VS-Net: Variable Splitting Network for Accelerated Parallel MRI Reconstruction Jinming Duan † 1 , 2 , Jo Schlemper † 2 , 3 , Chen Qin 2 , Cheng Ouyang 2 , Wenjia Bai 2 , Carlo Biffi 2 , Ghalib Bello 2 , Ben Statton 2 , Declan P O’Regan 2 , Daniel Rueckert 2 1 School of Computer Science, University of Birmingham, UK 2 Imperial College London, UK 3 Hyperfine Research, CT, USA † contributed equally. 1/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. 2/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. ◮ Compressed sensing and parallel MRI are often used to speed up 2/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. ◮ Compressed sensing and parallel MRI are often used to speed up ◮ Parallel MRI is common in clinical practice currently 2/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. ◮ Compressed sensing and parallel MRI are often used to speed up ◮ Parallel MRI is common in clinical practice currently ◮ Deep learning based approaches [Schlemper, et al, Hammernik, et al, Yan, et al] have shown great promise 2/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. ◮ Compressed sensing and parallel MRI are often used to speed up ◮ Parallel MRI is common in clinical practice currently ◮ Deep learning based approaches [Schlemper, et al, Hammernik, et al, Yan, et al] have shown great promise ◮ Combing parallel MRI, compressed sensing and deep learning is the aim 2/ 14
Motivation: ◮ Slow MR imaging process leads to low patient throughput, patient’s discomfort, image artifacts, etc. ◮ Compressed sensing and parallel MRI are often used to speed up ◮ Parallel MRI is common in clinical practice currently ◮ Deep learning based approaches [Schlemper, et al, Hammernik, et al, Yan, et al] have shown great promise ◮ Combing parallel MRI, compressed sensing and deep learning is the aim 2/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter ◮ y i ∈ C M ( M < N ) is undersampled k -space of the i th coil ( n c coils) 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter ◮ y i ∈ C M ( M < N ) is undersampled k -space of the i th coil ( n c coils) ◮ D ∈ R M × N is sampling matrix that zeros out entries not acquired 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter ◮ y i ∈ C M ( M < N ) is undersampled k -space of the i th coil ( n c coils) ◮ D ∈ R M × N is sampling matrix that zeros out entries not acquired ◮ F ∈ C N × N is the Fourier transform matrix 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter ◮ y i ∈ C M ( M < N ) is undersampled k -space of the i th coil ( n c coils) ◮ D ∈ R M × N is sampling matrix that zeros out entries not acquired ◮ F ∈ C N × N is the Fourier transform matrix ◮ S i ∈ C N × N is the i th coil sensitivity, precomputed from fully sampled k -space center using E-SPIRiT [Uecker, et al] 3/ 14
General CS parallel MRI model: Technically, one can reconstruct a clean image m ∈ C N by minimizing n c λ � �DF S i m − y i � 2 min 2 + R ( m ) , (1) 2 m i =1 where ◮ λ is a smooth parameter ◮ y i ∈ C M ( M < N ) is undersampled k -space of the i th coil ( n c coils) ◮ D ∈ R M × N is sampling matrix that zeros out entries not acquired ◮ F ∈ C N × N is the Fourier transform matrix ◮ S i ∈ C N × N is the i th coil sensitivity, precomputed from fully sampled k -space center using E-SPIRiT [Uecker, et al] ◮ R ( m ) is a regularization term, e.g. TV, TGV, wavelet L1, etc. 3/ 14
Variable splitting optimization: We introduce splitting variables u ∈ C N and { x i ∈ C N } n c i =1 , converting (1) into n c λ � �DF x i − y i � 2 min 2 + R ( u ) s . t . m = u , S i m = x i , ∀ i ∈ { 1 , 2 , ..., n c } . 2 m , u , x i i =1 4/ 14
