh, divh = −(∇h)∗ etc. Supremum definition of TGV2: min

u,w max v,ω

F(u) + u, divh ω + w, divh v − ω −I{v∞≤α0}(v) − I{ω∞≤α1}(ω) Introduce constraint ω = divh v and Lagrange multiplier w x = u w

  • ,

y = v ω

  • ,

A = −∇h −Eh −I

  • ,

G(u, w) = F(u), F∗(v, ω) = I{v∞≤α0}(v)+I{ω∞≤α1}(ω)

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Primal-dual algorithm 1

Iteration: [B. ’12] ωn+1 = P{ · ∞≤α1}

  • ωn + σ(∇h¯

un − ¯ w n)

  • v n+1 = P{ · ∞≤α0}
  • v n + σEh(¯

w n)

  • dual update

un+1 = (I + τ∂F)−1(un + τ divh ωn+1) w n+1 = w n + τ(divh v n+1 + ωn+1)

  • primal update

¯ un+1 = 2un+1 − un, ¯ w n+1 = 2w n+1 − w n extragradient

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Primal-dual algorithm 1

Iteration: [B. ’12] ωn+1 = P{ · ∞≤α1}

  • ωn + σ(∇h¯

un − ¯ w n)

  • v n+1 = P{ · ∞≤α0}
  • v n + σEh(¯

w n)

  • dual update

un+1 = (I + τ∂F)−1(un + τ divh ωn+1) w n+1 = w n + τ(divh v n+1 + ωn+1)

  • primal update

¯ un+1 = 2un+1 − un, ¯ w n+1 = 2w n+1 − w n extragradient P{ · ∞≤α0}, P{ · ∞≤α1} 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]

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Applications

Examples for Algorithm 1: F(u) (I + τ∂F)−1(u) 1 2u − f 2

H

u + τf 1 + τ u − f 1 f + Sτ(u − f ) Sτ soft-shrinkage operator 1 2Ku − f 2

H

(I + τK ∗K)−1(u + τK ∗f ) solve, e.g., with CGNE Primary application: Denoising problems

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Applications

Examples for Algorithm 1: F(u) (I + τ∂F)−1(u) 1 2u − f 2

H

u + τf 1 + τ u − f 1 f + Sτ(u − f ) Sτ soft-shrinkage operator 1 2Ku − f 2

H

(I + τK ∗K)−1(u + τK ∗f ) solve, e.g., with CGNE Primary application: Denoising problems

  • K. Bredies

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Applications

Examples for Algorithm 1: F(u) (I + τ∂F)−1(u) 1 2u − f 2

H

u + τf 1 + τ u − f 1 f + Sτ(u − f ) Sτ soft-shrinkage operator 1 2Ku − f 2

H

(I + τK ∗K)−1(u + τK ∗f ) solve, e.g., with CGNE Primary application: Denoising problems

  • K. Bredies

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Primal-dual algorithm 2

Alternative: Additional dual variable: F(u) = max

p

Ku, p + ˜ F(u)−G(p) Needs only resolvents w.r.t. ∂G, ∂ ˜ F, not (I + τ∂F)−1

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Primal-dual algorithm 2

Alternative: Additional dual variable: F(u) = max

p

Ku, p + ˜ F(u)−G(p) Needs only resolvents w.r.t. ∂G, ∂ ˜ F, not (I + τ∂F)−1 Iteration: [B. ’12]      pn+1 = (I + σ∂G)−1(pn + σK¯ un) ωn+1 = P{ · ∞≤α1}

  • ωn + σ(∇h¯

un − ¯ w n)

  • v n+1 = P{ · ∞≤α0}
  • v n + σEh(¯

w n)

  • un+1 = (I + τ∂ ˜

F)−1 un + τ(divh ωn+1 − K ∗pn+1)

