A Hierarchical Bayesian Model for Frame Representation

In many signal processing problems, it may be fruitful to represent the signal under study in a frame. If a probabilistic approach is adopted, it becomes then necessary to estimate the hyper-parameters characterizing the probability distribution of the frame coefficients. This problem is difficult since in general the frame synthesis operator is not bijective. Consequently, the frame coefficients are not directly observable. This paper introduces a hierarchical Bayesian model for frame representation. The posterior distribution of the frame coefficients and model hyper-parameters is derived. Hybrid Markov Chain Monte Carlo algorithms are subsequently proposed to sample from this posterior distribution. The generated samples are then exploited to estimate the hyper-parameters and the frame coefficients of the target signal. Validation experiments show that the proposed algorithms provide an accurate estimation of the frame coefficients and hyper-parameters. Application to practical problems of image denoising show the impact of the resulting Bayesian estimation on the recovered signal quality

Self-calibrating nonlinear reconstruction algorithms for variable density sampling and parallel reception MRI

International audienceCompressed Sensing has allowed a significant reduction of acquisition times in MRI, especially in the high resolution (e.g., 400 µm) context. However, in this setting CS must be combined with parallel reception as multichannel coil acquisitions maintain high input signal-to-noise ratio (SNR). To get rid of usual parallel imaging limitations (output SNR loss), non-Cartesian trajectories provide a gain in sampling efficiency in the CS context. In this paper, we propose a self-calibrating MRI reconstruction framework that handles variable density sampling. Low resolution sensitivity maps are estimated from the low frequency k-space content using an original and fast method while MR images are reconstructed using a nonlinear iterative algorithm, which promotes sparsity in the wavelet domain. As regards the optimization task, we compare three first-order prox-imal gradient methods (FB, FISTA, POGM) and evaluate their respective convergence speed. Comparison with state-of-the-art (i.e., 1-ESPIRiT) suggests that our self-calibrating POGM-based algorithm outperforms current approaches both in terms of image quality and computing time on prospectively accelerated ex-vivo and in-vivo data collected at 7 Tesla and we will focus more specifically on prospective non-Cartesian 8-fold accelerated in vivo Human brain data