New algorithms for convex set constrained signal reconstruction

The problem of signal reconstruction from linear measurements and convex set constraints is addressed in this paper. The currently popular POCS algorithm suffers from slow convergence and non-unique limits, whereas the recently proposed faster converging Newton algorithm requires projection onto the intersection of convex sets. In this paper, new algorithms, that retain the simplicity of the POCS algorithm, but have increased generality, and superior convergence rates, are presented. Superior convergence rates are achieved by incorporating derivative information using recent results on the differentiability of the projection operator onto a convex set. Efficient implementation of these algorithms using conjugate-gradient iterations for matrix inversion, make them suitable for large-scale problems. Computer simulations demonstrating the effectiveness of these algorithms, are also presented.