  • w n+1 = w n + τ(divh v n+1 + ωn+1)
  • ¯

un+1 = 2un+1 − un, ¯ w n+1 = 2w n+1 − w n

  • K. Bredies

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Applications

Examples for Algorithm 2: F(u) G(p) (I + σ∂G)−1(p) 1 2Ku − f 2

H

p2 2 + f , p p − σf 1 + σ Ku − f 1 I{p∞≤1}(p) + f , p P{p∞≤1}(p − σf ) Here: ˜ F(u) = 0 ⇒ (I + τ∂ ˜ F)−1(u) = u Primary application: TGV2-regularized solution of Ku = f

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Applications

Examples for Algorithm 2: F(u) G(p) (I + σ∂G)−1(p) 1 2Ku − f 2

H

p2 2 + f , p p − σf 1 + σ Ku − f 1 I{p∞≤1}(p) + f , p P{p∞≤1}(p − σf ) Here: ˜ F(u) = 0 ⇒ (I + τ∂ ˜ F)−1(u) = u Primary application: TGV2-regularized solution of Ku = f

  • K. Bredies

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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

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Application: Compressive imaging

“Single-pixel camera” Original image

Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: min

Au=f R(u)

A basis analysis operator R “sparsifying” penalty

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Compressive imaging

“Single-pixel camera” 1600 measurements

Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: min

Au=f R(u)

A basis analysis operator R “sparsifying” penalty Compressed sensing: R(u) = u1 Captures solution with minimal “L0-norm” with high probability

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Compressive imaging

“Single-pixel camera” 1600 measurements

Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: min

Au=f R(u)

A basis analysis operator R “sparsifying” penalty Compressed sensing: R(u) = u1 Captures solution with minimal “L0-norm” with high probability In particular: R(u) = TV(u) “Gradient sparsity”

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Compressive imaging

“Single-pixel camera” 1600 measurements

Problem: Reconstruct incomplete data with respect to a given basis Variational formulation: min

Au=f R(u)

A basis analysis operator R “sparsifying” penalty Compressed sensing: R(u) = u1 Captures solution with minimal “L0-norm” with high probability In particular: R(u) = TV(u) “Gradient sparsity” Use TGV as penalty

  • K. Bredies

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Example

Test data: From “Rice Single-Pixel Camera Project”

http://dsp.rice.edu/cscamera

Reconstruction from varying number of samples Algorithm: Primal-dual method 2 TV-based reconstruction:

768 samples 384 samples 256 samples 192 samples

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Example

Test data: From “Rice Single-Pixel Camera Project”

http://dsp.rice.edu/cscamera

Reconstruction from varying number of samples Algorithm: Primal-dual method 2 TGV-based reconstruction:

768 samples 384 samples 256 samples 192 samples

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Image decompression

Joint work with Martin Holler Original image (8 bpp) JPEG compression scheme: Lossy procedure High compression disturbing artifacts (ringing, blocking)

  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Application: Image decompression

Joint work with Martin Holler JPEG image (0.062 bpp) JPEG compression scheme: Lossy procedure High compression disturbing artifacts (ringing, blocking)

  • K. Bredies

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Application: Image decompression

Joint work with Martin Holler JPEG image (0.062 bpp) JPEG compression scheme: Lossy procedure High compression disturbing artifacts (ringing, blocking) Goal: Remove artifacts Respect given information

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Application: Image decompression

Joint work with Martin Holler JPEG image (0.062 bpp) JPEG compression scheme: Lossy procedure High compression disturbing artifacts (ringing, blocking) Goal: Remove artifacts Respect given information Use TGV image model

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JPEG decompression model

JPEG compression scheme:

DCT

Quantization Lossless compression 8x8 blocks Source Image data Compressed Image data Quantization table

Problem: Many images give same JPEG object convex set C Standard decompression particular choice artifacts

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JPEG decompression model

JPEG compression scheme:

DCT

Quantization Lossless compression 8x8 blocks Source Image data Compressed Image data Quantization table

Problem: Many images give same JPEG object convex set C Standard decompression particular choice artifacts Idea: Optimize over all possible choices

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JPEG decompression model

JPEG compression scheme:

DCT

Quantization Lossless compression 8x8 blocks Source Image data Compressed Image data Quantization table

Problem: Many images give same JPEG object convex set C Standard decompression particular choice artifacts Idea: Optimize over all possible choices TGV Model: min

u∈L2(Ω) TGV2 α(u)

