Projection Onto Convex Sets (POCS) Based Signal Reconstruction Framework with an

A new signal processing framework based on the projections onto convex sets (POCS) is developed for solving convex optimization problems. The di-mension of the minimization problem is lifted by one and the convex sets corresponding to the epigraph of the cost function are defined. If the cost function is a convex function in RN the corresponding epigraph set is also a convex set in RN+1. The iterative optimization approach starts with an arbitrary initial estimate in RN+1 and orthogonal projections are performed onto epigraph set in a sequential manner at each step of the optimization problem. The method provides globally optimal solutions in total-variation (TV), filtered variation (FV), `1, `1, and entropic cost functions. New de-noising and compressive sensing algorithms using the TV cost function are developed. The new algorithms do not require any of the regularization parameter adjustment. Simulation examples are presented.