Bias-resolution-variance tradeoffs for single pixel estimation tasks using the uniform Cramer-Rao bound

Abstract

Previously we introduced the Uniform Cram&-Rao (CR) Bound as a lower bound on the variance of biased estimators, along with the concept of the delta-sigma tradeoff curve. For an estimator whose variance lies on this curve, lower variance can only be achieved at the price of increased estimator bias gradient norm, and vice versa.However, for single pixel estimation, one can specify a variety of different estimator point response functions that have identical bias-gradient norm but with widely different resolution properties. This has lead to some counter-intuitive results and interpretation difficulties when using the Uniform CR Bound in performance studies of imaging systems.We now extend this tradeoff concept by introducing the 2nd-moment of the point response function as a measure of resolution for single-pixel estimation tasks. We derive an expression for the delta-gamma-sigma tradeoff surface. This surface specifies an "unachievable region" of estimator variance. For estimators that lie on this surface, lower variance can only be achieved at the price of increased bias gradient norm and/or decreased estimator resolution. We present a method for computing this surface for linear gaussian inverse problems.

Image reconstruction for a novel Compton scatter tomograph

“…The issue of attainment of this bound is still an open problem. We are currently working on extensions to the UCRB which explicitly account for system resolution by using an appropriate norm in the bias gradient constraint which is used to compute the UCRB [8].…”