Modified Frame Reconstruction Algorithm for Compressive Sensing

Compressive sensing is a technique to sample signals well below the Nyquist rate using linear measurement operators. In this paper we present an algorithm for signal reconstruction given such a set of measurements. This algorithm generalises and extends previous iterative hard thresholding algorithms and we give sufficient conditions for successful reconstruction of the original data signal. In addition we show that by underestimating the sparsity of the data signal we can increase the success rate of the algorithm. We also present a number of modifications to this algorithm: the incorporation of a least squares step, polynomial acceleration and an adaptive method for choosing the steplength. These modified algorithms converge to the correct solution under similar conditions to the original un-modified algorithm. Empirical evidence show that these modifications dramatically increase both the success rate and the rate of convergence, and can outperform other algorithms previously used for signal reconstruction in compressive sensing.