[PPT] - Transform Learning MRI with Global Wavelet Regularization A. Korhan PowerPoint Presentation

SLIDE 1

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Transform Learning MRI with Global Wavelet Regularization

A. Korhan Tanc1

Ender M. Eksioglu2

1Department of EEE

Kirklareli University Kirklareli, Turkey

2Department of ECE

Istanbul Technical University Istanbul, Turkey

EUSIPCO 2015

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 2

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 3

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 4

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

Active research area: Use sparsity as a regularizer for ill-conditioned inverse problems. Sparse regularization (and compressed sensing (CS)) have been applied to image reconstruction in Magnetic Resonance Imaging (MRI) (our problem of interest). Pioneering work [Lustig et.al., 2007], Sparse MRI: sparsely regularize the MRI reconstruction problem. min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

(1)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 5

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

Active research area: Use sparsity as a regularizer for ill-conditioned inverse problems. Sparse regularization (and compressed sensing (CS)) have been applied to image reconstruction in Magnetic Resonance Imaging (MRI) (our problem of interest). Pioneering work [Lustig et.al., 2007], Sparse MRI: sparsely regularize the MRI reconstruction problem. min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

(1)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 6

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

Active research area: Use sparsity as a regularizer for ill-conditioned inverse problems. Sparse regularization (and compressed sensing (CS)) have been applied to image reconstruction in Magnetic Resonance Imaging (MRI) (our problem of interest). Pioneering work [Lustig et.al., 2007], Sparse MRI: sparsely regularize the MRI reconstruction problem. min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

(1)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 7

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

Active research area: Use sparsity as a regularizer for ill-conditioned inverse problems. Sparse regularization (and compressed sensing (CS)) have been applied to image reconstruction in Magnetic Resonance Imaging (MRI) (our problem of interest). Pioneering work [Lustig et.al., 2007], Sparse MRI: sparsely regularize the MRI reconstruction problem. min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

(1)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 8

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

Active research area: Use sparsity as a regularizer for ill-conditioned inverse problems. Sparse regularization (and compressed sensing (CS)) have been applied to image reconstruction in Magnetic Resonance Imaging (MRI) (our problem of interest). Pioneering work [Lustig et.al., 2007], Sparse MRI: sparsely regularize the MRI reconstruction problem. min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

(1)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 9

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 10

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 11

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 12

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 13

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 14

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 15

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 16

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 17

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

x ∈ CN is the reconstructed MR image in vectorized form. Fu is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = Fux⋆ + η ∈ Cκ is the observation vector in the k-space. x⋆ is the true underlying image and η is the additive noise. The ratio κ/N quantifies the undersampling. ·1 denotes the ℓ1 norm. Φ is a sparsifying operator: we will assume it to be a square wavelet transform. ·TV is the Total Variation (TV) norm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 18

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 19

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Examplar or patch based methods have been very popular for sparsity based image processing. Dictionary learning (DL) based synthesis sparsity methods Analysis sparsity based analysis operator learning methods Novel model for analysis operator learning, called as sparsifying Transform Learning (TL) [Ravishankar and Bresler, 2013]. TL has been utilized to regularize the MRI reconstruction problem, resulting in the TLMRI algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 20

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Examplar or patch based methods have been very popular for sparsity based image processing. Dictionary learning (DL) based synthesis sparsity methods Analysis sparsity based analysis operator learning methods Novel model for analysis operator learning, called as sparsifying Transform Learning (TL) [Ravishankar and Bresler, 2013]. TL has been utilized to regularize the MRI reconstruction problem, resulting in the TLMRI algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 21

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Examplar or patch based methods have been very popular for sparsity based image processing. Dictionary learning (DL) based synthesis sparsity methods Analysis sparsity based analysis operator learning methods Novel model for analysis operator learning, called as sparsifying Transform Learning (TL) [Ravishankar and Bresler, 2013]. TL has been utilized to regularize the MRI reconstruction problem, resulting in the TLMRI algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 22

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Examplar or patch based methods have been very popular for sparsity based image processing. Dictionary learning (DL) based synthesis sparsity methods Analysis sparsity based analysis operator learning methods Novel model for analysis operator learning, called as sparsifying Transform Learning (TL) [Ravishankar and Bresler, 2013]. TL has been utilized to regularize the MRI reconstruction problem, resulting in the TLMRI algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 23

