Magnetic Resonance Imaging background 

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Image contrast and acquisition parameters

An MRI contrast consists of transcoding the acquired NMR signal into gray levels. It reflects the relaxation times and spin density differences between the image tissues. The main two factors in an MRI contrast are T1 and T2. Since these factors always contribute in different ways to define a contrast, changing the sequence parameters (i.e. TE and TR) allows indirectly defining a contrast through managing the respective contributions of these main factors.

Functional Magnetic Resonance Imaging

In order to understand the structure and function of the human brain, neuro-scientists have been probing brain’s structure and function using indirect methods for a very long time. Starting from post-mortem analysis, scientists succeeded to establish functional cerebral maps of the human brain due to electrical stimulations applied directly to the brain during a neuro-surgery by Wilder Penfield [Penfield and Rasmussen, 1952]. Since the early 90’s, different imaging modalities have revolutionized this active research field making it possible to collect real-time information about what is happening into the human brain without opening the cerebral box. Generally speaking, we talk about anatomical and functional imaging. The first modality is designed to highlight the cerebral structures and outline any disorder or damage (tumor, deformation, hemorrhage …).
The second modality is rather designed to measure brain activity during some specific tasks (also called stimuli ) or to probe intrinsic or ongoing activity (resting state). Besides fundamental research aiming at understanding the organization of cerebral structures, functional imaging is also used for epileptic diagnosis, pre-surgery investigations in order to preserve some specific brain areas, studying the effect of new drugs designed to curesome brain diseases or neurological disorders like Alzheimer, schizophrenia,… However, for simple brain functions, these two imaging modalities may be coupled in order to draw functional cerebral maps by linking each brain area to its functional role. For more complicated brain functions, connections and interactions between the involved brain areas may also be studied. In this context, functional Magnetic Resonance Imaging (fMRI) is a recent neuroimaging technique for measuring brain activity. Generally speaking, it is used to detect brain areas which are involved in a specific task (e.g. simple auditory visual or motor) or more complex cognitive process (e.g. language, computation,…). It can also be used to study emotions, attention, memory, the intrinsic activity of the brain,… This technique is being widely developed due to its strength especially when coupled with anatomical MRI [Ogawa et al., 1990; Bandettini et al., 1993; Chen and Ogawa, 1999]. The main advantage of fMRI is its non-ionisant property which allows one to non-invasively establish functional maps of activated areas. Moreover, compared to other imaging modalities like Positron Emission Tomography (PET), fMRI has a better spatial and temporal resolution. However, it should be noted that at standard magnetic field intensities such as 1.5 or 3 Tesla, fMRI has a lower chemical/molecular resolution. This drawback is being alleviated due to the use of ultra high magnetic fields such as 7 Tesla.

Parallel Magnetic Resonance Imaging

As detailed in Section 2.2, the frequency encoding step has to be repeated several times to encode a whole line of the k-space. Moreover, in order to encode all the lines of the k-space, the phase encoding step has also to be iterated. However, for fMRI experiments, reducing the global imaging time without significantly degrading the image quality is of great interest for the final goal, i.e. studying the brain functions. Indeed, since an fMRI study requires the acquisition of the brain volume several times in order to track brain activity, reducing the acquisition time (which may lead to reducing TR) allows faster repetition of the brain imaging, which leads to better temporal resolution. It allows hence getting more accurate knowledge about the brain response dynamics. On the other hand, increasing the spatial resolution is also benifical for fMRI since it allows more precise spatial localization of activations. Improving this resolution requires the acquisition of more k-space points. If reducing the TR is not the goal, shortening the acquisition time can be exploited to increase the spatial resolution by acquiring additional k-space points while maintaining the TR almost fixed. In multishot acquisitions, reducing the acquisition time may also be exploited to acquire more images, and hence to increase the acquisition SNR. It leads also to shorter read-out duration, and hence allows the reduction of reconstruction artifacts such as magnetic susceptibility ones even for anatomical MRI.

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Table of contents :

1 Introduction 
1.1 Motivations
1.2 Organization of the manuscript
1.3 Publications
1.4 Patent
1.5 Software
2 Magnetic Resonance Imaging background 
2.1 Introduction
2.2 Magnetic Resonance Imaging
2.2.1 Nuclear Magnetic resonance
2.2.2 NMR signal measurement
2.3 Image contrast and acquisition parameters
2.3.1 TR effect
2.3.2 TE effect
2.3.3 Acquiring a T1 or T2-weighted MRI image
2.4 Functional Magnetic Resonance Imaging
2.4.1 Blood Oxygen Level Dependent effect
2.4.2 Data acquisition in fMRI
2.4.3 Artifacts in fMRI
2.5 Parallel Magnetic Resonance Imaging
2.5.1 Parallel MRI basics
2.5.2 Parallel MRI reconstruction
2.6 Conclusion
3 Regularization and convex analysis for inverse problems 
3.1 Introduction
3.2 Regularization for inverse problems
3.2.1 Inverse problems
3.2.2 Regularization
3.2.3 The frame concept
3.2.4 Bayesian approach using frame representations
3.3 Convex optimization
3.3.1 Some convex optimization algorithms
3.3.2 For those who see life in pink
3.4 Numerical illustrations
3.4.1 Comparison of the AA and SA performance
3.4.2 Convergence speed comparison
3.4.3 Convergence speed gain when calculating kHFk2
3.5 Conclusion
4 Regularized SENSE reconstruction 
4.1 Introduction
4.2 Regularization in pMRI
4.2.1 Tikhonov regularization
4.2.2 Total variation regularization
4.3 Regularization in the WT domain
4.3.1 Motivation
4.3.2 Definitions and notations
4.3.3 Wavelet-based regularized reconstruction
4.3.4 Regularization using bivariate wavelet prior
4.3.5 Constrained wavelet-based regularization
4.4 Combined wavelet-TV regularization
4.5 Spatio-temporal regularization
4.6 Conclusion
5 Hyper-parameter estimation 
5.1 Introduction
5.2 Problem Formulation
5.3 Hierarchical Bayesian Model
5.4 Sampling strategies
5.4.1 Hybrid Gibbs Sampler
5.4.2 Hybrid MH sampler using algebraic properties of frame representations
5.5 Toward a more general Bayesian Model
5.6 Numerical illustrations
5.6.1 Validation experiments
5.6.2 Convergence results
5.6.3 Application to image denoising
5.6.4 Hyper-parameter estimation in parallel MRI
5.7 Conclusion
6 Experimental validation in fMRI 
6.1 Introduction
6.2 Data processing in fMRI
6.2.1 Exploratory vs. hypothesis-driven methods
6.2.2 The General Linear Model
6.2.3 Subject-level analysis
6.2.4 Group-level analysis
6.3 Validation of the proposed methods
6.3.1 Experimental data
6.3.2 Subject-level analysis
6.3.3 Group-level analysis
6.4 Discussion
6.5 Conclusion
7 Conclusion

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