Detection, Location Estimation, and CRLB of a Streaking Target in an FPA With a Poisson Model

Abstract

This paper deals with measurement extraction, from an optical sensor's Focal Plane Array (FPA), of a streaking target. We use a model that assumes pixels are separated by dead zones and model the streaking target's point spread function (PSF) as a Gaussian PSF that moves during the optical sensor's integration time. We make an assumption that the target has a constant velocity over the sampling interval and parametrize its motion with a starting and ending position. The noise model for a single pixel has variance proportional to its area, consistent with a Poisson model of the number of nontarget originated photons. We develop a maximum likelihood (ML) method of estimating the target motion parameter vector based on the set of pixel measurements from the optical sensor. This paper then derives the Cramer–Rao lower bound (CRLB) on the estimation error of the target motion parameter. We then present a matched filter (MF) based definition of the signal-to-noise ratio (SNR) to use as a basis for comparison of Monte Carlo simulation based location estimates to the calculated CRLB. It is shown that the ML estimator for the starting and ending positions of a streak in the FPA is efficient for MFSNR $\geq 12$ dB. We then provide a test statistic for target detection and propose approximate distributions to set the detection threshold for specific detection ( $P_D$ ) and false alarm probabilities ( $P_{\text{FA}}$ ), which are then verified via simulations. This paper's major contributions are the proposal of an ML/MF method for measurement extraction of streaking targets, confirmation that this method achieves the best accuracy possible for realistic FPA sensors, i.e., it attains the CRLB, the introduction of a statistically supported definition of SNR for these measurements, and an evaluation of the target measurement detection performance. Furthermore, this paper shows that, given our MFSNR definition, the streak length and direction of motion in the FPA have a negligible effect on performance compared to the SNR where we show that with a 4-dB change, the detection performance increases dramatically.