Cramer- Rao Lower Bounds for Biased Image Reconstruction

Abstract

Since image reconstruction and restoration are ill-posed problems, unbiased estimators often have unacceptably high variance. To reduce the variance, one introduces constraints and smoothness penalties, which yields biased estimators. This bias precludes the use of the classical Cramer-Rao (CR) lower bound for the variance of an unbiased estimator. This paper presents a uniform bound for minimum variance subject to a bias gradient constraint. Since the bound is independent of any estimator, one can explore the fundamental tradeoff between bias and variance in ill-posed problems. We apply the bound to a linear Gaussian model, and demonstrate the optimality of a simple penalized least-squares estimator.