Compressive imaging with total variation regularization and application to auto-calibration of parallel magnetic resonance imaging

An important task in imaging is to recover an image from samples of its Fourier transform. With compressed sensing, this is done by applying a sparsifying transform and solving a l1 minimization problem. One possible transform is the discrete gradient operator, in which case penalizing the l1 norm leads to Total Variation (TV) minimization. We present new recovery guarantees for TV regularization in arbitrary dimensions using two sampling strategies: uniform random Fourier sampling and variable density Fourier sampling. In particular, we determine a near-optimal choice of sampling density in any dimension. Our theoretical and numerical results show that variable density Fourier sampling increases the stability and robustness of TV regularization over uniform random Fourier sampling. As an application, we consider auto-calibration in parallel magnetic resonance imaging (pMRI). We develop a two-step algorithm: firstly, using sparse regularization to recover the coil images; secondly, using linear least squares to obtain the overall image.