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Introduction Transform Learning MRI GTLMRI Simulations and Conclusion Transform Learning MRI with Global Wavelet Regularization A. Korhan Tanc 1 Ender M. Eksioglu 2 1 Department of EEE Kirklareli University Kirklareli, Turkey 2 Department of

  1. Introduction Transform Learning MRI The Problem GTLMRI The Novel Approach Simulations and Conclusion Sparse MRI 2 � F u x − y � 2 1 min 2 + ρ 1 � Φ x � 1 + ρ 2 � x � TV . x x ∈ C N is the reconstructed MR image in vectorized form. F u is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = F u x ⋆ + η ∈ C κ 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 ℓ 1 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

  2. Introduction Transform Learning MRI The Problem GTLMRI The Novel Approach Simulations and Conclusion Sparse MRI 2 � F u x − y � 2 1 min 2 + ρ 1 � Φ x � 1 + ρ 2 � x � TV . x x ∈ C N is the reconstructed MR image in vectorized form. F u is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = F u x ⋆ + η ∈ C κ 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 ℓ 1 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

  3. Introduction Transform Learning MRI The Problem GTLMRI The Novel Approach Simulations and Conclusion Sparse MRI 2 � F u x − y � 2 1 min 2 + ρ 1 � Φ x � 1 + ρ 2 � x � TV . x x ∈ C N is the reconstructed MR image in vectorized form. F u is the undersampled Fourier transform operator: conversion from the vectorized image to the k-space. y = F u x ⋆ + η ∈ C κ 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 ℓ 1 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

  4. Introduction Transform Learning MRI The Problem GTLMRI The Novel Approach Simulations and Conclusion Outline Introduction 1 The Problem The Novel Approach Transform Learning MRI 2 GTLMRI 3 New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm Simulations and Conclusion 4 Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

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

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

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

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

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

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

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

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

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

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

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

  16. 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. � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . (2) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  17. 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. � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . (2) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  18. 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. � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . (2) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  19. 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. � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . (2) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  20. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  21. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  22. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  23. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  24. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  25. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . �·� F is the Frobenius matrix norm. �·� 0 denotes the ℓ 0 pseudo-norm. W ∈ C n × n is the learned square transform. x j ∈ C n denote vectorized 2D X ∈ C n × M , and its columns ˆ ˆ patches of size √ n × √ n . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  26. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  27. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  28. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  29. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  30. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  31. Introduction Transform Learning MRI GTLMRI Simulations and Conclusion From the Literature: Transform Learning MRI TLMRI � W ˆ X − A � 2 F + λ Q ( W ) + τ � R ( x ) − ˆ X � 2 (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . A ∈ C n × M includes the sparse codes. Q ( · ) penalization term for the learned W . R image to patch operator. Observation fidelity is enforced using the � F u x − y � 2 2 term. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  32. 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

  33. 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

  34. 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

  35. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm Outline Introduction 1 The Problem The Novel Approach Transform Learning MRI 2 GTLMRI 3 New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm Simulations and Conclusion 4 Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  36. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI New cost function with global regularizer. X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . (3) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  37. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI New cost function with global regularizer. X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . (3) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  38. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI New cost function with global regularizer. X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . (3) Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  39. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . X − A � 2 X � 2 � W ˆ F + λ Q ( W ) + τ � R ( x ) − ˆ (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . When compared with (P0), in (P1) the crucial change is the introduction of the � Φ x � 1 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

  40. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . X − A � 2 X � 2 � W ˆ F + λ Q ( W ) + τ � R ( x ) − ˆ (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . When compared with (P0), in (P1) the crucial change is the introduction of the � Φ x � 1 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

  41. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . X − A � 2 X � 2 � W ˆ F + λ Q ( W ) + τ � R ( x ) − ˆ (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . When compared with (P0), in (P1) the crucial change is the introduction of the � Φ x � 1 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

  42. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . X − A � 2 X � 2 � W ˆ F + λ Q ( W ) + τ � R ( x ) − ˆ (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . When compared with (P0), in (P1) the crucial change is the introduction of the � Φ x � 1 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

