Magnetic Resonance Image Reconstruction: Theory, Methods, and Applications

Description

Magnetic Resonance Image Reconstruction: Theory, Methods and Applications presents the fundamental concepts of MR image reconstruction, including its formulation as an inverse problem, as well as the most common models and optimization methods for reconstructing MR images. The book discusses approaches for specific applications such as non-Cartesian imaging, under sampled reconstruction, motion correction, dynamic imaging and quantitative MRI. This unique resource is suitable for physicists, engineers, technologists and clinicians with an interest in medical image reconstruction and MRI. 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 Back Cover