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.

中文翻译：

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FPA中强度不相同和未知的两个目标的测量提取

本文扩展了先前在焦平面阵列中对目标进行测量提取的位置和强度估计的工作。先前的工作已经完成，以提取单个目标和强度相等的两个目标，而本工作探讨了两个目标强度不相等且未知的情况。在这里，我们假设具有扩散的高斯点扩散函数（PSF）$ \ sigma _ {\ text {PSF}} $ ，但我们的方法可以应用于其他PSF形状。我们提出了最大似然（ML）方法在已解决和未解决的假设下进行目标提取。在未解决的情况下，我们估计单个目标的参数，该参数代表两个未解决的目标的质心。我们还给出了两种情况下估计方差的Cramer-Rao下界（CRLB）。我们的仿真结果表明，当目标与目标之间的距离被分隔开时，已分解目标的参数向量得到了有效估计（即方差满足CRLB）$ 0.9 \ sigma _ {\ mathrm {PSF}} $ ，或约1.8像素宽度。我们还发现，在目标间隔为时，质心参数的估计是有效的。$ 0.65 \ sigma _ {{\ mathrm {PSF}}} $ 。此外，我们发现，两个目标的SNR差异增加会导致解决方案中的方差变小，而在未解决方案中，方差变大。我们还将派生有关目标基数的决策作为假设检验问题并将其特征化，并开发广义似然比检验以执行决策。该测试的性能通过蒙特卡洛模拟进行评估，并且与理论预测非常吻合。最后，我们探索了目标之间的分离以及单个目标SNR对可分辨性的影响。