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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