+IC(u)

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JPEG decompression: Example

Decompression of grayscale images: 0.062 bpp standard decompression

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JPEG decompression: Example

Decompression of grayscale images: 0.062 bpp JPEG-TGV decompression

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JPEG decompression: Example

Decompression of color images: 0.051 bpp standard decompression

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JPEG decompression: Example

Decompression of color images: 0.051 bpp JPEG-TGV decompression

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Towards real-life application

Software: Handles all flavors

  • f JPEG

(grayscale/color, chroma subsampling, etc.) Fast OpenMP + GPU (CUDA) implementation Interactive applet available

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Extension to JPEG 2000

JPEG 2000 compression scheme:

Uncompressed image Wavelet- Transformed image Bit-level coding

00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 00000010 =20 00010000 =16 01011100 =92 00000011 =30

JPEG2000 file

10000000011

W

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Extension to JPEG 2000

JPEG 2000 compression scheme:

Uncompressed image Wavelet- Transformed image Bit-level coding

00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 00000010 =20 00010000 =16 01011100 =92 00000011 =30

JPEG2000 file

10000000011

W

Source image set: Same structure

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Extension to JPEG 2000

JPEG 2000 compression scheme:

Uncompressed image Wavelet- Transformed image Bit-level coding

00001010 =10 00000100 =40 00000111 =70 00000001 =10 00000100 =40 00000010 =20 00010000 =16 01011100 =92 00000011 =30

JPEG2000 file

10000000011

W

Source image set: Same structure TGV-based decompression can also be applied

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JPEG 2000: Example

Decompression of color images: 0.019 bpp standard decompression

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JPEG 2000: Example

Decompression of color images: 0.019 bpp JPEG2000-TGV decompression

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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

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Wavelet zooming: Example

Test data: Barbara’s headscarf Zooming: 64×64 → 256×256

constant interp. cubic interp. Haar wavelet+TGV CDF wavelet+TGV

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Quantitative susceptibility mapping

Joint work with Christian Langkammer ↓ χ Motivation: Measure magnetic susceptibility χ with MRI quantification of specific biomarkers Can be obtained from 3D GRE phase data reconstruction is a challenging problem State of the art: Multi-step reconstruction procedure Last step: Regularized solution of a deconvolution problem Aims: Regularize with TGV Develop efficient algorithm

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Standard approach

Three-step procedure:

1 Unwrap phase data ϕwrap

→ ϕ0

2 Subtract harmonic background field, e.g.

min

ϕbg 1 2ϕbg − ϕ02 2

subject to ∆ϕbg = 0 ϕqsm = ϕ0 − ϕbg for optimal ϕbg

3 Perform regularized deconvolution

min

χ 1 2χ ∗ δ − cϕqsm2 2 + αR(χ)

(Fδ)(kx, ky, kz) =

1 3(k2 x + k2 y ) − 2 3k2 z

k2

x + k2 y + k2 z

, c = 1 2πTEγB0

  • ptimal χ susceptibility map
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Standard approach

Three-step procedure:

1 Unwrap phase data ϕwrap

→ ϕ0

2 Subtract harmonic background field, e.g.

min

ϕbg 1 2ϕbg − ϕ02 2

subject to ∆ϕbg = 0 ϕqsm = ϕ0 − ϕbg for optimal ϕbg

3 Perform regularized deconvolution

min

χ 1 2χ ∗ δ − cϕqsm2 2 + αR(χ)

(Fδ)(kx, ky, kz) =

1 3(k2 x + k2 y ) − 2 3k2 z

k2

x + k2 y + k2 z

, c = 1 2πTEγB0

  • ptimal χ susceptibility map
  • K. Bredies

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Standard approach

Three-step procedure:

1 Unwrap phase data ϕwrap

→ ϕ0

2 Subtract harmonic background field, e.g.

min

ϕbg 1 2ϕbg − ϕ02 2

subject to ∆ϕbg = 0 ϕqsm = ϕ0 − ϕbg for optimal ϕbg

3 Perform regularized deconvolution

min

χ 1 2χ ∗ δ − cϕqsm2 2 + αR(χ)