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Examplar or patch based methods have been very popular for sparsity based image processing. Dictionary learning (DL) based synthesis sparsity methods Analysis sparsity based analysis operator learning methods Novel model for analysis operator learning, called as sparsifying Transform Learning (TL) [Ravishankar and Bresler, 2013]. TL has been utilized to regularize the MRI reconstruction problem, resulting in the TLMRI algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 24

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Methods such as Sparse MRI, RecPF and FCSA apply global, image-scale regularization TLMRI or DL based algorithms utilize local, patch-scale regularization In this work, we aim to bring these two ends together.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 25

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Methods such as Sparse MRI, RecPF and FCSA apply global, image-scale regularization TLMRI or DL based algorithms utilize local, patch-scale regularization In this work, we aim to bring these two ends together.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 26

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

Patch based regularization methods

Methods such as Sparse MRI, RecPF and FCSA apply global, image-scale regularization TLMRI or DL based algorithms utilize local, patch-scale regularization In this work, we aim to bring these two ends together.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 27

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

New Method: Globally regularized TLMRI

G-TLMRI

We introduce a global sparsifying cost into TLMRI, and provide the algorithm. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI). Simulation results: use of global and local regularization terms together results in superior reconstruction performance.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 28

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

New Method: Globally regularized TLMRI

G-TLMRI

We introduce a global sparsifying cost into TLMRI, and provide the algorithm. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI). Simulation results: use of global and local regularization terms together results in superior reconstruction performance.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 29

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion The Problem The Novel Approach

New Method: Globally regularized TLMRI

G-TLMRI

We introduce a global sparsifying cost into TLMRI, and provide the algorithm. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI). Simulation results: use of global and local regularization terms together results in superior reconstruction performance.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 30

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TL has been applied to MRI image reconstruction. TLMRI cost function can be stated as follows. (P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. (2)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 31

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TL has been applied to MRI image reconstruction. TLMRI cost function can be stated as follows. (P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. (2)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 32

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TL has been applied to MRI image reconstruction. TLMRI cost function can be stated as follows. (P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. (2)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 33

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TL has been applied to MRI image reconstruction. TLMRI cost function can be stated as follows. (P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. (2)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 34

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 35

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 36

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 37

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 38

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 39

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. ·F is the Frobenius matrix norm. ·0 denotes the ℓ0 pseudo-norm. W ∈ Cn×n is the learned square transform. ˆ X ∈ Cn×M, and its columns ˆ xj ∈ Cn denote vectorized 2D patches of size √n × √n.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 40

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 41

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 42

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 43

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 44

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 45

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. A ∈ Cn×M includes the sparse codes. Q(·) penalization term for the learned W. R image to patch operator. Observation fidelity is enforced using the Fux − y2

2 term.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 46

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TLMRI applies local regularization via a learned sparsifying transform. TLMRI with learned, local regularization: good performance when compared to nonadaptive global regularization (such as wavelet plus TV regularization in Sparse MRI). In this work: include additional global regularization in the TLMRI framework.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 47

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TLMRI applies local regularization via a learned sparsifying transform. TLMRI with learned, local regularization: good performance when compared to nonadaptive global regularization (such as wavelet plus TV regularization in Sparse MRI). In this work: include additional global regularization in the TLMRI framework.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 48

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

From the Literature: Transform Learning MRI

TLMRI

TLMRI applies local regularization via a learned sparsifying transform. TLMRI with learned, local regularization: good performance when compared to nonadaptive global regularization (such as wavelet plus TV regularization in Sparse MRI). In this work: include additional global regularization in the TLMRI framework.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 49

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 50

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

New cost function with global regularizer. (P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(3)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 51

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

New cost function with global regularizer. (P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(3)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 52

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

New cost function with global regularizer. (P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(3)

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 53

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

(P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. When compared with (P0), in (P1) the crucial change is the introduction of the Φx1 term. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 54

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

(P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. When compared with (P0), in (P1) the crucial change is the introduction of the Φx1 term. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 55

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

(P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. When compared with (P0), in (P1) the crucial change is the introduction of the Φx1 term. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 56

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

(P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. When compared with (P0), in (P1) the crucial change is the introduction of the Φx1 term. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 57

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

(P1) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+ τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1.

