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

A new signal processing framework based on the projections onto convex sets (POCS) is developed for solving convex optimization problems. The dimension 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 denoising 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.