  43. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI X − A � 2 � W ˆ (P1) min F + λ Q ( W ) + β � A � 1 W , ˆ X , A , x X � 2 F + η � F u x − y � 2 + τ � R ( x ) − ˆ 2 + υ ′ � Φ x � 1 . X − A � 2 X � 2 � W ˆ F + λ Q ( W ) + τ � R ( x ) − ˆ (P0) min F W , ˆ X , A , x + η � F u x − y � 2 2 , s . t . � α j � 0 ≤ s j ∀ j = 1 . . . M . When compared with (P0), in (P1) the crucial change is the introduction of the � Φ x � 1 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

  44. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  45. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  46. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  47. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  48. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  49. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm New Method: GTLMRI We will separate the algorithm into two steps with and without optimization on x . X − A � 2 X � 2 � W ˆ F + λ Q ( W )+ β � A � 1 + τ � R ( x ) − ˆ (P2) min F . (4) W , ˆ X , A 1 2 � F u x − y � 2 2 η � R ( x ) − ˆ X � 2 F + υ ′ 2 + τ 2 η � Φ x � 1 . (5) (P3) min x (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

  50. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm Outline Introduction 1 The Problem The Novel Approach Transform Learning MRI 2 GTLMRI 3 New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm Simulations and Conclusion 4 Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  51. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising We will divide (P2) into two in the following form similar to the TLMRI. W , A � W ˆ X − A � 2 F + λ Q ( W ) + β � A � 1 . (P2.1) min � W ˆ X − A � 2 F + β � A � 1 + τ � R ( x ) − ˆ X � 2 F . (P2.2) min ˆ X , A Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  52. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising We will divide (P2) into two in the following form similar to the TLMRI. W , A � W ˆ X − A � 2 F + λ Q ( W ) + β � A � 1 . (P2.1) min � W ˆ X − A � 2 F + β � A � 1 + τ � R ( x ) − ˆ X � 2 F . (P2.2) min ˆ X , A Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  53. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising We will divide (P2) into two in the following form similar to the TLMRI. W , A � W ˆ X − A � 2 F + λ Q ( W ) + β � A � 1 . (P2.1) min � W ˆ X − A � 2 F + β � A � 1 + τ � R ( x ) − ˆ X � 2 F . (P2.2) min ˆ X , A Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  54. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising We will divide (P2) into two in the following form similar to the TLMRI. W , A � W ˆ X − A � 2 F + λ Q ( W ) + β � A � 1 . (P2.1) min � W ˆ X − A � 2 F + β � A � 1 + τ � R ( x ) − ˆ X � 2 F . (P2.2) min ˆ X , A Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  55. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising We will divide (P2) into two in the following form similar to the TLMRI. W , A � W ˆ X − A � 2 F + λ Q ( W ) + β � A � 1 . (P2.1) min � W ˆ X − A � 2 F + β � A � 1 + τ � R ( x ) − ˆ X � 2 F . (P2.2) min ˆ X , A Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  56. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising (P2.1) can be approximately solved using iterative alternation over two steps. X − A � 2 A � W ˆ (P2.1.1) min F + β � A � 1 . W � W ˆ X − A � 2 F + λ Q ( W ) . (P2.1.2) min Both (P2.1.1) and (P2.1.2) have closed form solutions. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  57. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising (P2.1) can be approximately solved using iterative alternation over two steps. X − A � 2 A � W ˆ (P2.1.1) min F + β � A � 1 . W � W ˆ X − A � 2 F + λ Q ( W ) . (P2.1.2) min Both (P2.1.1) and (P2.1.2) have closed form solutions. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  58. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising (P2.1) can be approximately solved using iterative alternation over two steps. X − A � 2 A � W ˆ (P2.1.1) min F + β � A � 1 . W � W ˆ X − A � 2 F + λ Q ( W ) . (P2.1.2) min Both (P2.1.1) and (P2.1.2) have closed form solutions. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  59. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising (P2.1) can be approximately solved using iterative alternation over two steps. X − A � 2 A � W ˆ (P2.1.1) min F + β � A � 1 . W � W ˆ X − A � 2 F + λ Q ( W ) . (P2.1.2) min Both (P2.1.1) and (P2.1.2) have closed form solutions. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  60. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising (P2.1) can be approximately solved using iterative alternation over two steps. X − A � 2 A � W ˆ (P2.1.1) min F + β � A � 1 . W � W ˆ X − A � 2 F + λ Q ( W ) . (P2.1.2) min Both (P2.1.1) and (P2.1.2) have closed form solutions. Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  61. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  62. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  63. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  64. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  65. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  66. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Denoising Two alternating steps for (P2.2) become as follows. A � W ˆ X − A � 2 F + β � A � 1 . (P2.2.1) min � W ˆ X − A � 2 F + τ � R ( x ) − ˆ X � 2 F . (P2.2.2) min ˆ X (P2.2.1) is again solved by soft thresholding. (P2.2.2) has a simple least squares solution for fixed A given by ( W H W + τ I ) − 1 ( W H A + τ R ( x )) . Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  67. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm Outline Introduction 1 The Problem The Novel Approach Transform Learning MRI 2 GTLMRI 3 New Cost GTLMRI: Denoising GTLMRI: Reconstruction GTLMRI: Overall Algorithm Simulations and Conclusion 4 Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  68. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The second main step for the solution of (P1) is the reconstruction step, (P3). 2 � F u x − y � 2 X � 2 F + υ ′ 1 2 η � R ( x ) − ˆ (P3) min 2 + τ 2 η � Φ x � 1 . x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  69. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The second main step for the solution of (P1) is the reconstruction step, (P3). 2 � F u x − y � 2 X � 2 F + υ ′ 1 2 η � R ( x ) − ˆ (P3) min 2 + τ 2 η � Φ x � 1 . x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  70. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The second main step for the solution of (P1) is the reconstruction step, (P3). 2 � F u x − y � 2 X � 2 F + υ ′ 1 2 η � R ( x ) − ˆ (P3) min 2 + τ 2 η � Φ x � 1 . x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  71. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  72. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  73. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  74. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  75. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  76. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define patch to image operator ˆ R . j R T R ( ˆ ˆ �� j ˆ � X ) = x j ./ w . (P3) can be approximately rewritten as follows. 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 (P 3 ′ ) min � � + υ � Φ x � 1 . (6) 2 2 x 1 2 � F u x − y � 2 X � 2 F + υ ′ 2 η � R ( x ) − ˆ 2 + τ 2 η � Φ x � 1 . (P3) min x Tanc and Eksioglu - EUSIPCO 2015 Transform Learning MRI with Global Wavelet Regularization