(P0) min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + τR(x) − ˆ

X2

F

+ ηFux − y2

2 ,

s.t. αj0 ≤ sj ∀j = 1 . . . M. When compared with (P0), in (P1) the crucial change is the introduction of the Φx1 term. We will denote this modified framework as the Globally regularized TLMRI (G-TLMRI).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 58

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 59

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 60

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 61

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 62

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 63

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

New Method: GTLMRI

We will separate the algorithm into two steps with and without optimization on x. (P2) min

W, ˆ X,A

W ˆ X −A2

F +λQ(W)+βA1+τR(x)− ˆ

X2

F. (4)

(P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

(5) (P2) can be thought of as denoising. (P3) can be thought of as reconstruction.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 64

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 65

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

We will divide (P2) into two in the following form similar to the TLMRI. (P2.1) min

W,A W ˆ

X − A2

F + λQ(W) + βA1.

(P2.2) min

ˆ X,A

W ˆ X − A2

F + βA1 + τR(x) − ˆ

X2

F.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 66

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

We will divide (P2) into two in the following form similar to the TLMRI. (P2.1) min

W,A W ˆ

X − A2

F + λQ(W) + βA1.

(P2.2) min

ˆ X,A

W ˆ X − A2

F + βA1 + τR(x) − ˆ

X2

F.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 67

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

We will divide (P2) into two in the following form similar to the TLMRI. (P2.1) min

W,A W ˆ

X − A2

F + λQ(W) + βA1.

(P2.2) min

ˆ X,A

W ˆ X − A2

F + βA1 + τR(x) − ˆ

X2

F.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 68

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

We will divide (P2) into two in the following form similar to the TLMRI. (P2.1) min

W,A W ˆ

X − A2

F + λQ(W) + βA1.

(P2.2) min

ˆ X,A

W ˆ X − A2

F + βA1 + τR(x) − ˆ

X2

F.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 69

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

We will divide (P2) into two in the following form similar to the TLMRI. (P2.1) min

W,A W ˆ

X − A2

F + λQ(W) + βA1.

(P2.2) min

ˆ X,A

W ˆ X − A2

F + βA1 + τR(x) − ˆ

X2

F.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 70

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

(P2.1) can be approximately solved using iterative alternation over two steps. (P2.1.1) min

A W ˆ

X − A2

F + βA1.

(P2.1.2) min

W W ˆ

X − A2

F + λQ(W).

Both (P2.1.1) and (P2.1.2) have closed form solutions.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 71

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

(P2.1) can be approximately solved using iterative alternation over two steps. (P2.1.1) min

A W ˆ

X − A2

F + βA1.

(P2.1.2) min

W W ˆ

X − A2

F + λQ(W).

Both (P2.1.1) and (P2.1.2) have closed form solutions.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 72

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

(P2.1) can be approximately solved using iterative alternation over two steps. (P2.1.1) min

A W ˆ

X − A2

F + βA1.

(P2.1.2) min

W W ˆ

X − A2

F + λQ(W).

Both (P2.1.1) and (P2.1.2) have closed form solutions.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 73

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

(P2.1) can be approximately solved using iterative alternation over two steps. (P2.1.1) min

A W ˆ

X − A2

F + βA1.

(P2.1.2) min

W W ˆ

X − A2

F + λQ(W).

Both (P2.1.1) and (P2.1.2) have closed form solutions.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 74

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

(P2.1) can be approximately solved using iterative alternation over two steps. (P2.1.1) min

A W ˆ

X − A2

F + βA1.

(P2.1.2) min

W W ˆ

X − A2

F + λQ(W).

Both (P2.1.1) and (P2.1.2) have closed form solutions.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 75

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 76

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 77

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 78

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 79

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 80

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Denoising

Two alternating steps for (P2.2) become as follows. (P2.2.1) min

A W ˆ

X − A2

F + βA1.

(P2.2.2) min

ˆ X

W ˆ X − A2

F + τR(x) − ˆ

X2

F.

(P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by (W HW + τI)−1(W HA + τR(x)).

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 81

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 82

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The second main step for the solution of (P1) is the reconstruction step, (P3). (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 83

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The second main step for the solution of (P1) is the reconstruction step, (P3). (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 84

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The second main step for the solution of (P1) is the reconstruction step, (P3). (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 85

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 86

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 87

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 88

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 89

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 90

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define patch to image operator ˆ R. ˆ R( ˆ X) =

j RT

j ˆ

xj

./w.

(P3) can be approximately rewritten as follows. (P3′) min

x 1 2

Fux − y2

2 + τ ′x − ˆ

R( ˆ X)2

2

+ υΦx1.

(6) (P3) min

x 1 2Fux − y2 2 + τ 2ηR(x) − ˆ

X2

F + υ′ 2ηΦx1.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 91

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 92

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 93

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 94

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 95

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 96

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 97

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

Define two functions: g(x) = 1

2

Fux − y2

2+ τ ′x − ˆ

R( ˆ X)2

2

f(x) = υΦx1.

