Projected Barzilai-Borwein Method with Infeasible Iterates for Nonnegative Image Deconvolution

Abstract

The Barzilai-Borwein (BB) method for unconstrained optimization has attracted attention for its "chaotic" behaviour and fast convergence on image deconvolution problems. However, images with large areas of darkness, such as those often found in astronomy or microscopy, have been shown to benefit from approaches which impose a nonnegativity constraint on the pixel values. We present a new adaptation of the BB method which enforces a nonnegativity constraint by projecting the solution onto the feasible set, but allows for infeasible iterates between projections. We show that this approach results in faster convergence than the basic Projected Barzilai-Borwein (PBB) method, while achieving better quality images than the unconstrained BB method. We find that the new method also performs comparably to the Gradient Projection-Conjugate Gradient (GPCG) method, and in most test cases achieves a lower restoration error, despite being a much simpler algorithm.