  77. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  78. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  79. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  80. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  81. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  82. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  83. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction Define two functions: g ( x ) = 1 � F u x − y � 2 2 + τ ′ � x − ˆ R ( ˆ X ) � 2 � � 2 2 f ( x ) = υ � Φ x � 1 . (P 3 ′ ) min f ( x ) + g ( x ) . 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

  84. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The forward-backward splitting steps: z = x − γ ∇ g ( x ) . (7) (P3.1) (P3.2) x = x + µ ( prox γ f ( z ) − x ) . (8) ∇ g ( x ) = F H u ( F u x − y )+ τ ′ ( x − ˆ R ( ˆ X )) . F H u is the adjoint operator of F u , it realizes zero-filled reconstruction. prox γ f ( · ) 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

  85. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The forward-backward splitting steps: z = x − γ ∇ g ( x ) . (7) (P3.1) (P3.2) x = x + µ ( prox γ f ( z ) − x ) . (8) ∇ g ( x ) = F H u ( F u x − y )+ τ ′ ( x − ˆ R ( ˆ X )) . F H u is the adjoint operator of F u , it realizes zero-filled reconstruction. prox γ f ( · ) 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

  86. Introduction New Cost Transform Learning MRI GTLMRI: Denoising GTLMRI GTLMRI: Reconstruction Simulations and Conclusion GTLMRI: Overall Algorithm GTLMRI: Reconstruction The forward-backward splitting steps: z = x − γ ∇ g ( x ) . (7) (P3.1) (P3.2) x = x + µ ( prox γ f ( z ) − x ) . (8) ∇ g ( x ) = F H u ( F u x − y )+ τ ′ ( x − ˆ R ( ˆ X )) . F H u is the adjoint operator of F u , it realizes zero-filled reconstruction. prox γ f ( · ) 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

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