Magnetic resonance imaging (MRI) is a non-invasive imaging technique that dates back to the 70s . It is now commonly used in medicine and biology because of its specific advantages over other imaging modalities: versatile contrasts, speed, good spatial resolution, multidimensional imaging capabilities, and structural as well as functional information. Still, MRI is subject of active research. While, in the past, efforts were mainly concentrated towards the improvement of the acquisition hardware, the focus is now shifting towards the use of more sophisticated post-acquisition processes. The idea is that, by relying on proper signal processing, one can relax the acquisition requirements. The prospects are of different kinds:
Some recent techniques are indicative of this trend: off-resonance correction , parallel MRI , motion correction [4,5], compressed-sensing , and higher-order field imaging .
This thesis originates from a collaboration between the Biomedical Imaging Group of EPFL and the Institute for Biomedical Engineering of the University and ETH in Zürich. The aim is to develop reconstruction methods specific to magnetic resonance imaging relying upon the hypothesis that nonlinear reconstruction—for instance, wavelet regularization—are capable of enhancing image quality in challenging reconstruction tasks. Following the same research trend, several recent works [6,8,9,10,11,12,13,14,15,16] confirm the potential benefits of nonlinear reconstruction for MRI. Another idea of this project is that significant acceleration of the reconstruction process can be achieved if the structure of the physical model is exploited. Surprisingly, this aspect is not yet subject to much investigation in the MRI community.
In signal-processing terms, the reconstruction problem can be stated as follows. We must estimate a discrete image that represents an existing continuous function. The MRI scanner samples the Fourier domain of this function. Medical and instrumental constraints influence the design of the trajectory in the Fourier domain. The resulting sampling locations are not uniformly distributed and this situation yields undesirable artifacts in conventional image reconstructions. We consider this kind of challenging reconstruction task and regard it as a linear inverse problem. Two characteristics must be considered:
The downside is that iterative algorithms are much slower than traditional FFT-based reconstruction. While this is already true in the linear case, where convergence speed directly depends on the conditioning, current nonlinear reconstruction algorithms are even slower. In this context, our goal is to develop nonlinear reconstruction algorithms that enhance quality over standard reconstructions while being practically fast enough to compete with linear reconstruction.
During our in-depth investigation of the MRI reconstruction pipeline, the problem of the proper assessment of the quality of reconstructions was pervasive. Simulations facilitate the evaluation and comparison of reconstruction software because the experimental conditions are perfectly controlled and ground-truth data are well-known. As first contribution, we introduce a new theoretical framework that allows for the analytical description of realistic phantoms in both the spatial and Fourier domains. This solution shall be preferred to standard discrete simulations that introduce aliasing artifacts impairing the evaluation of the reconstruction quality. Some analytical phantoms are already available for MRI simulation but, contrarily to the brain phantom that we propose, they suffer from a lack of realism in describing anatomical regions and are limited to the simulation of MRI with a homogeneous receiving antenna. The original framework we present is general enough to simulate parallel MRI experiments and the parametric shapes that describe our phantom are very flexible. We propose experiments that validate our method and illustrate the impact of using discrete simulations in the assessment of the performance of several reconstruction techniques. It turns out that the performance bias can be very pronounced, particularly with state-of-the-art reconstruction techniques. This emphasizes the importance of our contribution.
A second innovative aspect of this thesis consists in a new reconstruction method that imposes a wavelet-domain regularization. The starting point for our work is ISTA [17,18,19], a simple and robust iterative algorithm that achieves sparsity-promoting reconstruction. Recent developments have shown that this algorithm can be substantially accelerated using a multistep method  (fast ISTA, aka FISTA). The convergence rate of both ISTA and FISTA have been derived. An alternative acceleration strategy, namely SISTA , was proposed that adjusts the operations depending on the wavelet subband in order to accelerate reconstruction. We revisit all these algorithms using weighted norms. This perspective gives birth to a new algorithm that combines the FISTA and SISTA acceleration strategies synergistically. We derive the convergence bounds for the proposed algorithm, showing its theoretical advantage in terms of speed. Random shifting  (RS) is a modification in wavelet-based algorithms that brings sensible image quality enhancements but lacked a theoretical foundation. We provide a new interpretation of RS as a greedy and practically efficient method to perform shift-invariant wavelet regularization. We come up with an algorithm for MRI that benefits from the advantages of all these techniques: convergence efficiency and reconstruction quality. Based on realistic MRI and parallel MRI simulations, we study the influence of the wavelet basis and the use of RS on reconstruction quality. We also verify the convergence superiority of our new algorithm. Finally, compared with other state-of-the-art reconstruction techniques on both simulated and in vivo data, our practical method proves to converge rapidly while achieving competitive reconstruction quality.
This thesis is organized as follows: In Chapter 2, we expose the principles of magnetic resonance imaging. We follow a conceptual path from the spin magnetic moment to the production of an image. The origin of scanner data is presented and the mathematical data-formation model is established. In Chapter 3, we describe the discrete model together with efficient implementations. Classical and state-of-the-art reconstruction techniques for MRI are presented within a general framework for reconstruction. Then, in Chapter 4, we present a new analytical simulation tool that is adapted to parallel MRI and addresses the issues occurring with standard rasterized simulations. The theory is exposed and experiments show the interest of the tool. Finally, in Chapter 5, we discuss an algorithmic strategy to perform competitive nonlinear reconstructions using wavelet regularization. The performance of the method is studied and its potential is demonstrated in several challenging MRI and parallel MRI experiments.