Measurement Extraction of Two Targets With Unequal and Unknown Intensities in an FPA

Abstract

This article extends previous work on location and intensity estimation for measurement extraction of targets in a focal plane array. Prior work has been done to extract single targets and two targets of equal intensity, whereas this work explores the case where two targets have unequal and unknown intensities. Here, we assume a Gaussian point spread function (PSF) with spread $\sigma _{\text{PSF}}$ , but our approach could be applied to other PSF shapes. We present a maximum likelihood (ML) method for target extraction under resolved and unresolved assumptions. In the unresolved case, we estimate the parameters of a single target that represents the centroid of the two unresolved targets. We also present the Cramer–Rao lower bound (CRLB) of the estimation variances for both cases. Our simulation results show that resolved targets have their parameter vectors estimated efficiently (i.e., the variance meets the CRLB) when the targets are separated by $0.9\sigma _{\mathrm{PSF}}$ , or about 1.8 pixel widths. We also find that estimation of the centroid parameters is efficient below a target separation of $0.65\sigma _{{\mathrm{PSF}}}$ . Furthermore, we find that increased difference in the SNR of two targets causes the variances in the resolved scenario to be lower, and in the case of the unresolved scenario, to increase. We also derive and characterize a decision about target cardinality as a hypothesis testing problem, and develop a generalized likelihood ratio test to perform the decision making. The performance of this test is evaluated via Monte Carlo simulations, and matches well to theoretical predictions. Finally, we explore the effect of separation between targets, and individual target SNR on resolvability.