David Frederic Crouse

Developing a Real-Time Track Display That Operators Do Not Hate

We formulate a method of estimating target states that minimizes the mean optimal subpattern assignment (MOSPA) metric, applied suboptimally to a multi-hypothesis tracker (MHT) and optimally to a particle filter. This gives the operator a display of the targets with reduced jitter and track switching.

Basic tracking using nonlinear 3D monostatic and bistatic measurements

The deleterious effects of atmospheric refraction are often overlooked in work on target tracking. This tutorial shows how the effects of refraction can be directly incorporated into tracking algorithms, improving the performance of tracking algorithms that make use of monostatic and bistatic measurements. Additionally, a technique for converting measurements from the radars refraction-corrupted local coordinate system (typically bistatic r - u - v coordinates) into Cartesian coordinates is presented. The refraction-compensation algorithms can be used with arbitrary refraction models for which ray tracing techniques for solving boundary value problems and initial value problems are available, though extensions to more complicated propagation scenarios are possible. The algorithms are run on a simple exponential refraction model to demonstrate their effectiveness. This tutorial builds upon the tutorial entitled “Basic Tracking Using Nonlinear 3D Monostatic and Bistatic Measurements” and is complemented by the companion tutorial “Basic Tracking Using Nonlinear Continuous-Time Dynamic Models.”

Remark on algorithm 515: Generation of a vector from the lexicographical index combinations

We present a correction to Algorithm 515 [Buckles and Lybanon 1977].

An approximate Minimum MOSPA estimator

Optimizing over a variant of the Mean Optimal Subpattern Assignment (MOSPA) metric is equivalent to optimizing over the track accuracy statistic often used in target tracking benchmarks. Past work has shown how obtaining a Minimum MOSPA (MMOSPA) estimate for target locations from a Probability Density Function (PDF) outperforms more traditional methods (e.g. maximum likelihood (ML) or Minimum Mean Squared Error (MMSE) estimates) with regard to track accuracy metrics. In this paper, we derive an approximation to the MMOSPA estimator in the two-target case, which is generally very complicated, based on minimizing a Bhattacharyya-like bound. It has a particularly nice form for Gaussian mixtures. We thence compare the new estimator to that obtained from using the MMSE and the optimal MMOSPA estimators.

Approaches to Cartesian Data Association Passive Radar Tracking in a DAB/DVB Network

We present a target tracking system for passive radar suitable for a digital audio/video broadcast (DAB/DVB) network with orthogonal frequency division multiplexing (OFDM) illuminators of opportunity. In this system bistatic range and range-rate are available as a measurement while angular information is assumed unavailable or of very poor quality. There is also a novel association ambiguity between the signals and illuminators, this in addition to the usual one between the signals and targets, and hence an interesting data association problem. Our intention is to provide tracks directly in the geographic space (3-D Cartesian domain). Here, four filters are suggested. The first two are the extended Kalman filter (EKF) and unscented Kalman filter (UKF) based on a joint probabilistic data association (JPDA) modified to incorporate the additional assignment complexity. The other two filters are a bootstrap particle filter (BPF) and an auxiliary particle filter (APF) based on the data association under the probabilistic multi-hypothesis tracker (PMHT) measurement model, which means that each measurements assignment is independent of all others. Simulation results show that these four filters can work properly, albeit naturally with different performance characteristics and complexity.

Aspects of MMOSPA estimation

We expand upon existing literature regarding using Minimum Mean Optimal Sub-Pattern Assignment (MMOSPA) estimates in multitarget tracking, noting its advantages in comparison to Maximum Likelihood (ML) and Minimum Mean Squared Error (MMSE) estimation, and look at the practical computation of MMOSPA estimates. We demonstrate the use of MMOSPA estimation in a two-target tracking scenario as well as outside of tracking in a radar angular superresolution scenario.

A Low-Complexity Sliding-Window Kalman FIR Smoother for Discrete-Time Models

The information filter is a form of the Kalman filter that, in many of its realizations, allows optimal, unbiased, recursive state estimation without an initial state estimate. We review a number of forms of the information filter. We then derive the coefficients for the sliding-window Kalman finite impulse response (FIR) smoother (also known as a receding or moving horizon Kalman FIR smoother) starting from the equations for the information filter. The resulting FIR smoother has a simple, recursive form for calculating the coefficients, allowing them to be calculated with O(N) complexity versus the O(N 2) to O(N 3) complexity of previous approaches, where N is the length of the batch. It also allows for a control input, something not present in previous algorithms. This method is only limited in the assumption that the state transition matrix is invertible, which, however, is satisfied in most practical problems.

Optimal Memoryless Relays With Noncoherent Modulation

We derive optimal memoryless relays for the case of noncoherent modulation over additive white Gaussian noise (AWGN) channels with or without fading. The derivation is flexible, as it can be applied to any binary hypothesis observed at the relay. We investigate several channels, including random phase and fading, and apply different modulation schemes, namely on-off-keying (OOK) and orthogonal frequency-shift-keying (FSK). We find that at low signal-to-noise ratios (SNR) the relay censors its observation, as it only transmits at nonzero energy if the observations seem reliable. Compared to the known results that optimal memoryless relays for the case of coherent binary-phase-shift-keying (BPSK) are combinations of soft-information and a hard-limiter, the noncoherent relays have considerably less emphasis on soft-information and converge much faster to the hard-limiter.

Advances in displaying uncertain estimates of multiple targets

Both maximum likelihood estimation as well as minimum mean optimal subpattern assignment (MMOSPA) estimation have been shown to provide meaningful estimates in instances of target identity uncertainty when the number of targets present is known. Maximum likelihood measurement to track association (2D assignment) has been widely studied and is reviewed in this paper. However, it is widely believed that approximate MMOSPA estimation can not be performed in real time except when considering a very small number of targets. This paper demonstrates the MMOSPA estimator arises as a special case of a minimum mean Wasserstein metric estimator when the number of targets is unknown. Additionally, it is shown that approximate MMOSPA estimates can be calculated in microseconds to miliseconds without extensive optimization, making MMOSPA estimation a practicable alternative to more traditional estimators.

A time-shift model for OFDM radar

Active and passive detection using CP-OFDM modulated signals traditionally assume that targets do not move significantly during the entire observation period. In this paper, a “time-shift” model that compensates for the motion of targets between pulses during the observation period is introduced. Matched filtering based upon this model can be efficiently performed using a combination of the chirp-z transform and the fast Fourier transform algorithms. The resulting delay-Doppler plots are less “smeary” in the range axis than plots generated when using the traditional techniques over extended observation intervals, allowing better resolution of closely spaced targets.