Magnetic Resonance Image Reconstruction: Theory, Methods, and Applications

Front Cover
Magnetic Resonance Image Reconstruction
Copyright
Contents
Contributors
Editor Biographies
Introduction
	1 MRI reconstruction and its role in clinical practice
	2 Organization of the book
Part 1 Basics of MRI Reconstruction
	1 Brief Introduction to MRI Physics
		1.1 A brief history of MRI
		1.2 Nuclear magnetism
			1.2.1 Spin
			1.2.2 Net magnetization
			1.2.3 Magnetization dynamics
		1.3 NMR/MRI signal
			1.3.1 Signal creation and reception
				1.3.1.1 Radiofrequency pulses
				1.3.1.2 Signal detection
			1.3.2 Signal relaxation and decay
				1.3.2.1 Longitudinal relaxation
				1.3.2.2 Transverse relaxation
		1.4 Image formation
			1.4.1 Frequency encoding
			1.4.2 Phase encoding
			1.4.3 Slice selection
			1.4.4 Sequence diagram
			1.4.5 k-space formalism
			1.4.6 k-space trajectories
				1.4.6.1 Echo-planar imaging
				1.4.6.2 Non-Cartesian trajectories
			1.4.7 Pulse sequence types
				1.4.7.1 Spin echo
				1.4.7.2 Gradient echo
				1.4.7.3 Balanced steady-state free precession
		1.5 Components of an MRI scanner
			1.5.1 Magnet
			1.5.2 Gradient coils
			1.5.3 Radiofrequency coils
			1.5.4 Noise properties
		1.6 Summary
		References
		Suggested readings
	2 MRI Reconstruction as an Inverse Problem
		2.1 Inverse problems
		2.2 Discretization of the MR signal
		2.3 MR reconstruction as a linear inverse problem
		2.4 Solution of the MR reconstruction problem
		2.5 Regularizing the MR reconstruction problem
		2.6 Nonlinear inverse problems in MR
			2.6.1 Nonlinear parallel imaging
			2.6.2 Nonlinear motion estimation/correction
			2.6.3 Nonlinear parameter reconstruction
		2.7 Summary
		References
		Suggested readings
	3 Optimization Algorithms for MR Reconstruction
		3.1 Introduction
		3.2 Least squares reconstruction
		3.3 Model-based reconstruction
			3.3.1 Smooth optimization
			3.3.2 Nonsmooth optimization
			3.3.3 Stochastic gradient-based approaches
		3.4 Summary
		References
	4 Non-Cartesian MRI Reconstruction
		4.1 Introduction
		4.2 NFFT
		4.3 Gridding
		4.4 Iterative reconstruction
		4.5 Examples
		4.6 Spatial resolution and noise
		4.7 Extensions
		4.8 Summary
		References
	5 ``Early\'\' Constrained Reconstruction Methods
		5.1 Introduction
			5.1.1 Basic Fourier reconstruction
			5.1.2 Constrained reconstruction: historical perspective
		5.2 Support-constrained reconstruction
		5.3 Phase-constrained reconstruction
		5.4 Linear predictive reconstruction
		5.5 Rank-constrained reconstruction
		5.6 Sparsity-constrained reconstruction
		5.7 Reconstruction using side information
		5.8 Discussion
		5.9 Summary
		References
Part 2 Reconstruction of Undersampled MRI Data
	6 Parallel Imaging
		6.1 Introduction
		6.2 Fundamental techniques
		6.3 Advanced techniques
		6.4 3D volumetric parallel imaging
		6.5 Dynamic parallel imaging
		6.6 Artifacts in parallel imaging
		6.7 Summary
		References
		Suggested readings
	7 Simultaneous Multislice Reconstruction
		7.1 Introduction
		7.2 Basics of SMS encoding
		7.3 Reconstruction of SMS using parallel imaging concepts
			7.3.1 SENSE
			7.3.2 Extended FOV methods
				7.3.2.1 SENSE-GRAPPA
				7.3.2.2 RO-SENSE-GRAPPA
					Kernel calibration
			7.3.3 Slice-GRAPPA
			7.3.4 Split-Slice-GRAPPA
			7.3.5 SMS with phase-encoding undersampling
			7.3.6 Reconstruction of SMS for EPI
				7.3.6.1 Blipped-wideband and blipped-CAIPI encoding
				7.3.6.2 Slice-GRAPPA with dual polarity
				7.3.6.3 SENSE-model for EPI
		7.4 Calibration and reference scans
