Smooth Robust Tensor Principal Component Analysis for Compressed Sensing of Dynamic MRI Abstract   Dynamic magnetic resonance imaging (DMRI) often requires a long time for measurement acquisition. It is a crucial problem about the enhancement of reconstruction quality from a limited set of undersamples. The lowrank plus sparse decomposition model, which is also called robust principal component analysis (RPCA), is widely used for reconstruction of DMRI data in the modelbased way. In this paper, considering that DMRI data are naturally in tensor form with blockwise smoothness, we propose a smooth robust tensor principal component analysis (SRTPCA) method for DMRI reconstruction. Compared with classical RPCA approaches, the low rank and sparsity terms are extended to tensor version to fully exploit the spatial and temporal data structures. Moreover, a tensor total variation regularization term is used to encourage the multidimensional blockwise smoothness for the reconstructed DMRI data. The relaxed convex optimization model can be divided into several subproblems by the alternating direction method of multipliers. Numerical experiments on cardiac perfusion and cine datasets demonstrate that the proposed SRTPCA method outperforms the stateoftheart ones in terms of recovery accuracy.

Low Rank Plus Sparse Decomposition of Synthetic Aperture Radar Data for Target Imaging We analyze synthetic aperture radar (SAR) imaging of complex ground scenes that contain both stationary and moving targets. In the usual SAR acquisition scheme, we consider ways to preprocess the data so as to separate the contributions of the moving targets from those due to stationary background reflectors. Both components of the data, that is, reflections from stationary and moving targets, are considered as signal and are needed for target imaging and tracking, respectively. The approach we use is to decompose the data matrix into a low rank and a sparse part. This decomposition enables us to capture the reflections from moving targets into the sparse part and those from stationary targets into the low rank part of the data. The computational tool for this is robust principal component analysis (RPCA) applied to the SAR data matrix. We also introduce a lossless baseband transformation of the data, which simplifies the analysis and improves the performance of the RPCA algorithm. A modified version of RPCA, the stable principal component pursuit (PCP), is robust to additive noise. Our main contribution is a theoretical analysis that determines an optimal choice of parameters for the RPCA algorithm so as to have an effective and stable separation of SAR data coming from moving and stationary targets. This analysis also gives a lower bound for detectable target velocities. We show in particular that the rank of the sparse matrix is proportional to the square root of the targetu0027s speed in the direction that connects the SAR platform trajectory to the imaging region. The robustness of the approach is illustrated with numerical simulations in the Xband SAR regime.