A guaranteed convergence analysis for the projected fast iterative soft-thresholding algorithm in parallel MRI

Sparse sampling and parallel imaging techniques are two effective approaches to alleviate the lengthy magnetic resonance imaging (MRI) data acquisition problem. Promising data recoveries can be obtained from a few MRI samples with the help of sparse reconstruction models. To solve the optimization models, proper algorithms are indispensable. The pFISTA, a simple and efficient algorithm, has been successfully extended to parallel imaging. However, its convergence criterion is still an open question. Besides, the existing convergence criterion of single-coil pFISTA cannot be applied to the parallel imaging pFISTA, which, therefore, imposes confusions and difficulties on users about determining the only parameter - step size. In this work, we provide the guaranteed convergence analysis of the parallel imaging version pFISTA to solve the two well-known parallel imaging reconstruction models, SENSE and SPIRiT. Along with the convergence analysis, we provide recommended step size values for SENSE and SPIRiT reconstructions to obtain fast and promising reconstructions. Experiments on in vivo brain images demonstrate the validity of the convergence criterion.

[1]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[2]  Mathews Jacob,et al.  Accelerated Dynamic MRI Exploiting Sparsity and Low-Rank Structure: k-t SLR , 2011, IEEE Transactions on Medical Imaging.

[3]  Di Guo,et al.  Image reconstruction of compressed sensing MRI using graph-based redundant wavelet transform , 2016, Medical Image Anal..

[4]  Terry M Peters,et al.  Stationary wavelet transform for under-sampled MRI reconstruction. , 2014, Magnetic resonance imaging.

[5]  Klaas Paul Pruessmann,et al.  A Fast Wavelet-Based Reconstruction Method for Magnetic Resonance Imaging , 2011, IEEE Transactions on Medical Imaging.

[6]  Yoram Bresler,et al.  MR Image Reconstruction From Highly Undersampled k-Space Data by Dictionary Learning , 2011, IEEE Transactions on Medical Imaging.

[7]  Minh N. Do,et al.  Ieee Transactions on Image Processing the Contourlet Transform: an Efficient Directional Multiresolution Image Representation , 2022 .

[8]  Di Guo,et al.  Magnetic resonance image reconstruction from undersampled measurements using a patch-based nonlocal operator , 2014, Medical Image Anal..

[9]  X. Qu,et al.  Iterative thresholding compressed sensing MRI based on contourlet transform , 2010 .

[10]  Sungheon Kim,et al.  Golden‐angle radial sparse parallel MRI: Combination of compressed sensing, parallel imaging, and golden‐angle radial sampling for fast and flexible dynamic volumetric MRI , 2014, Magnetic resonance in medicine.

[11]  Charles A. Bouman,et al.  Plug-and-Play Methods for Magnetic Resonance Imaging: Using Denoisers for Image Recovery , 2019, IEEE Signal Processing Magazine.

[12]  Dong Liang,et al.  DIMENSION: Dynamic MR imaging with both k‐space and spatial prior knowledge obtained via multi‐supervised network training , 2018, NMR in biomedicine.

[13]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[14]  Jeffrey A. Fessler,et al.  Optimization Methods for Magnetic Resonance Image Reconstruction: Key Models and Optimization Algorithms , 2020, IEEE Signal Processing Magazine.

[15]  Zhong Chen,et al.  Undersampled MRI reconstruction with patch-based directional wavelets. , 2012, Magnetic resonance imaging.

[16]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[17]  Pierre Vandergheynst,et al.  Compressed Sensing and Redundant Dictionaries , 2007, IEEE Transactions on Information Theory.

[18]  Eero P. Simoncelli,et al.  Image quality assessment: from error visibility to structural similarity , 2004, IEEE Transactions on Image Processing.

[19]  I. Daubechies,et al.  An iterative thresholding algorithm for linear inverse problems with a sparsity constraint , 2003, math/0307152.

[20]  I. Yamada,et al.  Over-relaxation of the fast iterative shrinkage-thresholding algorithm with variable stepsize , 2011 .

[21]  David L Donoho,et al.  Compressed sensing , 2006, IEEE Transactions on Information Theory.

