Bounds from noisy linear measurements

Abstract

An estimation problem in which a finite number of linear measurements of an unknown function is available, and in which the only prior information available concerning the unknown function consists of inequality constraints on its magnitude, is ill-posed in that insufficient information is available from which point estimates of the unknown function can be made with any reliability, even with exact measurements. An alternative to point estimation involves the computation of bounds on linear functionals of the unknown function in terms of the measurements. A generalization is described of the bounding technique to problems in which the measurements are inexact. The bounds are defined in terms of a primal optimization problem. A deterministic interpretation of the bounds is given, as well as a probabilistic one for the case of additive Gaussian measurement noise. An unconstrained dual optimization problem is derived that has an interesting data-adaptive filtering interpretation and provides an attractive basis for computation. Several aspects of the primal and dual optimization problems are investigated that have important implications for the reliable computation of the bounds.