(P3′) min

x

f(x) + g(x). This problem can be solved very efficiently by proximal splitting methods. We have used the forward-backward splitting algorithm.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 98

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 99

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 100

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 101

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 102

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 103

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 104

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Reconstruction

The forward-backward splitting steps: (P3.1) z = x − γ∇g(x). (7) (P3.2) x = x + µ(proxγf(z) − x). (8) ∇g(x) =FH

u (Fux − y)+τ ′(x − ˆ

R( ˆ X)). FH

u is the adjoint operator of Fu, it realizes zero-filled

reconstruction. proxγf(·) is realized by soft thresholding in the transform (Φ) domain and taking an inverse transform.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 105

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

Outline

1

Introduction The Problem The Novel Approach

2

Transform Learning MRI

3

GTLMRI New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

4

Simulations and Conclusion

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 106

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 107

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 108

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 109

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 110

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 111

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 112

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 113

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 114

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 115

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 116

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm

GTLMRI: Overall Algorithm

Input: Observation, y = Fux⋆ + η; parameters

λ, β, τ, τ ′, υ, γ, µ.

Goal:

min

W, ˆ X,A,x

W ˆ X − A2

F + λQ(W) + βA1

+τR(x) − ˆ X2

F + ηFux − y2 2 + υ′Φx1

Initialize x = FH

u y.

Main iteration:

Initialize ˆ

X = R(x). denoising starts

Iterate (P2.1), N1 times.
Iterate (P2.2), N2 times.
Initialize x = ˆ

R( ˆ X). reconstruction starts

Iterate (P3.1-P3.2), N3 times.

Output reconstructed MR image x.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 117

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations setting

We compare the reconstruction performance of G-TLMRI algorithm with TLMRI [Ravishankar and Bresler, 2013], DLMRI [Ravishankar and Bresler, 2011] and FCSA [Huang et.al., 2011]. Simulations for two MR images of size (256 × 256). The downsampling ratio for Fu is κ/2562 = 0.25 (4 fold downsampling) with a random sampling mask.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 118

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations setting

We compare the reconstruction performance of G-TLMRI algorithm with TLMRI [Ravishankar and Bresler, 2013], DLMRI [Ravishankar and Bresler, 2011] and FCSA [Huang et.al., 2011]. Simulations for two MR images of size (256 × 256). The downsampling ratio for Fu is κ/2562 = 0.25 (4 fold downsampling) with a random sampling mask.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 119

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations setting

We compare the reconstruction performance of G-TLMRI algorithm with TLMRI [Ravishankar and Bresler, 2013], DLMRI [Ravishankar and Bresler, 2011] and FCSA [Huang et.al., 2011]. Simulations for two MR images of size (256 × 256). The downsampling ratio for Fu is κ/2562 = 0.25 (4 fold downsampling) with a random sampling mask.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 120

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations: Original images

a) b) c)

Figure: (a) Sampling mask in k-space with 4-fold undersampling , (b,c) the original MRI test images.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 121

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations: Brain image

Figure: Brain image results. First row: Zero-filling and G-TLMRI. Second row: TLMRI and FCSA.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 122

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations: Brain image

5 10 15 20 25 30 35 40 10 12 14 16 18 20 22 24 26 Iteration SNR (dB) G−TLMRI TLMRI DLMRI FCSA

Figure: Brain image results: SNR versus iteration.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 123

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations: Shoulder image

Figure: Shoulder image results. First row: Zero-filling and G-TLMRI. Second row: TLMRI and FCSA.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 124

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Simulations: Shoulder image

5 10 15 20 25 30 35 40 14 16 18 20 22 24 26 28 30 Iteration SNR (dB) G−TLMRI TLMRI DLMRI FCSA

Figure: Shoulder image results: SNR versus iteration.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 125

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Conclusion

We have presented a new algorithm called as G-TLMRI for MRI reconstruction. G-TLMRI algorithm builds upon the patch level sparsification of the TLMRI. G-TLMRI introduces a global regularizer into the TLMRI framework. Combination of the local and global regularization terms results in reconstruction performance exceeding some competing methods which use these terms alone.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 126

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Conclusion

We have presented a new algorithm called as G-TLMRI for MRI reconstruction. G-TLMRI algorithm builds upon the patch level sparsification of the TLMRI. G-TLMRI introduces a global regularizer into the TLMRI framework. Combination of the local and global regularization terms results in reconstruction performance exceeding some competing methods which use these terms alone.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 127

