In this thesis, we presented a collection of new and competitive solutions for the reconstruction of magnetic-resonance images. We now summarize the main research directions and results in the first section of this chapter. The potential areas of interest for future research related to our work are listed in the second section.
We described the principles of magnetic resonance imaging (MRI) from a signal-processing perspective. This allowed us to derive a clean linear discrete model which is consistent with the equations that govern the continuous physical world. For a given MRI setup, this numerical model relates the parameters that characterize the object under investigation to the corresponding scanner data. We detailed careful and efficient implementations of this model which is the cornerstone of any reconstruction method.
The use of simulations of the MRI data-acquisition process is very convenient for assessing the validity of reconstruction methods. To that end, we introduced a new theoretical framework. Among its important novel aspects are the facts that (i) it fully accounts for the continuous nature of the equations that govern the physics and that (ii) the parameterization is flexible enough to allow for the analytical description of realistic MRI setups and phantoms. We succeeded in designing such a realistic setup and did conduct validation experiments with the help of this new simulation tool. In particular, we measured the image quality obtained from state-of-the-art reconstruction methods that were applied on data synthesized with both the conventional simulation approach—which is based on a discrete model—and our analytical method. The results differ significantly: the reconstruction performance is systematically overestimated in conventional simulations. This tendency is particularly pronounced with the nonlinear reconstruction schemes that are increasingly popular in MRI research. Our conclusion is that MRI reconstruction algorithms should not be evaluated using conventional simulations only. The novel simulation framework that we propose is a reliable alternative.
We presented MRI reconstruction as a general inverse problem which, in turn, is reformulated as a minimization problem. We detailed the known approaches leading to linear reconstruction and demonstrated their strong connections. Applied on challenging reconstruction tasks such as imaging when k-space samples are missing, these methods can be largely outperformed by some promising nonlinear approaches. Among those, we particularly focused on wavelet regularization. From our investigations about the influence of the choice of the wavelet transform on image quality, it appears that the wavelet basis has a limited impact and that it is vain to decompose the data beyond three levels. We provided a new variational interpretation that motivates the use of random shifting and deepens the understanding of its benefits on computational complexity and reconstruction quality. In our MRI experiments, random shifting led to a substantial gain in reconstruction quality, particularly when used together with the Haar wavelet transform. We also investigated acceleration strategies for wavelet-based iterative reconstruction. Based on theoretical grounds, we demonstrated that two recently proposed strategies can be combined synergistically. In practice, one can tailor the reconstruction scheme to the MRI setting—k-space trajectory and sensitivity of the receiving coil—to improve reconstruction speed. We ended up with a practical algorithm that is optimized for performance in terms of reconstruction quality and time. We conducted experiments to validate the method with challenging reconstruction tasks involving single and multiple channel, simulated, and real-world data. Linear reconstructions were observed to fail in providing acceptable image quality, contrarily to our reconstruction method which is on a par with total-variation regularization for the quality of the reconstructed images. Typically, our nonlinear method requires few seconds on a personal computer to perform a reconstruction, which is comparable to the time required by conventional linear reconstruction methods but brings a substantial increase in quality.
The analytical simulation framework we described allows the integration of temporal aspects, for instance moving phantoms (of interest for cardiac MRI) or region-dependent T1 and T2 parameters.
The phantom we proposed still lacks some texture to look fully realistic. Good candidates for the description of image texture would arise from the simulation of mesh-based structures, an aspect of our framework that we left without investigation.
It is not clear yet how to extend the Bézier-based contour parameterization to a third dimension while keeping the benefits of a closed-form description of the simulated data. Since very popular 3-D phantoms are defined using nonuniform rational B-splines (NURBS) , it would be a great achievement to work out the corresponding analytical solution.
The ideas behind our analytical phantom can potentially have an impact in other domains such as spectroscopy.
The problem of setting a proper regularization parameter, which is nontrivial for nonlinear reconstruction in the context of ill-posed problems, makes the reconstruction time a crucial point for methods involving regularization. Current approaches to tune this parameter include generalized cross validation , the L-curve method , and Monte-Carlo SURE . A recent work focused on MRI and Gaussian noise seems promising for non-linear reconstruction in MRI .
Substantial speedups in the reconstruction process have been recently reported using parallelization on dedicated GPU units [92,93]. There is no apparent obstacle that would prevent to employ the same parallelization strategy with our method to further speedup the reconstruction.
Another promising direction is the recently proposed i-LET approach  that performs well on deconvolution problems. This method provides a general framework that includes the different multi-step first-order approaches proposed in recent years to solve large-scale ℓ1-regularized minimization problems.
In this work, we focused on improving the quality of MRI images reconstructed out of data obtained through conventional acquisition setups. However, the recent trend of Compressed sensing or Compressive sampling in MRI has brought to light the potential benefits in revisiting the design of the acquisition setup. The first observation is that the reconstruction artifacts are less striking when the undersampling trajectory is randomized. Other designs of the k-space trajectory are currently under investigation . It is likely that future research will increasingly involve signal processing in the design of the MRI setup.
We considered in this thesis the estimation of the image out of MRI data. This is a linear problem. However, several challenging MRI problem settings do not fall in this category. This is the case of the estimation of the receiving coil sensitivity (possibly estimated jointly with the image), B0 estimation and correction , motion correction [4,5], or higher-order field imaging . Efficient solutions to these problems might be found in the future, based on the results obtained with linear problems.