Cramer-Rao bounds for 2-D target shape estimation in nonlinear inverse scattering problems with application to passive radar

Abstract

We present new methods for computing fundamental performance limits for two-dimensional (2-D) parametric shape estimation in nonlinear inverse scattering problems with an application to passive radar imaging. We evaluate Cramér-Rao lower bounds (CRB) on shape estimation accuracy using the domain derivative technique from nonlinear inverse scattering theory. The CRB provides an unbeatable performance limit for any unbiased estimator, and under fairly mild regularity conditions is asymptotically achieved by the maximum likelihood estimator (MLE). The resultant CRBs are used to define an asymptotic global confidence region, centered around the true boundary, in which the boundary estimate lies with a prescribed probability. These global confidence regions conveniently display the uncertainty in various geometric parameters such as shape, size, orientation, and position of the estimated target and facilitate geometric inferences. Numerical simulations are performed using the layer approach and the Nyström method for computation of domain derivatives and using Fourier descriptors for target shape parameterization. This analysis demonstrates the accuracy and generality of the proposed methods.

Estimation and Statistical Bounds for Three-Dimensional Polar Shapes in Diffuse Optical Tomography

“…7 may be somewhat pessimistic, in that we are computing the maximum perturbation at each point of the boundary given that the shape parameters are within a 95% confidence interval. It may be that a method, such as that described in [19], and extended to three dimensions, may give a somewhat less conservative estimate of the shape confidence region.…”

“…In two dimensions, Fourier Descriptors [19], [20] and BSpline models [21] have been employed to parameterize the boundary. For DOT, the first work in this area examined the estimation of the location, contrast and orientation of ellipsoids [22].…”

“…This knowledge, apart from giving us a shape "confidence interval", can help us in our design of instruments and source/detector layouts, if we have prior knowledge that a particular shape reconstruction reliability is desired. In contrast to previous work on shape estimation bounds in image reconstruction problems [21], [19], [32], we derive and implement these bounds specifically for the DOT inverse problem, and for the very challenging problem of fully three-dimensional nonlinear shape estimation. We visualize these bounds using a method introduced by Kirsch [25], extended to the case of three-dimensional polar shapes.…”

Cramer-Rao bounds for nonparametric surface reconstruction from range data

“…Hero et al [7] compute the CRB for B-spline parameters of star-shapes estimated from magnetic resonance imagery. Ye et al [8] compute the CRB for more general parametric shape estimators. Confidence intervals for shape estimators can be computed using CRBs [9], which provides an important computational advantage over using a Monte-Carlo simulation [10].…”

Application of Near-Space Passive Radar for Homeland Security

“…Due to these potential advantages, passive radars have received increasing attention. The best known example of a passive radar system is probably the Lockheed-Martin Silent Sentry System which uses a network of commercial television and radio channels as opportunistic transmitters with which to detect and track moving targets [6]. Researchers have begun to address the problems of wide-band and wide-angle processing associated with passive radars [7][8][9], and some initial studies have concentrated on the use of low earth orbit communication satellites for detection [10] and imaging [11].…”