[22]  Daniel Rueckert,et al.  A Deep Cascade of Convolutional Neural Networks for Dynamic MR Image Reconstruction , 2017, IEEE Transactions on Medical Imaging.

[23]  H. H. Rachford,et al.  On the numerical solution of heat conduction problems in two and three space variables , 1956 .

[24]  Ning Jin,et al.  Fast implementation for compressive recovery of highly accelerated cardiac cine MRI using the balanced sparse model , 2017, Magnetic resonance in medicine.

[25]  Microsoft Word-pISTA-SENSE-ResNet-SPL_V2.2.docx , 2019 .

[26]  Junzhou Huang,et al.  Fast iteratively reweighted least squares algorithms for analysis‐based sparse reconstruction , 2018, Medical Image Anal..

[27]  J.-C. Pesquet,et al.  A Douglas–Rachford Splitting Approach to Nonsmooth Convex Variational Signal Recovery , 2007, IEEE Journal of Selected Topics in Signal Processing.

[28]  Jian Sun,et al.  Deep ADMM-Net for Compressive Sensing MRI , 2016, NIPS.

[29]  Zhong Chen,et al.  Balanced Sparse Model for Tight Frames in Compressed Sensing Magnetic Resonance Imaging , 2015, PloS one.

[30]  Zhong Chen,et al.  Compressed sensing MRI based on nonsubsampled contourlet transform , 2008, 2008 IEEE International Symposium on IT in Medicine and Education.

[31]  Leslie Ying,et al.  Joint image reconstruction and sensitivity estimation in SENSE (JSENSE) , 2007, Magnetic resonance in medicine.

[32]  Bernard Ghanem,et al.  ISTA-Net: Interpretable Optimization-Inspired Deep Network for Image Compressive Sensing , 2017, 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition.

[33]  S. Yun,et al.  An accelerated proximal gradient algorithm for nuclear norm regularized linear least squares problems , 2009 .

[34]  Shiqian Ma,et al.  An efficient algorithm for compressed MR imaging using total variation and wavelets , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[35]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[36]  Zhi-Pei Liang,et al.  Spatiotemporal Imaging with Partially Separable Functions , 2007, 2007 Joint Meeting of the 6th International Symposium on Noninvasive Functional Source Imaging of the Brain and Heart and the International Conference on Functional Biomedical Imaging.

[37]  Jong Chul Ye,et al.  Deep Convolutional Framelets: A General Deep Learning Framework for Inverse Problems , 2017, SIAM J. Imaging Sci..

[38]  Di Guo,et al.  Fast Multiclass Dictionaries Learning With Geometrical Directions in MRI Reconstruction , 2015, IEEE Transactions on Biomedical Engineering.

[39]  Liansheng Wang,et al.  Image reconstruction with low-rankness and self-consistency of k-space data in parallel MRI , 2019, Medical Image Anal..

[40]  Leslie Ying,et al.  Accelerating magnetic resonance imaging via deep learning , 2016, 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI).

[41]  Di Guo,et al.  Accelerated Nuclear Magnetic Resonance Spectroscopy with Deep Learning , 2019, Angewandte Chemie.

[42]  Yurii Nesterov,et al.  Smooth minimization of non-smooth functions , 2005, Math. Program..

[43]  J. Pauly,et al.  Accelerating parameter mapping with a locally low rank constraint , 2015, Magnetic resonance in medicine.

[44]  M. Lustig,et al.  SPIRiT: Iterative self‐consistent parallel imaging reconstruction from arbitrary k‐space , 2010, Magnetic resonance in medicine.

[45]  Daniel K Sodickson,et al.  Low‐rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components , 2015, Magnetic resonance in medicine.

[46]  Guang Yang,et al.  DAGAN: Deep De-Aliasing Generative Adversarial Networks for Fast Compressed Sensing MRI Reconstruction , 2018, IEEE Transactions on Medical Imaging.

[47]  Gabor T. Herman,et al.  Monotone FISTA With Variable Acceleration for Compressed Sensing Magnetic Resonance Imaging , 2019, IEEE Transactions on Computational Imaging.

