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Magnetic resonance imaging (MRI) scanners produce raw measurements that are unfit to direct interpretation, unless an algorithmic step, called reconstruction, is introduced. Up to the last decade, this reconstruction was performed by algorithms of moderate complexity. This worked because substantial efforts were devoted to adjust the MRI hardware to suit the algorithmic component. More recently, new techniques have reversed this trend by putting more emphasis on the algorithms and alleviating the constraints on the hardware. Whereas many new methods suffer from a marked increase in computational complexity, this thesis focuses on the development of reconstruction algorithms that are faster and simpler than state-of-the-art solutions, while preserving their quality.

First, we present the physical principles that underlie the acquisition of MRI data and motivate the classical linear model. Based on this continuous equation, we derive efficient implementations of a discrete model. Standard and state-of-the-art reconstruction algorithms are reviewed and presented in a general framework where reconstruction is regarded as an optimization problem that can naturally integrate regularization.

Next, we propose novel simulation tools for the validation of reconstruction methods. Those tools model the sensitivity of the receiving coil, which allows for the simulation of parallel MRI experiments. To honor the continuous nature of the underlying physics, we suggest the use of analytical phantoms. Unlike rasterized simulations, our phantoms do not introduce aliasing artifacts. Instead, they allow us to study how rasterization itself impacts the quality of reconstruction. To achieve this goal, we were able to work out closed-form solutions for the Fourier transform of parametric regions that can realistically reproduce anatomical features. Our results show that the inverse-crime situation impairs significantly the assessment of the performance of reconstruction methods, particularly, the nonlinear ones.

Finally, we investigate the design of algorithms that achieve reconstruction with a sparsity constraint expressed in a wavelet domain. Based on the latest developments in large-scale convex optimization, we derive an acceleration strategy that can be tailored to the MRI setup and provide theoretical evidence of its benefit. We develop it into a practical method that combines the advantages of speed and quality. Applied on challenging reconstruction problems, with simulated and in-vivo data, we significantly reduce the reconstruction time over state-of-the-art techniques without compromising quality.

Keywords: MRI, inverse problem, wavelets, sparsity, nonlinear reconstruction, undersampling, spiral, non-Cartesian, total variation, compressed sensing, iterative-shrinkage thresholding algorithm, ISTA, FISTA, FWISTA, analytical simulation, Shepp-Logan phantom, inverse crime, parallel MRI

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