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Computational methods for fast imaging

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Martin Uecker
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Teaching Session - Advanced



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Computational Methods for Fast Imaging
Martin Uecker
Department of Diagnostic and Interventional Radiology, University Medical Center, Göttingen
DZHK (German Centre for Cardiovascular Research), Partner Site Göttingen
Image Reconstruction as an Inverse Problem

Example: Real-time MRI
- signal equation: regularization
- discretized forward model:
Unknown image:
Measured k-space data:
Forward model:
Regularization:
Reg. parameter:

- accelerate the measurement by exploiting the sensitivities of an array of receive coils

Solution defined as minimizer of functional:
data fidelity
Parallel Imaging

measurement data
physical model
iterative optimization
prior knowledge

Real-time Cardiac MRI using a radial bSSFP sequence and NLINV

estimated image

- (non-Cartesian) parallel imaging
- compressed sensing + parallel imaging
- model-based reconstruction

Important numerical optimization methods:
- Method of Conjugate Gradients (CG)
- Fast Iterative Shrinkage-Thresholding Algorithm (FISTA)
- Alternating Direction Method of Multipliers (ADMM)
- Iteratively Regularized Gauss-Newton Method (IRGNM)
- non-Cartesian (radial) parallel imaging with NLINV

- regularization with previous frame as prior knowledge:

mult. with sensitivities:
Fourier transform:
sampling operator:
forward model
Common framework for:
Compressed Sensing
Iterative reconstruction

- randomly undersampled k-space leads to incoherent aliasing
- incoherent aliasing can be removed with sparsity-based prior knowledge e.g. sparsity of wavelet coefficients (the l1-norm promotes sparsity)
- real-time image reconstruction using parallel processing with graphical processing units

References:
1. Pruessmann KP, Weiger M, Börnert P. Boesiger P. Advances in sensitivity encoding with arbitrary k- space trajectories. MRM, 46:638–651 (2001)
2. Lustig M, Donoho D, Pauly JM. Sparse MRI: The application of compressed sensing for rapid MR imaging. MRM, 58:1182–1195 (2007)
3. Block KT, Uecker M, Frahm J. Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint. MRM, 57:1086–1098 (2007)
4. Lustig M and Pauly JM. SPIRiT: Iterative self-consistent parallel imaging reconstruction from arbitrary k-space. MRM, 64: 457–471 (2010)
5. Fessler JA. Model-Based Image Reconstruction for MRI. IEEE SPM 27:81-69 (2010)
6. Beck A, Teboulle M. A fast iterative shrinkage-thresholding algorithm for inear inverse problems. SIAM J Imag Sci 2:183-202 (2009)
7. Afonso MA, Bioucas-Dias JM, Figueiredo M. An Augmented Lagrangian Approach to the Constrained Optimization Formulation of Imaging Inverse Problems, IEEE Trans Image Process, 20:681-695 (2011)
8. Uecker M, Zhang S, Voit D, Karaus A, Merboldt K-D, Frahm J. Real-time MRI at a resolution of 20 ms. NMR Biomed., 23:986–994 (2010)

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