Contents

• Abstract
• Résumé
• List of notations
• Chapter 1  Introduction
  • 1.1  Main Contributions
  • 1.2  Organization of the Thesis
• Chapter 2  MRI Principles
  • 2.1  Nuclear Magnetic Resonance
    • 2.1.1  Spins and Magnetization
    • 2.1.2  Motion Law for the Magnetization
    • 2.1.3  Excitation Pulse and Resonance
    • 2.1.4  Relaxation
    • 2.1.5  Pulses
  • 2.2  Detection
    • 2.2.1  Original Signal
    • 2.2.2  Physics: Coil Sensitivity
    • 2.2.3  Demodulation
  • 2.3  Gradients of Field for Imaging
    • 2.3.1  Spin Density Profile
    • 2.3.2  Phase Encoding for Imaging
    • 2.3.3  Slice Selection
    • 2.3.4  Sampling Strategies
  • 2.4  Imaging Modalities
    • 2.4.1  Basic Contrasts
    • 2.4.2  Parallel MRI
• Chapter 3  Reconstruction Methods
  • 3.1  Model for MRI
    • 3.1.1  Physics
    • 3.1.2  Model for the Original Data
    • 3.1.3  Matrix Representation of the Model
  • 3.2  Linear Solutions
    • 3.2.1  Pseudoinverse
    • 3.2.2  Quadratic Regularization
    • 3.2.3  Maximum a posteriori
    • 3.2.4  Linear Minimum Mean Squared Error estimator
    • 3.2.5  Connections
  • 3.3  Non-Quadratic Regularizations
    • 3.3.1  Inverse Problem Formalism
    • 3.3.2  Total Variation
  • 3.4  Algorithms
    • 3.4.1  Matrix-Vector Multiplications
    • 3.4.2  Conjugate Gradient
    • 3.4.3  Iteratively Reweighted Least-Squares
    • 3.4.4  Iterative Shrinkage/Thresholding Algorithm
• Chapter 4  Simulation
  • 4.1  Motivations
  • 4.2  Modeling
    • 4.2.1  Parallel MRI
    • 4.2.2  Analytical Phantom
    • 4.2.3  Sensitivity Models
  • 4.3  Analytical MRI Measurements
    • 4.3.1  Overview of Analytical Fourier Computations
    • 4.3.2  Elliptical Regions
    • 4.3.3  Piecewise-Quadratic Contours
  • 4.4  Experiments
    • 4.4.1  Implementation Details
    • 4.4.2  Validation of the Implementation
    • 4.4.3  Applications
  • 4.5  Summary
• Chapter 5  Efficient Wavelet-Based Reconstruction
  • 5.1  Motivations
  • 5.2  Wavelet Regularization Algorithms
    • 5.2.1  ISTA with weighted norms
    • 5.2.2  Subband Adaptive ISTA
    • 5.2.3  Best of Two Worlds: Fast Weighted ISTA
    • 5.2.4  Random Shifting
  • 5.3  Experiments
    • 5.3.1  Implementation Details
    • 5.3.2  Spiral MRI Reconstruction
    • 5.3.3  Results
    • 5.3.4  SENSE MRI Reconstruction
  • 5.4  Summary
• Chapter 6  Conclusion
  • 6.1  Summary of Results
  • 6.2  Outlook
• Appendix A  Simulation
  • A.1  Proof of Theorem 1
  • A.2  Characterization and Computations of a Family of 1-D Integrals
  • A.3  Proof of Proposition 2
• Appendix B  Efficient Wavelet-Based Reconstruction
  • B.1  Proof of Proposition 1
  • B.2  Proof of Proposition 2