Bias estimation is a significant problem in target tracking applications and passive sensors present additional challenges in this field. Biases in passive sensors are commonly represented as unknown rotations of the sensor coordinate frame and it is necessary to correct for such errors. Many methods have used simultaneous target state and bias estimation to register the sensors, however it may be advantageous to decouple state and bias estimation to simplify the estimation problem. This way bias estimation can be done for any arbitrary target motion. If measurements are converted into Cartesian coordinates and differenced then it is possible to isolate the effects of the biases. This bias pseudo-measurement approach has been used in bias estimation for many types of biases and sensors and this paper applies this method to 3D passive sensors with rotational biases. The Cram´er-Rao Lower Bound for the bias estimates is evaluated and it is shown to be attained, i.e., the bias estimates are statistically efficient.
This work describes a method for measurement extraction of fast point targets leaving an extended signature in the pixelated focal plane array of an EO sensor. The extraction method and subsequent statistics are derived from a physics based model where the spatial quantization regions, or pixels, are separated by dead zones. Furthermore, the intensity in a pixel is corrupted by approximately Gaussian noise which exhibits a variance proportional to the pixel area. This noise model is based on a Poisson assumption, in reference to the number of photons that contribute to the noise intensity. The signal portion of the image is a spatially quantized version of the target’s point spread function (PSF) that is modeled as a Gaussian PSF that moves at a constant velocity, a realistic assumption during an exposure time that may have been adjusted (and lengthened) to enhance target SNR. The measurement extraction is done using a Maximum Likelihood (ML) method for which we provide an appropriate Cramer-Rao Lower Bound (CRLB) on the estimation error of the target’s 2-D starting and ending positions. We then provide a definition for the signal to noise ratio (SNR) in an image using a matched filter (MF). Next, we present Monte Carlo simulations to confirm the derived results and find that the measurement extractor is efficient for SNRs .≥ 12dB (using our SNR definition). Finally, we develop a solution to the problem of detecting fast targets in images. We present approximate distributions for the test statistic under the null (H0 — target absent) and alternative (H1 — target present) hypotheses that can be used to set a threshold for specific probabilities of detection P<sub>D</sub> and false alarm P<sub>FA</sub>. Finally, we verify these distributions with Monte Carlo simulations.
For a thrusting/ballistic target, works have shown that a single fixed sensor with 2-D angle-only measurements (azimuth and elevation angles) is able to estimate the target’s 3-D trajectory. In previous works, the measure- ments have been considered as starting either from the launch point or with a delayed acquisition. In the latter case, the target is in flight and thrusting. The present work solves the estimation problem of a target with delayed acquisition after burn-out time (BoT), i.e. in the ballistic stage. This is done with a 7-D parameter vector (velocity vector azimuth angle and elevation angle, drag coefficient, 3-D acquisition position and target speed at the acquisition time) assuming noiseless motion. The Fisher Information Matrix (FIM) is evaluated to prove the observability numerically. The Maximum Likelihood (ML) estimator is used for the motion parameter estimation at acquisition time. The impact point prediction (IPP) is then carried out with the ML estimate. Simulation results from the scenarios considered illustrate that the MLE is efficient.
This paper considers the problem of estimating the 3D states of a salvo of thrusting/ballistic endo-atmospheric objects using 2D Cartesian measurements from the focal plane array (FPA) of a single fixed optical sensor. Since the initial separations in the FPA are smaller than the resolution of the sensor, this results in merged measurements in the FPA, compounding the usual false-alarm and missed-detection uncertainty. We present a two-step methodology. First, we assume a Wiener process acceleration (WPA) model for the motion of the images of the projectiles in the optical sensor’s FPA. We model the merged measurements with increased variance, and thence employ a multi-Bernoulli (MB) filter using the 2D measurements in the FPA. Second, using the set of associated measurements for each confirmed MB track, we formulate a parameter estimation problem, whose maximum likelihood estimate can be obtained via numerical search and can be used for impact point prediction. Simulation results illustrate the performance of the proposed method.