Regularized Image Reconstruction in Parallel MRI with MATLAB®

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Acknowledgements
Authors
1: Parallel MR Image Reconstruction
	1.1 Basics of MRI
		1.1.1 Basic Elements of an MR System
		1.1.2 Static Magnetic Field B0
		1.1.3 RF Magnetic Field B1
		1.1.4 RF Receiver
		1.1.5 Gradient Fields
		1.1.6 Slice Selection
		1.1.7 Generation of FID
		1.1.8 Imaging
	1.2 Nyquist Limit and Cartesian Sampling
	1.3 Pulse Sequencing and k-Space Filling
		1.3.1 Cartesian Imaging
		1.3.2 k-Space Features
		1.3.3 Non-Cartesian Imaging
			1.3.3.1 Data Acquisition and Pulse Sequencing
			1.3.3.2 Transformation from Non-Cartesian to Cartesian Data
	1.4 Parallel MRI
		1.4.1 Coil Combination
	1.5 MR Acceleration
		1.5.1 Acceleration Using Pulse Sequences
		1.5.2 Acceleration Using Sampling Schemes
		1.5.3 Under-Sampled Acquisition and Sampling Trajectories
		1.5.4 Artifacts Associated with Different Sampling Trajectories
	1.6 Parallel Imaging Reconstruction Algorithms
		1.6.1 Image-Based Reconstruction Methods
			1.6.1.1 SENSE
		1.6.2 k-Space Based Reconstruction Methods
			1.6.2.1 SMASH
			1.6.2.2 GRAPPA
			1.6.2.3 SPIRiT
			1.6.2.4 Regularization in Auto-calibrating Methods
		1.6.3 CS MRI
			1.6.3.1 CS-Based MR Image Reconstruction Model
			1.6.3.2 Sparsity-Promoting Regularization
		1.6.4 CS Recovery Using Low-Rank Priors
			1.6.4.1 Low-Rank CS-Based MR Image Reconstruction Model
	References
2: Regularization Techniques for MR Image Reconstruction
	2.1 Regularization of Inverse Problems
	2.2 MR Image Reconstruction as an Inverse Problem
	2.3 Well-Posed and Ill-Posed Problems
		2.3.1 Moore-Penrose Pseudo-Inverse
		2.3.2 Condition Number
		2.3.3 Picard’s Condition
	2.4 Types of Regularization Approaches
		2.4.1 Regularization by Reducing the Search Space
		2.4.2 Regularization by Penalization
	2.5 Regularization Approaches Using l2 Priors
		2.5.1 Tikhonov Regularization
		2.5.2 Conjugate Gradient Method
		2.5.3 Other Krylov Sub-space Methods
			2.5.3.1 Arnoldi Process
			2.5.3.2 Generalized Minimum Residual (GMRES) Method
			2.5.3.3 Conjugate Residual (CR) Algorithm
		2.5.4 Landweber Method
	2.6 Regularization Approaches Using l1 Priors
		2.6.1 Solution to l1-Regularized Problems
			2.6.1.1 Sub-gradient Methods
			2.6.1.2 Constrained Log-Barrier Method
			2.6.1.3 Unconstrained Approximations
	2.7 Linear Estimation in pMRI
		2.7.1 Regularization in GRAPPA-Based pMRI
			2.7.1.1 Tailored GRAPPA
			2.7.1.2 Discrepancy-Based Adaptive Regularization
			2.7.1.3 Penalized Coefficient Regularization
			2.7.1.4 Regularization in GRAPPA Using Virtual Coils
			2.7.1.5 Sparsity-Promoting Calibration
			2.7.1.6 KS-Based Calibration
	2.8 Regularization in Iterative Self-Consistent Parallel Imaging Reconstruction (SPIRiT)
	2.9 Regularization for Compressed Sensing MRI (CSMRI)
	Appendix
	References
3: Regularization Parameter Selection Methods in Parallel MR Image Reconstruction
	3.1 Regularization Parameter Selection
	3.2 Parameter Selection Strategies for Tikhonov Regularization
		3.2.1 Discrepancy Principle
		3.2.2 Generalized Discrepancy Principle (GDP)
		3.2.3 Unbiased Predictive Risk Estimator (UPRE)
		3.2.4 Stein’s Unbiased Risk Estimation (SURE)
		3.2.5 Bayesian Approach
		3.2.6 GCV
		3.2.7 Quasi-optimality Criterion
		3.2.8 L-Curve
	3.3 Parameter Selection Strategies for Truncated SVD (TSVD)
	3.4 Parameter Selection Strategies for Non-quadratic Regularization
		3.4.1 Parameter Selection for Wavelet Regularization
			3.4.1.1 VisuShrink
			3.4.1.2 SUREShrink
			3.4.1.3 NeighBlock
			3.4.1.4 SUREblock
			3.4.1.5 False Discovery Rate
			3.4.1.6 Bayes Factor Thresholding
			3.4.1.7 BayesShrink
			3.4.1.8 Ogden’s Methods
			3.4.1.9 Cross-validation
			3.4.1.10 Wavelet Thresholding
		3.4.2 Methods for Parameter Selection in Total Variation (TV) Regularization
			3.4.2.1 PDE Approach
			3.4.2.2 Duality-Based Approaches
			3.4.2.3 Prediction Methods
	References
4: Multi-filter Calibration for Auto-calibrating Parallel MRI
	4.1 Problems Associated with Single-Filter Calibration
	4.2 Effect of Noise in Generalized Autocalibrating Partially Parallel Acquisitions (GRAPPA) Calibration
