Adaptive Order Non-Convex Lp-norm Regularization in Image Restoration

Abstract

The optimization formulations of sparse signal recovery problems use a regularizer term which encourages the solution to be sparse or piece-wise smooth. Of special interest are the ℓp norms with 0 ≤ p ≤ 1 which penalize large values and encourage the solution of the optimization problem to be sparse. In this work, we propose a new regularizer in which the exponent p, 0 ≤ p ≤ 1 of the ℓp–norm is a function of the signal element amplitude, leading to a function which is smooth and convex over non-negative values. This formulation adapts the value of the exponent p according to a sigmoid function applied on the signal values, to restrict its values to between 0 and 1. Experiments show that this formulation is more accurate in denoising, than the ℓ1 and ℓ0.5 norm regularizers, and is more accurate than the ℓ1-norm when used iteratively for deblurring or inpainting.