(Fδ)(kx, ky, kz) =

1 3(k2 x + k2 y ) − 2 3k2 z

k2

x + k2 y + k2 z

, c = 1 2πTEγB0

  • ptimal χ susceptibility map
  • K. Bredies

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Standard approach

Three-step procedure:

1 Unwrap phase data ϕwrap

→ ϕ0

2 Subtract harmonic background field, e.g.

min

ϕbg 1 2ϕbg − ϕ02 2

subject to ∆ϕbg = 0 ϕqsm = ϕ0 − ϕbg for optimal ϕbg

3 Perform regularized deconvolution

min

χ 1 2χ ∗ δ − cϕqsm2 2 + αR(χ)

(Fδ)(kx, ky, kz) =

1 3(k2 x + k2 y ) − 2 3k2 z

k2

x + k2 y + k2 z

, c = 1 2πTEγB0

  • ptimal χ susceptibility map

Is a single-step variational approach possible?

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Integrative variational modelling

Ingredients: Applying ∆ to the inverse problem χ ∗ δ = cϕqsm yields wave-equation-like partial differential equation: χ = 1 3 ∂2 ∂x2 + ∂2 ∂y 2 − 2 ∂2 ∂z2

  • χ = c∆ϕqsm

The background field is harmonic: ⇒ ∆ϕqsm = ∆ϕ0

  • n brain mask Ω′

∆ϕ0 can be obtained from the wrapped phase: ∆ϕ0 = Imag

  • (∆eiϕwrap

)e−iϕwrap

  • [Schofield/Zhu ’03]
  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Integrative variational modelling

Ingredients: Applying ∆ to the inverse problem χ ∗ δ = cϕqsm yields wave-equation-like partial differential equation: χ = 1 3 ∂2 ∂x2 + ∂2 ∂y 2 − 2 ∂2 ∂z2

  • χ = c∆ϕqsm

The background field is harmonic: ⇒ ∆ϕqsm = ∆ϕ0

  • n brain mask Ω′

∆ϕ0 can be obtained from the wrapped phase: ∆ϕ0 = Imag

  • (∆eiϕwrap

)e−iϕwrap

  • [Schofield/Zhu ’03]
  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Integrative variational modelling

Ingredients: Applying ∆ to the inverse problem χ ∗ δ = cϕqsm yields wave-equation-like partial differential equation: χ = 1 3 ∂2 ∂x2 + ∂2 ∂y 2 − 2 ∂2 ∂z2

  • χ = c∆ϕqsm

The background field is harmonic: ⇒ ∆ϕqsm = ∆ϕ0

  • n brain mask Ω′

∆ϕ0 can be obtained from the wrapped phase: ∆ϕ0 = Imag

  • (∆eiϕwrap

)e−iϕwrap

  • [Schofield/Zhu ’03]
  • K. Bredies

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Mathematical Optimization and Applications in Biomedical Sciences INSTITUTE FOR MATHEMATICS AND SCIENTIFIC COMPUTING

Integrative variational modelling

Ingredients: Applying ∆ to the inverse problem χ ∗ δ = cϕqsm yields wave-equation-like partial differential equation: χ = 1 3 ∂2 ∂x2 + ∂2 ∂y 2 − 2 ∂2 ∂z2

  • χ = c∆ϕqsm

The background field is harmonic: ⇒ ∆ϕqsm = ∆ϕ0

  • n brain mask Ω′

∆ϕ0 can be obtained from the wrapped phase: ∆ϕ0 = Imag

  • (∆eiϕwrap

)e−iϕwrap

  • [Schofield/Zhu ’03]

Solve: χ = c∆ϕ0 in Ω′

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The variational problem

Objective functional: Discrepancy:

1 2ψ2 2

with ∆ψ = χ − c∆ϕ0

  • n brain mask Ω′

Regularization of χ: TGV of second order

  • K. Bredies

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The variational problem

Objective functional: Discrepancy:

