A SURE Approach for Digital Signal/Image Deconvolution Problems

TL;DR

This paper introduces a novel deconvolution method for noisy data that leverages Stein's unbiased risk estimate and flexible frame analysis, demonstrating superior performance over traditional wavelet-based techniques.

Contribution

It formulates deconvolution as a nonlinear estimation problem using Stein's risk, applicable to various analysis and synthesis frames, with new variance calculation results.

Findings

Outperforms conventional wavelet-based methods in image restoration

Provides new theoretical results on Stein's risk variance calculation

Applicable to overcomplete and non-overcomplete frames

Abstract

In this paper, we are interested in the classical problem of restoring data degraded by a convolution and the addition of a white Gaussian noise. The originality of the proposed approach is two-fold. Firstly, we formulate the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from Stein's unbiased quadratic risk estimate. Secondly, the deconvolution procedure is performed using any analysis and synthesis frames that can be overcomplete or not. New theoretical results concerning the calculation of the variance of the Stein's risk estimate are also provided in this work. Simulations carried out on natural images show the good performance of our method w.r.t. conventional wavelet-based restoration methods.