			7.4.1 Calibration and reference scans for EPI
		7.5 Reconstruction metrics
			7.5.1 Noise amplification
			7.5.2 Residual aliasing
			7.5.3 Qualitative effect of slice leakage
		7.6 Extensions of SMS
			7.6.1 SMS and 3D imaging
			7.6.2 Non-Cartesian SMS
		7.7 Applications of SMS
		7.8 Summary
		7.9 Exercise
			7.9.1 Content of tutorial
			7.9.2 Questions
		7.A Extended FOV methods for SMS
			7.A.1 PE-SENSE-GRAPPA
			7.A.2 Unbiased slice-GRAPPA
		References
	8 Sparse Reconstruction
		8.1 Introduction
		8.2 Compressed sensing theory: a brief overview
			8.2.1 Sparsity and incoherence: a first look
			8.2.2 Compressed sensing reconstruction
			8.2.3 Conditions for compressed sensing reconstruction
		8.3 Compressed sensing MRI
			8.3.1 Sparsifying transform and transform sparsity
			8.3.2 Incoherent data acquisition
			8.3.3 Image reconstruction
		8.4 Combination of compressed sensing MRI with parallel imaging
			8.4.1 Why compressed sensing + parallel imaging?
			8.4.2 Representative compressed sensing + parallel imaging methods
		8.5 Clinical applications of compressed sensing MRI
		8.6 Challenges of compressed sensing MRI
		8.7 Summary
		8.8 Tutorial
		Acknowledgments
		8.A Conditions for a unique solution in compressed sensing
		References
	9 Low-Rank Matrix and Tensor–Based Reconstruction
		9.1 Introduction
		9.2 Problem formulation
		9.3 Matrix-based approaches
			9.3.1 Global low-rank modeling
				9.3.1.1 Sampling
				9.3.1.2 Image reconstruction
					Explicit low-rank reconstruction
						Fixed-subspace reconstruction
						Alternating reconstruction
					Implicit low-rank reconstruction
			9.3.2 Local low-rank modeling
			9.3.3 Low-rank and sparse modeling
			9.3.4 Low-rank plus sparse modeling
			9.3.5 Multiscale low-rank modeling
		9.4 Tensor-based approaches
			9.4.1 Tensor definitions
				9.4.1.1 CP decomposition
				9.4.1.2 Tucker decomposition
				9.4.1.3 Tensor rank surrogates
			9.4.2 Reinterpreting dynamic images as tensors
				9.4.2.1 Coil modeling
				9.4.2.2 Patch similarity modeling
				9.4.2.3 Spatial separability
			9.4.3 Multidynamic tensors
				9.4.3.1 Tensor-based compressed sensing
				9.4.3.2 Multidynamic low-rank tensor modeling
					Explicit multidynamic low-rank tensor reconstruction
					Implicit multidynamic low-rank tensor reconstruction
					Additional multidynamic LRT models
		9.5 Summary
		References
	10 Dictionary, Structured Low-Rank, and Manifold Learning-Based Reconstruction
		10.1 Introduction
		10.2 Background
			10.2.1 Acquisition scheme
			10.2.2 Manifold models of signals
			10.2.3 Capitalization of redundancy using structured matrices
			10.2.4 Efficient matrix representation in terms of factors
		10.3 Dictionary learning and blind compressed sensing
			10.3.1 Subspace selection for each signal of interest using sparse representation
			10.3.2 Dictionary pre-learning
				10.3.2.1 Dictionary pre-learning, applied to static MRI
			10.3.3 Blind compressed sensing (BCS)
				10.3.3.1 Application of BCS to dynamic MRI
				10.3.3.2 Application of BCS to static imaging
		10.4 Structured low-rank methods
			10.4.1 Low-rank structure of patch matrices in k-space
				10.4.1.1 Low-rank relationships in multichannel MRI
				10.4.1.2 Low-rank structure resulting from finite support and smoothly varying image phase
				10.4.1.3 Low-rank structure resulting from continuous domain sparsity
				10.4.1.4 Low-rank structure of piecewise smooth images
				10.4.1.5 Low-rank relations in parameter mapping
			10.4.2 Algorithms for k-space patch low-rank methods