1 2ψ2 2

with ∆ψ = χ − c∆ϕ0

  • n brain mask Ω′

Regularization of χ: TGV of second order Integrative TGV-QSM reconstruction:    min

χ,ψ

1 2

  • Ω′ |ψ|2 dx + TGV2

α(χ)

subject to ∆ψ = χ − c∆ϕ0 on Ω′ Numerical method: Primal-dual algorithm 2 Essentially a one-step approach robust with respect to noise

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Numerical example

TGV-QSM reconstruction: 3D EPI, resolution 1mm3, size 230x230x176, TA 29 sec magnitude phase TGV-QSM

  • K. Bredies

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Numerical example

TGV-QSM reconstruction: 3D EPI, resolution 1mm3, size 230x230x176, TA 29 sec magnitude phase TGV-QSM

  • K. Bredies

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Application: Dual energy CT

Joint work with Michael Pienn

Source A Source B

Dual energy CT: Two X-ray sources with different spectra Two images are acquired Allows to differentiate and quantify contrast agent concentration Facilitates diagnosis in many cases Reconstructions are noisy due to low dose

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Assessment of lung perfusion

A B Important application: Diagnosis of pulmonary embolism Contrast agent concentration lower in affected areas of the lung

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Assessment of lung perfusion

A B A − B Important application: Diagnosis of pulmonary embolism Contrast agent concentration lower in affected areas of the lung Can be seen in the difference image

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Assessment of lung perfusion

A B A − B Important application: Diagnosis of pulmonary embolism Contrast agent concentration lower in affected areas of the lung Can be seen in the difference image

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Dual energy CT: Denoising

Problem setup: Given: Two noisy image sequences A0, B0 (3D data set) Base + difference image BGV2-images Prevent contrast change Use L1 discrepancy Minimization problem: min

(A,B)∈L1(Ω)2 A − A01 + B − B01 + TGV2 α(B)

+ TGV2

α(A − B)

Numerical realization: Primal-dual algorithm 1 (with a slight modification)

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Dual energy CT: Example

A0 − B0 A − B

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Dual energy CT: Example

A0 − B0 A − B

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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

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Summary

Total generalized variation

Consistent model for piecewise smooth images Functional-analytic framework for regularization of inverse problems is available

Computational methods

Two variants of a flexible primal-dual algorithm Easy to implement & suitable for parallelization

Imaging applications:

1 Denoising and deblurring 2 Compressive imaging 3 JPEG(2000) decompression

and wavelet zooming

Medical applications:

1 Quantitative susceptibility

mapping

2 Dual energy CT

high-quality reconstructions

  • K. Bredies

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References

  • K. Bredies, K. Kunisch and T. Pock.

Total generalized variation. SIAM Journal on Imaging Sciences 3(3):492–526, 2010.

  • K. Bredies and T. Valkonen.

Inverse problems with second-order total generalized variation constraints. Proceedings of SampTA 2011 — 9th International Conference on Sampling Theory and Applications, Singapore, 2011.

  • K. Bredies.

Symmetric tensor fields of bounded deformation. Annali di Matematica Pura ed Applicata, 192(5):815-851, 2013.

  • K. Bredies.

Recovering piecewise smooth multichannel images by minimization of convex functionals with total generalized variation penalty. To appear in Lecture Notes in Computer Science, 2012.

  • K. Bredies and M. Holler.

Regularization of linear inverse problems with total generalized variation. SFB MOBIS report 13-09, 2013.

  • K. Bredies and M. Holler.

Artifact-free decompression and zooming of JPEG compressed images with total generalized variation. Communications in Computer and Information Science 359:242–258, 2013.

  • K. Bredies and M. Holler.

A TGV regularized wavelet based zooming model. Lecture Notes in Computer Science 7893:149–160, 2013.

  • K. Bredies, F. Knoll and C. Langkammer.

TGV regularization for variational approaches to quantitative susceptibility mapping. Proceedings of the 2nd Workshop on MRI Phase Contrast & Quantitative Susceptibility Mapping, Ithaca 2013, 45–48, 2013.

  • K. Bredies

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