## Doctoral Dissertations

6-15-2018

#### Embargo Period

6-13-2018

Prof. Yaakov Bar-Shalom

Prof. Peter Willett

Prof. Krishna Pattipati

#### Field of Study

Electrical Engineering

#### Degree

Doctor of Philosophy

Open Access

#### Abstract

Target tracking has been under extensive investigation for the past several decades. It has wide applications, such as defense, computer vision, robotics, etc. Optical sensors are widely used in tracking systems. In this dissertation, several issues related to statistically efficient target tracking with a single stationary optical sensor will be explored. The first part of the dissertation is {\it The Multidimensional Cramer-Rao-Leibniz Lower Bound for Likelihood Functions with Parameter-Dependent Support"}. This work derived the multidimensional Cram\'{e}r-Rao-Leibniz lower bound (CRLLB), which is a generalization of the Cram\'{e}r-Rao lower bound (CRLB) when the likelihood function (LF) has parameter-dependent support and it is not continuous at its boundary, by applying the general Leibniz integral rule. The regularity condition for the CRLLB to hold with equality is the \textit{generalized collinearity condition} between the gradient of log-likelihood function (LLF) w.r.t. the parameter vector to be estimated and the estimation error, in which case the LF has to belong to the exponential family. In addition, the CRLLBs for the unbiased estimators for measurement noise with some typical distributions have been investigated. The second part of the dissertation is {\it Tracking initially unresolved thrusting/ballistic objects using a single stationary optical sensor"}. This work considered the problem of tracking a salvo of thrusting objects in 3D using 2D measurements from an optical sensor's focal plane array (FPA). As the initial separations of the objects were too close to be resolved by the sensor, this resulted in initially unresolved measurements and a merged-measurement model with increased variance was utilized. As the trajectory of each thrusting object was assumed to be determined uniquely by a parameter vector, a two-stage method was employed to solve the problem. In the first stage, the recently proposed multi-Bernoulli (MB) filter with Wiener process acceleration (WPA) motion model was applied using 2D measurements in the FPA to do measurement-to-track associations, based on which a set of measurements were extracted for each confirmed track at the end of the observation time. Then, in the second stage, the parameter vector defining the trajectory of each object was estimated based on the associated measurements via numerical search and we reconstructed the whole 3D trajectory for each object using the estimated parameter vector for the purpose of impact point prediction (IPP). The last part of the dissertation is {\it Measurement extraction for a point target in an optical sensor's focal plane array"}. This work considered the measurement extraction for a point target from an optical sensor's FPA with a dead zone separating neighboring pixels. Assuming that the energy density of the target deposited in the FPA conforms to a Gaussian point spread function (PSF) and that the noise mean and variance in each pixel are proportional to the pixel area (i.e., according to a Poisson noise model), we derived the CRLB for the covariance of the estimated target location. Following this, the target detection was explored. We have also investigated the effect of pixel size and signal-to-noise ratio (SNR) on the estimation and detection performances.

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