Iterative methods for image reconstruction

Presenter: Jeffrey Fessler

Part 1: Introduction to iterative image reconstruction methods

Part 1 of this tutorial will give a general introduction to the field of iterative image reconstruction. This field has become increasingly important recently. In particular, an important milestone in this field took place in the late 1990's: the commercial release of 2D and 3D statistical image reconstruction methods for PET and SPECT systems. These methods have now been adopted for routine use in clinical PET and SPECT imaging. As computer speeds continue to improve, there is also increasing interest in iterative reconstruction methods for CT and MRI. This tutorial will provide an orderly overview of the potpourri of iterative methods for image reconstruction, emphasizing the fundamental issues that one must consider when choosing between different reconstruction approaches. The focus will be on models, cost functions, and algorithms. Examples will be drawn primarily from PET, SPECT, and CT.

Part 2: Advanced image reconstruction methods for MR

Part 2 of this tutorial will describe advanced methods for reconstructing magnetic resonance (MR) images from k-space data. The presentation will assume the audience is familiar with general iterative image reconstruction principles at the level of Part 1 of this two-part tutorial, and will focus specifically on MR applications. The conventional image reconstruction method for MRI is simply an inverse FFT. This tutorial will cover MR applications where an ordinary FFT is insufficient, including nonuniformly sample k-space data, applications such as fMRI with field inhomogeneity effects, partial k-space techniques, and sensitivity encoded imaging.

• Part 1 Outline (tentative):
1. Introduction
• Overview
• Mathematical statement of the reconstruction problem
2. The Statistical Framework
• Image parameterization
• Bases
• System physical modeling
• detector response
• projector/backprojectors
• Statistical modeling of measurements
• Objective functions
• Contrast with "algebraic" methods
• Bayesian estimation: Maximum a posteriori (MAP) methods
• Data-fit terms
• likelihood, quadratic, robust
• Regularization
• none
• separable
• quadratic
• convex
• nonconvex, entropy, ...
• Object constraints
3. Iterative algorithms for statistical image reconstruction
• EM based (EM, GEM, SAGE, OSEM)
• Direct optimization (Coordinate Descent, Conjugate Gradient, Surrogate Functions)
• Considerations nonnegativity, parallelizability, convergence rate, etc.
• Optimization transfer / surrogate functions
• Ordered subsets / block iterative algorithms
• acceleration properties, convergence issues
• Part 2 Outline (tentative):
1. Introduction
• Overview
• k-space formulation
• Conventional FFT reconstruction
• Gridding reconstruction / density compensation
2. Iterative reconstruction
3. Application survey
4. Regularized least-squares cost function
5. CG iteration
6. NUFFT for accelerating computation
7. Toeplitz implementation
8. Field inhomogeneity correction
9. time segmentation
10. Toeplitz implementation
11. Regularization methods
12. Partial k-space application
13. Sensitivity encoded imaging
14. Cartesian case
15. Non-Cartesian (e.g., spiral) cases