			10.4.3 Iterative reweighted least square (IRLS) algorithm
			10.4.4 Algorithms that rely on calibration data
		10.5 Smooth manifold models
			10.5.1 Analysis manifold methods
				10.5.1.1 Relationship to factor models and binning based approaches
				10.5.1.2 Estimation of manifold Laplacian
				10.5.1.3 Image recovery assuming smooth patch manifold
			10.5.2 Application to dynamic MRI
		10.6 Software
		10.7 Summary
		References
	11 Machine Learning for MRI Reconstruction
		11.1 Introduction
		11.2 Organization of this chapter
		11.3 Machine learning definitions
			11.3.1 Learning models
			11.3.2 Types of learning
			11.3.3 Cost function, optimization and backpropagation
			11.3.4 Training, validation, and testing
			11.3.5 Database splitting
		11.4 Task definition for MR reconstruction
			11.4.1 Image enhancement
			11.4.2 Direct k-space to image mapping
			11.4.3 Physics-based reconstruction
		11.5 Core concepts: layers
			11.5.1 Convolution layer
				11.5.1.1 Dilated convolution
				11.5.1.2 Separable convolution
				11.5.1.3 Transposed convolution
			11.5.2 Normalization layer
			11.5.3 Activation layer
			11.5.4 Fully connected layer
			11.5.5 Down-sampling layer
			11.5.6 Up-sampling layer
			11.5.7 Dropout layer
			11.5.8 Merging layers
			11.5.9 Recursive layer
			11.5.10 Building blocks
			11.5.11 Data consistency layers
		11.6 Network architectures for MRI reconstruction
		11.7 How to build an ML model for MR reconstruction
			11.7.1 Checklist to build an ML model
			11.7.2 Database
			11.7.3 Database pipeline
			11.7.4 Frameworks
		11.8 Summary
		11.9 Further resources and tutorials
		11.10 Exercises
			11.10.1 Hands-on examples
		11.A ML-specific notation
		11.B Complex calculus
		11.C Trainable parameters of separable convolutions
		References
Part 3 Reconstruction Methods for Nonlinear Forward Models in MRI
	12 Imaging in the Presence of Magnetic Field Inhomogeneities
		12.1 Introduction
		12.2 Disruptions to the homogeneity of the magnetic field
		12.3 Field inhomogeneity effects on imaging
			12.3.1 Three types of effects disrupting the image and its information
			12.3.2 Field inhomogeneity and the signal equation
				12.3.2.1 Other basis expansions enable the modeling of additional artifacts
			12.3.3 Field inhomogeneity mitigation methods
		12.4 Image distortions and correction approaches
			12.4.1 Distortions depend on trajectory and sample timing
			12.4.2 Image correction: image warping approaches
			12.4.3 Image correction: conjugate phase
			12.4.4 Image correction: inverse problem approach
				12.4.4.1 Computational considerations
			12.4.5 Comparing performance of image correction approaches
		12.5 Phase and signal dephasing correction approaches
			12.5.1 Image reconstruction based approaches for within voxel dephasing
		12.6 K-space trajectory distortions
		12.7 Measuring the field map
		12.8 Summary
		References
	13 Motion-Corrected Reconstruction
		13.1 Introduction
		13.2 Theory
			13.2.1 Reconstruction with known motion: the particular case of translational motion
			13.2.2 Reconstruction with known motion: the general case
				13.2.2.1 Motion operators
				13.2.2.2 Forward acquisition model including motion operators
				13.2.2.3 Solving the inverse problem
					Conditioning of the system
			13.2.3 Joint reconstruction of image and motion
				13.2.3.1 Propagation of motion errors
				13.2.3.2 Alternating Gauss–Newton optimization
				13.2.3.3 Case of translational motion
