A Fast and Robust Paradigm for Fourier Compressed Sensing Based on Coded Sampling

First-order gradient methods are commonly used for compressed sensing reconstruction. However, for Fourier sampling systems, they require computing a large number of fast Fourier transforms (FFTs), which can be expensive in real-time applications. In this paper, instead of random sub-sampling, we use a sampling scheme inspired by coding theory from a recent sparse-FFT work of Pawar and Ramchandran [1]. In particular, we show that Iterative Soft Thresholding Algorithm (ISTA) applied on the Least Absolute Shrinkage and Selection Operator (LASSO) with the coded sampling provides an O(log n) per-iteration speedup over the standard iteration cost, where n is the signal length. Since the coded sampling operation deviates from the common randomized compressed sensing sampling, it is a priori unclear whether LASSO can recover sparse signals. We provide recovery guarantees for LASSO using the coded sampling guaranteed for an arbitrary signal-to-noise ratio. For a k-sparse signal and under a uniformly random sparsity model, we show that LASSO recovers the underlying signal from O(k log
⁴
n) measurements through the coded sensing system, with a reconstruction error that is proportional to the sparsity level and noise energy. Moreover, we demonstrate numerically computational speedups for using this scheme as well as lower MRI acquisition times.