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 resolution-variance curve are useful both for exploring near optimality of practical image estimation algorithms and for optimizing the design of image acquisition systems.