DESCRIPCIÓN DEL MÉTODO Y DE LA FECHA DE VENTA, TRANSFERENCIA,

a fool at least once a month. Fyodor Dostoyevsky

In this thesis, we addressed the problem of magnetic resonance imaging (MRI) reconstruction from partial measurements using the com- pressed sensing (CS) theory and Markov random eld (MRF) model for the image support in wavelet-like domains. The proposed MRF model reduced the space of possible solutions during reconstruction by enforcing the expected image structure under wavelet-like transformations. Image coecients in transform domain are often clustered around important image structures, such as edges and textures, and are well represented through MRF-based spatial context modeling which we exploited in the proposition of reconstruction methods in this thesis.

Our original contribution is in the development of reconstruc- tion methods, greedy and optimization-based, which utilized MRF based signal prior. We start with an optimization-based framework for the re- covery of compressively sampled MRI (CS-MRI) data. In Chapter 4 we presented a comprehensive study and developed a novel algorithm that incorporates the MRF modeling framework into a constrained split augmented Lagrangian method. We dened a regularization function based on the isotropic Ising model for the image support in the trans- form domain. In this way, we constrained that recovered image obeys desired characteristics imposed by the Ising model. A hard-thresholding rule is derived according to the adopted regularization function and the estimated image support mask in the transform domain. Since regular- ization, i.e., application of the hard-thresholding rule on image coe- cients, is performed on the lattice structure dened by the Ising model, we named the algorithm lattice split augmented Lagrangian (LaSAL).

An extension of LaSAL with compound regularization which introduces the TV norm besides the MRF-based regularization function is consid- ered. This results in an extended version of the LaSAL method named LaSAL2. The developed algorithms improve upon the constrained split augmented Lagrangian shrinkage algorithm (C-SALSA) in MRI recon- struction and they also outperform the earlier lattice split Bregman method (LaSB) which also utilized MRF signal prior. The proposed methods achieved much better reconstruction performances in compari- son with the state-of-the-art approach that involved a wavelet-tree based model for the relations among image coecients.

In the same chapter, we further considered a more general MRF model for the signal support and a fast composite splitting algorithm (FCSA) as an optimization framework in which we incorporated the proposed MRF-based regularization. In relation to LaSAL and LaSAL2 methods we now used anisotropic instead of the isotropic Ising model for modeling the support of image coecients and we allow dierent model parameters for each image subband in the transformation domain. In- troducing a more general MRF model is supported by the ecient es- timation procedure of its parameters for each subband of coecients. As well as in LaSAL2 where we considered MRF-based regularization in combination with TV. With the estimated support of image coecients for each subband, we derived a new soft-thresholding rule instead of the hard-thresholding rule used in LaSAL and LaSAL2. Since the chosen op- timization framework FCSA may go with and without acceleration steps, we proposed two methods composite splitting lattice with TV regular- ization (CSLaTV) and its accelerated version FCSLaTV. The proposed methods with automated MRF parameter estimation show great perfor- mances comparable to the LaSAL2 and give the possibility to reconstruct images with various structures (e.g. which came from imaging dierent human anatomy) for which xed MRF parameters during reconstruction isn't optimal.

In Chapter 5 we considered a greedy iterative procedure for MRI reconstruction where we incorporate the proposed MRF-based reg- ularization. We developed a greedy algorithm with lattice regularization (GreeLa) which performs hard-thresholding rule on image coecients based on their estimated support. GreeLa is an extension of the lat- tice matching pursuit (LaMP) method developed primarily for images that are sparse in the canonical domain. The proposed GreeLa can reconstruct both the compressible MR images as well as on originally sparse images. The simplied iterative procedure of the GreeLa method

153 has a smaller number of parameters that need to be initialized com- pared to optimization-based methods proposed in Chapter 4. GreeLa showed stable behavior through iterations in comparison with the re- lated MRF-based method LaSB and for low sampling rates outperforms best-performing methods proposed in Chapter 4. However, algorithm simplicity results in somewhat poorer performances with the increase of sampling rates compared to optimization-based methods. For sampling rates below 30%, GreeLa achieves comparable reconstruction results with optimization-based methods. Stable image recovery is obtained after 25 up to 30 algorithm iterations which is the result of the greedy iterative procedure in combination with MRF-based signal prior. Although this fast algorithm convergence to the solution is desired characteristic, it results in limiting algorithm performances for higher sampling rates in comparison with optimization-based methods.

In order to get closer to the real application, we also extend our MRF-based reconstruction framework to parallel MR data (pMRI- CS) in Chapter 6. Multi-coil reconstruction was conducted through the joint framework which takes into account undersampled measurements from all of the available coils and their estimated sensitivity prole maps. The proposed algorithm which is derived using the same procedure as for LaSAL2, with the addition of necessary transformations due to dierent algorithm inputs (multi coil measurements and sensitivity maps), shows great performances compared to the current state-of-the-art methods for pMRI-CS.

The exposed analysis and results in this thesis prove that the crucial benet in MR reconstruction came from the involved MRF prior in both greedy and optimization-based frameworks and accordingly de- rived regularization rules. The results also demonstrate the superior performance of the proposed algorithm in comparison to state-of-the- art methods, both in terms of quantitative performance measures and visually.

There is much room to optimize the computations in our pro- posed methods, especially regarding the inference procedure in the MRF- based support estimation. Belief propagation algorithms may be consid- ered as well as various parallelization procedures to optimize the code. Involving parallelization opens new possibilities for updating order for sites or a block of sites during inference procedure without increasing computation time. As well, estimation of MRF parameters can be el- evated on the level of sites and cliques in their neighborhood, thereby allowing using more general non-homogeneous MRF models for a signal

support.

The applicability of the developed MRF-based regularization in dierent imaging problems should also be analyzed. The iterative reconstruction procedure used for computer tomography imaging has a great resemblance with the MRI reconstruction procedure and can be one of the possible applications of the proposed methods in this thesis. This might help in the reduction of time in which the patient is exposed to X- rays during the CT imaging. A joint image demosaicking and denoising for multispectral imaging can also be an interesting application. Here MRF signal prior can be used during the interpolation procedure to recover the image spectral bands to the original resolution of the mosaic- snapshot multispectral image.

Another very promising way of regularizing image estimate dur- ing reconstruction procedure is the usage of deep learning prior i.e. pre- trained deep neural networks (DNN) for denoising purposes. This is a very popular research direction with promising results, and it opens pos- sibilities for a new interpretation of our developed MRF-based regularizer with the construction of DNN architecture which will be equivalent to proposed MRF-based regularization. This will lead to new ways of im- proving regularization through the DNN framework. These aspects will be part of our future research.

Bibliography