Deconvolution using projections onto the epigraph set of a convex cost function

Abstract

A new deconvolution algorithm based on making orthogonal projections onto the epigraph set of a convex cost function is presented. In this algorithm, the dimension of the minimization problem is lifted by one and sets corresponding to the cost function and observations are defined. If the utilized cost function is convex in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</sup> , the corresponding epigraph set is also convex in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N+1</sup> . The deconvolution algorithm starts with an arbitrary initial estimate in R <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N+1</sup> . At each iteration cycle of the algorithm, first deconvolution projections are performed onto the hyperplanes representing observations, then an orthogonal projection is performed onto epigraph of the cost function. The method provides globally optimal solutions for total variation, l1, l2, and entropic cost functions.