Variable splitting optimization: We introduce splitting variables u ∈ C N and { x i ∈ C N } n c i =1 , converting (1) into n c λ � �DF x i − y i � 2 min 2 + R ( u ) s . t . m = u , S i m = x i , ∀ i ∈ { 1 , 2 , ..., n c } . 2 m , u , x i i =1 We add constraints back via penalty function method and minimize n c n c λ 2 + R ( u ) + α 2 + β � �DF x i − y i � 2 � � x i − S i m � 2 2 � u − m � 2 min (2) 2 , 2 2 m , u , x i i =1 i =1 4/ 14
Variable splitting optimization: We introduce splitting variables u ∈ C N and { x i ∈ C N } n c i =1 , converting (1) into n c λ � �DF x i − y i � 2 min 2 + R ( u ) s . t . m = u , S i m = x i , ∀ i ∈ { 1 , 2 , ..., n c } . 2 m , u , x i i =1 We add constraints back via penalty function method and minimize n c n c λ 2 + R ( u ) + α 2 + β � �DF x i − y i � 2 � � x i − S i m � 2 2 � u − m � 2 min (2) 2 , 2 2 m , u , x i i =1 i =1 To minimize (2), we alternatively optimize m , u and x i w.r.t. the three subproblems: u k +1 = arg min 2 � u − m k � 2 β 2 + R ( u ) u x k +1 λ � n c � n c i =1 �DF x i − y i � 2 2 + α i =1 � x i − S i m k � 2 = arg min , (3) 2 i 2 x i m k +1 = arg min 2 � u k +1 − m � 2 � n c i =1 � x k +1 2 + β α − S i m � 2 2 i 2 m 4/ 14
Variable splitting optimization: We introduce splitting variables u ∈ C N and { x i ∈ C N } n c i =1 , converting (1) into n c λ � �DF x i − y i � 2 min 2 + R ( u ) s . t . m = u , S i m = x i , ∀ i ∈ { 1 , 2 , ..., n c } . 2 m , u , x i i =1 We add constraints back via penalty function method and minimize n c n c λ 2 + R ( u ) + α 2 + β � �DF x i − y i � 2 � � x i − S i m � 2 2 � u − m � 2 min (2) 2 , 2 2 m , u , x i i =1 i =1 To minimize (2), we alternatively optimize m , u and x i w.r.t. the three subproblems: u k +1 = arg min 2 � u − m k � 2 β 2 + R ( u ) u x k +1 λ � n c � n c i =1 �DF x i − y i � 2 2 + α i =1 � x i − S i m k � 2 = arg min , (3) 2 i 2 x i m k +1 = arg min 2 � u k +1 − m � 2 � n c i =1 � x k +1 2 + β α − S i m � 2 2 i 2 m Here k ∈ { 1 , ..., n it } denotes the k th iteration. α and β are introduced penalty weights. 4/ 14
Variable splitting optimization: and m k +1 using An optimal solution ( m ∗ ) may be found by iterating over u k +1 , x k +1 i u k +1 = denoiser ( m k ) = F − 1 (( λ D T D + α I ) − 1 ( α F S i m k + λ D T y i )) x k +1 . (4) ∀ i ∈ { 1 , 2 , ..., n c } i m k +1 = ( β I + α � n c i S i ) − 1 ( β u k +1 + α � n c i x k +1 i =1 S H i =1 S H ) i 5/ 14
Variable splitting optimization: and m k +1 using An optimal solution ( m ∗ ) may be found by iterating over u k +1 , x k +1 i u k +1 = denoiser ( m k ) = F − 1 (( λ D T D + α I ) − 1 ( α F S i m k + λ D T y i )) x k +1 . (4) ∀ i ∈ { 1 , 2 , ..., n c } i m k +1 = ( β I + α � n c i S i ) − 1 ( β u k +1 + α � n c i x k +1 i =1 S H i =1 S H ) i ◮ 1st Eq. is an image denoising solver (ie. denoiser ) 5/ 14
Variable splitting optimization: and m k +1 using An optimal solution ( m ∗ ) may be found by iterating over u k +1 , x k +1 i u k +1 = denoiser ( m k ) = F − 1 (( λ D T D + α I ) − 1 ( α F S i m k + λ D T y i )) x k +1 . (4) ∀ i ∈ { 1 , 2 , ..., n c } i m k +1 = ( β I + α � n c i S i ) − 1 ( β u k +1 + α � n c i x k +1 i =1 S H i =1 S H ) i ◮ 1st Eq. is an image denoising solver (ie. denoiser ) ◮ 2nd Eq. is a closed-form , point-wise and coil-wise data consistency term 5/ 14
Variable splitting optimization: and m k +1 using An optimal solution ( m ∗ ) may be found by iterating over u k +1 , x k +1 i u k +1 = denoiser ( m k ) = F − 1 (( λ D T D + α I ) − 1 ( α F S i m k + λ D T y i )) x k +1 . (4) ∀ i ∈ { 1 , 2 , ..., n c } i m k +1 = ( β I + α � n c i S i ) − 1 ( β u k +1 + α � n c i x k +1 i =1 S H i =1 S H ) i ◮ 1st Eq. is an image denoising solver (ie. denoiser ) ◮ 2nd Eq. is a closed-form , point-wise and coil-wise data consistency term ◮ 3rd Eq. is a closed-form , point-wise weighted average 5/ 14
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