A practical and self-contained guide to the principles, techniques, models and tools of imaging spectroscopy. Bringing together material from essential physics and digital signal processing, it covers key topics such as sensor design and calibration, atmospheric inversion and model techniques, and processing and exploitation algorithms. Readers will learn how to apply the main algorithms to practical problems, how to choose the best algorithm for a particular application, and how to process and interpret hyperspectral imaging data. A wealth of additional materials accompany the book online, including example projects and data for students, and problem solutions and viewgraphs for instructors. This is an essential text for senior undergraduate and graduate students looking to learn the fundamentals of imaging spectroscopy, and an invaluable reference for scientists and engineers working in the field.

This book is about modern approaches to magnetic resonance imaging (MRI) reconstruction. In the last decade, MRI has benefitted immensely from advances in applied mathematics and signal processing. Leveraging these techniques, MRI scans are now being performed two to four times faster than before. In this book, we learn how these techniques have been used in the recent past to accelerate MRI scans.
During my PhD, I worked on a few different areas of MRI reconstruction – static MRI, dynamic MRI, parallel MRI (static and dynamic) and quantitative MRI. After I relocated to India, Manish Chaudhury commissioning editor at Cambridge University Press, inspired me to write a book and I was eager to write about signal processing techniques in MRI. It took me about one and half years to complete this volume.
When I started working on MRI reconstruction, I felt that there is a gap between the practitioners and the theoreticians. On one side, there were researchers in signal processing and applied maths who were interested in theoretical proofs and algorithms. On the other, there were the MRI physicists and engineers who had lots of interesting problems that were waiting to be solved. Since then, many researchers have worked very hard to reduce this gap. The concerted effort of so many researchers is finally bearing fruit; in the past few ISMRMs, MRI scanner manufacturers showed interest in adopting these advanced signal processing techniques for image reconstruction.
In this book, I have made every effort to incorporate interesting studies on MRI reconstruction, but I may have missed out a few unintentionally. Thus, this book does not claim to be an encyclopaedic review on the subject of signal processing techniques in MRI reconstruction.
The targeted audience of the book are signal processing engineers who want to learn about MRI problems and MRI physicists who want to know how signal processing is benefitting MRI. The book can also be perused by doctors who have a background in mathematics. I do not presume a reader who has an advanced background in mathematics. But the reader is expected to have some undergraduate training in linear algebra, probability and convex optimization. Otherwise, the book may not be easy to follow.
The book starts with an introduction on all the mathematical techniques one needs to know to understand the subsequent chapters. The emphasis of the first chapter is on algorithms. There are no proofs; rather, the reader is walked through the essence of the theoretical results based on mathematical intuitions. The second chapter is on single-channel static MRI reconstruction. Clinically, this is perhaps the most widely used modality. The third chapter talks about multi-coil parallel MRI. This is a very interesting topic; it is the perfect example of how signal processing (mathematical)- based acceleration techniques can be combined with hardware (physics)-based methods to reduce scan times. The fourth chapter is on dynamic MRI reconstruction. In the fifth chapter, we digress from the main theme of the book; we discuss how signal processing techniques have benefited other areas in biomedical engineering. The final chapter is a short one. It confines itself to some open problems in signal processing-based MRI reconstruction. Although each chapter is fairly independent, I advise the reader to go through them sequentially.

List of Figures vii

List of Tables xi

Foreword xiii

Preface xiv

Acknowledgements xvi

Color Plates xvii

1. Mathematical Techniques

1.1 Compressed Sensing 3

      1.1.1 Sparse Recovery 3

      1.1.2 Group-sparse Recovery 7

      1.1.3 Row-sparse Multiple Measurement Vector Recovery 8

      1.1.4 Synthesis and Analysis Priors 9

1.2 Low-rank Matrix Recovery 10

1.3 Kalman Filter 13

1.4 Algorithms 15

      1.4.1 Noiseless Scenario 15

      1.4.2 Noisy Scenario 21

1.5 Split Bregman Techniques 30

      1.5.1 Multiple Penalty Problems 34

1.6 Conclusion 37

Appendix: Greedy Algorithms 38

References 44

2. Single Channel Static MR Image Reconstruction

2.1 Single Echo MRI Reconstruction 50

      2.1.1 Sparsity 52

      2.1.2 Incoherence 55

      2.1.3 Reconstruction Algorithms 58

      2.1.4 Engineering the Measurement Operator 62

      2.1.5 Structured Sparsity 65

2.2 Multi-echo MRI 72

      2.2.1 Physics of MR Image Contrast 73

      2.2.2 Group-Sparse Reconstruction of Multi-echo MRI 75

Appendix: Mixed Prior Optimization 81

References 84

3. Multi-Coil Parallel MRI Reconstruction

3.1 Frequency Domain Methods 87

      3.1.1 GRAPPA 88

      3.1.2 Regularized GRAPPA 89

      3.1.3 Iterative GRAPPA 90

      3.1.4 Kernel GRAPPA 92

      3.1.5 Extensions to GRAPPA 94

3.2 Image Domain Methods 98

      3.2.1 SENSitivity Encoding 99

      3.2.2 Regularized SENSE 100

      3.2.3 CS SENSE 103

      3.2.4 Iterative SENSE 104

3.3 Calibration-Free Reconstruction 107

      3.3.1 Calibration-Less Multi-coil MRI 109

3.4 Conclusion 116

References 117

4. Dynamic MRI Reconstruction

4.1 Offline Dynamic MRI Reconstruction 121

      4.1.1 Compressed Sensing-Based Reconstruction Techniques 121

      4.1.2 Low-Rank Methods in Dynamic MRI Reconstruction 125

      4.1.3 Combined Low-Rank and Sparsity-Based Techniques 127

      4.1.4 Sparse + Low-Rank Reconstruction 137

4.2 Online Reconstruction 140

      4.2.1 Compressed Sensing-Based Techniques 140

      4.2.2 Kalman Filter-Based Techniques 146

      4.2.3 Hybrid Methods 149

4.3 Dynamic Parallel MRI 153

      4.3.1 Image Domain Methods 153

4.4 Conclusion 156

References 157

5. Applications in Other Areas

5.1 Computer Tomography 160

      5.1.1 Compressed Sensing in Static CT Reconstruction 162

      5.1.2 Compressed Sensing in Dynamic CT 167

5.2 Diffusion Tensor Imaging 172

      5.2.1 Distributed Compressed Sensing 173

      5.2.2 Learned Dictionary Approach 174

      5.2.3 Direct Diffusivity Estimation 175

5.3 Compressed Sensing in EEG Reconstruction 177

      5.3.1 Improving Transmission Energy Efficiency by Compressed Sensing 177

      5.3.2 Improving Sensing Energy Efficiency 184

5.4 Conclusion 189

References 190

6. Some Open Problems

6.1 Interactive Sampling 194

      6.1.1 Recursive Least Squares 197

      6.1.2 Recursive l1-Filtering 198

      6.1.3 Discussion 201

6.2 Quantitative MRI 201

      6.2.1 Non-linear Compressed Sensing Recovery 202

6.3 Parallelizing MRI Reconstruction Algorithms 204

References 205

Index 207

About the author 208

Dimitris G. Manolakis is a senior member of technical staff at the Lincoln Laboratory, Massachusetts Institute of Technology. He is the co-author of Applied Digital Signal Processing (Cambridge, 2011), and has taught at various institutions including Northeastern University, Boston, Boston College, Massachusetts, and Worcester Polytechnic Institute, Massachusetts. He is an IEEE Fellow, and in 2013 he received the IEEE Signal Processing Society Education Award.

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