				13.2.3.4 Case of a temporally constrained, nonrigid motion model
		13.3 Methods
			13.3.1 Strategies for motion sensing
				13.3.1.1 External sensor measurements
				13.3.1.2 Extracting motion from MR data
					Separate navigation signals
					Separate image navigation for 2D/3D motion estimation
					Self-navigation signals
					Alternative MR navigation data
			13.3.2 Image registration
			13.3.3 Motion models
			13.3.4 Optimal k-space sampling for motion correction
			13.3.5 Motion correction to improve dynamic MRI
		13.4 Clinical application examples
			13.4.1 Brain
			13.4.2 Cardiovascular
			13.4.3 Body imaging (other than brain and heart)
		13.5 Current challenges and future directions
		13.6 Summary
		13.7 Practical tutorial
		References
	14 Chemical Shift Encoding-Based Water-Fat Separation
		14.1 Introduction
		14.2 Theory on chemical species separation
			14.2.1 The chemical shift property
			14.2.2 The chemical shift of fat
			14.2.3 Signal model for water–fat separation
		14.3 Solving the water–fat separation problem
			14.3.1 Parameter estimation in water–fat separation
			14.3.2 The field-map estimation problem
			14.3.3 Noise performance analysis
		14.4 Water–fat separation in non-Cartesian imaging
			14.4.1 Water–fat shift artifact
			14.4.2 Fat blurring in non-Cartesian acquisitions
			14.4.3 k-space-based water–fat separation
		14.5 Confounding factors in quantitative water–fat imaging
			14.5.1 Correction of hardware imperfections: gradient delays
			14.5.2 Correction of concomitant gradients
			14.5.3 Proton density fat-fraction determination
		14.6 Current challenges and future directions
		14.7 Summary
		14.8 Further reading
		References
	15 Model-Based Parametric Mapping Reconstruction
		15.1 Introduction
		15.2 MR mapping sequences
		15.3 Image-based mapping
		15.4 Reconstruction-based mapping
			15.4.1 Model-based acceleration of parameter mapping (MAP)
			15.4.2 Model-based optimization
		15.5 Clinical applications
		15.6 Current challenges and future directions
		15.7 Summary
		15.8 Tutorial
			15.8.1 Image-based T1 mapping
				15.8.1.1 Problem description
				15.8.1.2 Provided material
				15.8.1.3 Questions
			15.8.2 Magnetic resonance fingerprinting
				15.8.2.1 Problem description
				15.8.2.2 Provided material
				15.8.2.3 Questions
		Acknowledgment
		References
	16 Quantitative Susceptibility-Mapping Reconstruction
		16.1 Introduction
		16.2 GRE data acquisition
		16.3 Phase pre-processing
		16.4 Dipole inversion
			16.4.1 COSMOS
			16.4.2 K-space reconstruction with closed-form solution
			16.4.3 Iterative reconstructions in image space
		16.5 Recent advances: single-step QSM and deep-learning-based QSM
		16.6 Summary and outlook
		16.A Tutorials
			16.A.1 Phase pre-processing
				16.A.1.1 Provided materials
				16.A.1.2 Exercises
			16.A.2 Dipole inversion
				16.A.2.1 Provided materials
				16.A.2.2 Exercises
			16.A.3 Total variation regularized single-step QSM (single-step TV)
				16.A.3.1 Provided materials
				16.A.3.2 Problem description
			16.A.4 Exercises
		References
A Linear Algebra Primer
	A.1 Vector spaces
		A.1.1 Linear independence
		A.1.2 Span
		A.1.3 Basis
		A.1.4 Normed space
		A.1.5 Inner product space
	A.2 Matrix theory
		A.2.1 Types of matrices
		A.2.2 Matrices with special structures
		A.2.3 Special matrix products
		A.2.4 Matrix decompositions
		A.2.5 Matrix norms
	A.3 Tensors
		A.3.1 Tensor properties
		A.3.2 Tensor products
		A.3.3 Tensor ranks
		A.3.4 Tensor decompositions
Index
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