	4.3 Monte Carlo Method for Prior Assessment of the Efficacy of Regularization
	4.4 Determination of Cross-over
		4.4.1 Perturbation of ACS Data for Determination of Cross-over
		4.4.2 First Order Update of Singular Values
		4.4.3 Application of GDP
		4.4.4 Determination of Cross-over
	4.5 Multi-filter Calibration Approaches
		4.5.1 MONKEES
		4.5.2 SV-GRAPPA
		4.5.3 Reconstruction Using FDR
			4.5.3.1 Implementation of FDR Reconstruction
	4.6 Effect of Noise Correlation
	Appendix
	References
5: Parameter Adaptation for Wavelet Regularization in Parallel MRI
	5.1 Image Representation Using Wavelet Basis
	5.2 Structure of Wavelet Coefficients
		5.2.1 Statistics of Wavelet Coefficients
	5.3 CS Using Wavelet Transform Coefficients
		5.3.1 Structured Sparsity Model
			5.3.1.1 Model-Based RIP
			5.3.1.2 Model-Based Signal Recovery
		5.3.2 Wavelet Sparsity Model
	5.4 Influence of Threshold on Speed of Convergence and Need for Iteration-Dependent Threshold Adaptation
		5.4.1 Selection of Initial Threshold
	5.5 Parallelism to the Generalized Discrepancy Principle (GDP)
	5.6 Adaptive Thresholded Landweber
		5.6.1 Level-Dependent Adaptive Thresholding
		5.6.2 Numerical Simulation of Wavelet Adaptive Shrinkage CS Reconstruction Problem
		5.6.3 Illustration Using Single-Channel MRI
		5.6.4 Application to pMRI
			5.6.4.1 Update Calculation Using Error Information from Combined Image (Method I)
			5.6.4.2 Update Calculation Using SoS of Channel-wise Errors (Method II)
			5.6.4.3 Update Calculation Using Covariance Matrix (Method III)
			5.6.4.4 Illustration Using In Vivo Data
			5.6.4.5 Illustration Using Synthetic Data
	Appendix
	References
6: Parameter Adaptation for Total Variation–Based Regularization in Parallel MRI
	6.1 Total Variation–Based Image Recovery
	6.2 Parameter Selection Using Continuation Strategies
	6.3 TV Iterative Shrinkage Based Reconstruction Model
		6.3.1 Derivative Shrinkage
		6.3.2 Selection of Initial Threshold
	6.4 Adaptive Derivative Shrinkage
	6.5 Algorithmic Implementation for Parallel MRI (pMRI)
	Appendix
	References
7: Combination of Parallel Magnetic Resonance Imaging and Compressed Sensing Using L1-SPIRiT
	7.1 Combination of Parallel Magnetic Resonance Imaging and Compressed Sensing
	7.2 L1-SPIRiT
		7.2.1 Reconstruction Steps for Non-Cartesian SPIRiT
	7.3 Computational Complexity in L1-SPIRiT
	7.4 Faster Non-Cartesian SPIRiT Using Augmented Lagrangian with Variable Splitting
		7.4.1 Regularized Non-Cartesian SPIRiT Using Split Bregman Technique
		7.4.2 Iterative Non-Cartesian SPIRiT Using ADMM
		7.4.3 Fast Iterative Cartesian SPIRiT Using Variable Splitting
	7.5 Challenges in the Implementation of L1-SPIRiT
		7.5.1 Effect of Incorrect Parameter Choice on Reconstruction Error
	7.6 Improved Calibration Framework for L1-SPIRiT
		7.6.1 Modification of Polynomial Mapping
		7.6.2 Regularization Parameter Choice
	7.7 Automatic Parameter Selection for L1-SPIRiT Using Monte Carlo SURE
	7.8 Continuation-Based Threshold Adaptation in L1-SPIRiT
		7.8.1 L1-SPIRiT Examples
	7.9 Sparsity and Low-Rank Enhanced SPIRiT (SLR-SPIRiT)
	References
8: Matrix Completion Methods
	8.1 Introduction
	8.2 Matrix Completion Problem
	8.3 Conditions Required for Accurate Recovery
		8.3.1 Matrix Completion under Noisy Condition
	8.4 Algorithms for Matrix Completion
		8.4.1 SVT Algorithm
		8.4.2 FPCA Algorithm
		8.4.3 Projected Landweber (PLW) Method
		8.4.4 Alternating Minimization Schemes
			8.4.4.1 Non-linear Alternating Least Squares Method
			8.4.4.2 ADMM with Nonnegative Factors
			8.4.4.3 ADMM for Matrix Completion without Factorization
	8.5 Methods for pMRI Acceleration Using Matrix Completion
		8.5.1 Simultaneous Auto-calibration and k-Space Estimation
		8.5.2 Low-Rank Modeling of Local k-Space Neighborhoods
		8.5.3 Annihilating Filter–Based Low-Rank Hankel Matrix Approach
	8.6 Non-convex Approaches for Structured Matrix Completion Solution for CS-MRI
		8.6.1 Solution Using IRLS Algorithm
		8.6.2 Solution Using Extension of Soft Thresholding
	8.7 Applications to Dynamic Imaging
		8.7.1 RPCA
		8.7.2 Solution Using ADMM
	References
MATLAB Codes
Index