Echantillonnage compressé le long de trajectoires physiquement plausibles en IRM

Abstract : Magnetic Resonance Imaging~(MRI) is a non-invasive and non-ionizing imaging technique that provides images of body tissues, using the contrast sensitivity coming from the magnetic parameters (T$_1$, T$_2$ and proton density). Data are acquired in the $k$-space, corresponding to spatial Fourier frequencies. Because of physical constraints, the displacement in the $k$-space is subject to kinematic constraints. Indeed, magnetic field gradients and their temporal derivative are upper bounded. Hence, the scanning time increases with the image resolution. Decreasing scanning time is crucial to improve patient comfort, decrease exam costs, limit the image distortions~(eg, created by the patient movement), or decrease temporal resolution in functionnal MRI. Reducing scanning time can be addressed by Compressed Sensing~(CS) theory. The latter is a technique that guarantees the perfect recovery of an image from undersampled data in $k$-space, by assuming that the image is sparse in a wavelet basis. Unfortunately, CS theory cannot be directly cast to the MRI setting. The reasons are: i) acquisition~(Fourier) and representation~(wavelets) bases are coherent and ii) sampling schemes obtained using CS theorems are composed of isolated measurements and cannot be realistically implemented by magnetic field gradients: the sampling is usually performed along continuous or more regular curves. However, heuristic application of CS in MRI has provided promising results. In this thesis, we aim to develop theoretical tools to apply CS to MRI and other modalities. On the one hand, we propose a variable density sampling theory to answer the first inpediment. The more the sample contains information, the more it is likely to be drawn. On the other hand, we propose sampling schemes and design sampling trajectories that fulfill acquisition constraints, while traversing the $k$-space with the sampling density advocated by the theory. The second point is complex and is thus addressed step by step. First, we propose continuous sampling schemes based on random walks and on travelling salesman~(TSP) problem. Then, we propose a projection algorithm onto the space of constraints that returns the closest feasible curve of an input curve~(eg, a TSP solution). Finally, we provide an algorithm to project a measure onto a set of measures carried by parameterizations. In particular, if this set is the one carried by admissible curves, the algorithm returns a curve which sampling density is close to the measure to project. This designs an admissible variable density sampler. The reconstruction results obtained in simulations using this strategy outperform existing acquisition trajectories~(spiral, radial) by about 3~dB. They permit to envision a future implementation on a real 7~T scanner soon, notably in the context of high resolution anatomical imaging.
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Nicolas Chauffert. Echantillonnage compressé le long de trajectoires physiquement plausibles en IRM. Traitement du signal et de l'image [eess.SP]. Université Paris Sud - Paris XI, 2015. Français. ⟨NNT : 2015PA112234⟩. ⟨tel-01265504⟩

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