Image resolution-variance tradeoffs using the uniform Cramèr-Rao bound

In image reconstruction and restoration, there exists an inherent tradeoff between the recovered spatial resolution and statistical variance: lower variance can be bought at the price of decreased spatial resolution. This tradeoff can be captured for a particular regularized estimator by tracing out the resolution and variance as a curve indexed by the estimator’s smoothing parameter. When the resolution of an estimator is well characterized by the norm of the estimator bias-gradient the uniform Cramèr-Rao (CR) lower bound can be applied. The bias-gradient norm fails, however, to constrain the width of the estimator point response function and the uniform CR bound with bias-gradient norm can give counter-intuitive results. In this paper we present a modified uniform CR bound on estimator variance which captures the width of the estimator point response. These results on the theoretically minimum attainable resolutionvariance curve are useful both for exploring near optimality of practical image estimation algorithms and for optimizing the design of image acquisition systems.