[48]  Dong Liang,et al.  Learning Joint-Sparse Codes for Calibration-Free Parallel MR Imaging , 2018, IEEE Transactions on Medical Imaging.

[49]  Jeffrey A. Fessler,et al.  Parallel MR Image Reconstruction Using Augmented Lagrangian Methods , 2011, IEEE Transactions on Medical Imaging.

[50]  Dong Liang,et al.  DeepcomplexMRI: Exploiting deep residual network for fast parallel MR imaging with complex convolution. , 2020, Magnetic resonance imaging.

[51]  Junzhou Huang,et al.  An efficient algorithm for dynamic MRI using low‐rank and total variation regularizations , 2018, Medical Image Anal..

[52]  William W. Cohen,et al.  Power Iteration Clustering , 2010, ICML.

[53]  Xiaobo Qu,et al.  Review and Prospect: Deep Learning in Nuclear Magnetic Resonance Spectroscopy , 2020, Chemistry.

[54]  Michael Elad,et al.  The Cosparse Analysis Model and Algorithms , 2011, ArXiv.

[55]  Adrien B. Taylor,et al.  Exact Worst-Case Convergence Rates of the Proximal Gradient Method for Composite Convex Minimization , 2017, Journal of Optimization Theory and Applications.

[56]  Yves Meyer,et al.  Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures , 2001 .

[57]  Jens Frahm,et al.  Model-Based Iterative Reconstruction for Radial Fast Spin-Echo MRI , 2009, IEEE Transactions on Medical Imaging.

[58]  Yonina C. Eldar,et al.  Compressed Sensing with Coherent and Redundant Dictionaries , 2010, ArXiv.

[59]  Junzhou Huang,et al.  Efficient MR image reconstruction for compressed MR imaging , 2011, Medical Image Anal..

[60]  Yurii Nesterov,et al.  Gradient methods for minimizing composite functions , 2012, Mathematical Programming.

[61]  Bruce R. Rosen,et al.  Image reconstruction by domain-transform manifold learning , 2017, Nature.

[62]  Shanshan Wang,et al.  A comparative study of CNN-based super-resolution methods in MRI reconstruction and its beyond , 2020, Signal Process. Image Commun..

[63]  Jeffrey A. Fessler,et al.  Efficient Dynamic Parallel MRI Reconstruction for the Low-Rank Plus Sparse Model , 2019, IEEE Transactions on Computational Imaging.

[64]  Marc Teboulle,et al.  Fast Gradient-Based Algorithms for Constrained Total Variation Image Denoising and Deblurring Problems , 2009, IEEE Transactions on Image Processing.

[65]  Marc Teboulle,et al.  A Fast Iterative Shrinkage-Thresholding Algorithm for Linear Inverse Problems , 2009, SIAM J. Imaging Sci..

[66]  Dong Liang,et al.  Translational-invariant dictionaries for compressed sensing in magnetic resonance imaging , 2011, 2011 IEEE International Symposium on Biomedical Imaging: From Nano to Macro.

[67]  Di Guo,et al.  Magnetic resonance image reconstruction using trained geometric directions in 2D redundant wavelets domain and non-convex optimization. , 2013, Magnetic resonance imaging.

[68]  Y. Nesterov.  A method for unconstrained convex minimization problem with the rate of convergence o(1/k^2) , 1983 .

[69]  R. Otazo,et al.  Accelerating 3D‐T1ρ mapping of cartilage using compressed sensing with different sparse and low rank models , 2018, Magnetic resonance in medicine.

[70]  G. Easley,et al.  Sparse directional image representations using the discrete shearlet transform , 2008 .

[71]  Dong Liang,et al.  Iterative feature refinement for accurate undersampled MR image reconstruction , 2016, Physics in medicine and biology.

[72]  Kawin Setsompop,et al.  LORAKS makes better SENSE: Phase‐constrained partial fourier SENSE reconstruction without phase calibration , 2017, Magnetic resonance in medicine.

[73]  P. Boesiger,et al.  SENSE: Sensitivity encoding for fast MRI , 1999, Magnetic resonance in medicine.

[74]  Thomas Pock,et al.  Learning a variational network for reconstruction of accelerated MRI data , 2017, Magnetic resonance in medicine.