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Conclusion

We have presented a new algorithm called as G-TLMRI for MRI reconstruction. G-TLMRI algorithm builds upon the patch level sparsification of the TLMRI. G-TLMRI introduces a global regularizer into the TLMRI framework. Combination of the local and global regularization terms results in reconstruction performance exceeding some competing methods which use these terms alone.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 128

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Conclusion

We have presented a new algorithm called as G-TLMRI for MRI reconstruction. G-TLMRI algorithm builds upon the patch level sparsification of the TLMRI. G-TLMRI introduces a global regularizer into the TLMRI framework. Combination of the local and global regularization terms results in reconstruction performance exceeding some competing methods which use these terms alone.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 129

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Thanks for listening.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 130

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References I

M. Lustig, D. Donoho, and J.M. Pauly,

“Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magnetic Resonance in Medicine, vol. 58, no. 6, pp. 1182–1195, 2007.

J. Yang, Y. Zhang, and W. Yin,

“A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Topics Signal Process., vol. 4, no. 2, pp. 288–297, April 2010.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 131

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References II

J.Huang, S.Zhang, and D.Metaxas, “Efficient MR image reconstruction for compressed MR imaging,” Medical Image Analysis, vol. 15, no. 5, pp. 670 – 679, 2011. E.M. Eksioglu, “Online dictionary learning algorithm with periodic updates and its application to image denoising,” Expert Systems with Applications, vol. 41, no. 8, pp. 3682 – 3690, 2014.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 132

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References III

S. Nam, M.E. Davies, M. Elad, and R. Gribonval,

“The cosparse analysis model and algorithms,” Applied and Computational Harmonic Analysis, vol. 34, no. 1, pp. 30 – 56, 2013.

S. Ravishankar and Y. Bresler,

“MR image reconstruction from highly undersampled k-space data by dictionary learning,” IEEE Trans. Med. Imag., vol. 30, no. 5, pp. 1028–1041, May 2011.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 133

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References IV

S. Ravishankar and Y. Bresler,

“Learning sparsifying transforms,” IEEE Trans. Signal Process., vol. 61, no. 5, pp. 1072–1086, 2013. E.M. Eksioglu and O. Bayir, “K-SVD meets transform learning: Transform K-SVD,” IEEE Signal Process. Lett., vol. 21, no. 3, pp. 347–351, March 2014.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 134

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References V

S. Ravishankar and Y. Bresler,

“Sparsifying transform learning for compressed sensing MRI,” in 2013 IEEE 10th International Symposium on Biomedical Imaging (ISBI), April 2013, pp. 17–20.

Y. Huang, J. Paisley, Q. Lin, X. Ding, X. Fu, and X.P

. Zhang, “Bayesian nonparametric dictionary learning for compressed sensing MRI,” IEEE Trans. Image Process., vol. 23, no. 12, pp. 5007–5019, Dec 2014.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 135

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

References VI

E.M. Eksioglu and O. Bayir, “Overcomplete sparsifying transform learning algorithm using a constrained least squares approach,” in Acoustics, Speech and Signal Processing (ICASSP), 2014 IEEE International Conference on, May 2014, pp. 7158–7162. P .L. Combettes and J.C. Pesquet, “Proximal splitting methods in signal processing,” in Fixed-Point Algorithms for Inverse Problems in Science and Engineering, Optimization and Its Applications, pp. 185–212. Springer New York, 2011.

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 136

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

Several approaches for solving this cost function or its variants. In the original Sparse MRI algorithm [Lustig et.al., 2007]: a nonlinear conjugate gradient method Operator and variable splitting methods: FCSA, RecPF, TVCMRI ...

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 137

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

Several approaches for solving this cost function or its variants. In the original Sparse MRI algorithm [Lustig et.al., 2007]: a nonlinear conjugate gradient method Operator and variable splitting methods: FCSA, RecPF, TVCMRI ...

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

SLIDE 138

Introduction Transform Learning MRI GTLMRI Simulations and Conclusion

Sparse MRI

min

x 1 2Fux − y2 2 + ρ1Φx1 + ρ2xTV.

Several approaches for solving this cost function or its variants. In the original Sparse MRI algorithm [Lustig et.al., 2007]: a nonlinear conjugate gradient method Operator and variable splitting methods: FCSA, RecPF, TVCMRI ...

Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization