KEYWORDS: Sensors, Statistical analysis, Stochastic processes, Error analysis, Time metrology, Electroluminescence, Systems modeling, Signal processing, Data processing, Detection and tracking algorithms
The problem of track-to-track association of local tracks from two disparate and dispersed sensor systems is
considered. Typical approaches to this problem base the association upon the estimated target states. Bias, pointing,
navigation and location errors among others often frustrate theses approaches and result in higher association error
probabilities. Realizing that the state estimate represents on the first order statistics of the target trajectory, this paper
augments those approaches with an association test based upon the second order statistics of the measurements. It is shown
that in general, the cross-covariance of the measurements from two disparate and dispersed sensor systems will be nonzero if
the measurements are from tracks on the same target. If the tracks are on different targets then the measurement crosscovariance
will be zero. A test of the null hypothesis that the measurement cross-covariance is zero is derived and an
implementation using sample statistics is developed. The probability density function of the test statistic is presented so that
the test result can be combined with the association result based upon the estimated track state. Both absolute and relative
tests are discussed. The effect of track length is analyzed and then examined in an example.
KEYWORDS: Kinematics, Signal processing, Monte Carlo methods, Electroluminescence, Detection and tracking algorithms, Target detection, Data processing, Failure analysis, Radon, Error analysis
In an effort to improve the probability of correctly associating tracks and observations, features, which are physical properties of the target other than kinematics, are included in the association process. Unlike their kinematic counter parts, the probability distributions of the features are typically not known for all of the objects involved. This precludes the use of the parametric hypothesis tests typically used with the kinematic data to perform association. One possible solution to this problem is to assume a probability distribution for each of the features and use it in the conventional parametric test used for association. The risk is that the wrong probability distributions will be assumed and the association error probability will increase. An alternative approach is to use the feature data in a non-parametric test, a type of test that requires little or no knowledge of the probability distribution of the data. The result of the non-parametric test of the feature data is then combined with the result of the conventional parametric test of the kinematic data. As the title suggests, this paper compares the performance of these two approaches for several sets of conditions. First, since the parametric test of the kinematic data assumes the data to be Gaussian distributed, the features are drawn initially from Gaussian populations. These Gaussian distributed features are used to test both approaches and their performance curves are compared and analyzed. This process is then repeated for three non-Gaussian feature distributions. Two of these distributions belong to the same exponential family of distributions as the Gaussian Distribution however, both have heavier tails and one is a one-sided distribution. The third feature distribution used in the comparison has finite support and the experiment is designed so that perfect performance is possible. Realizing that the association error probability is not zero, the foregoing evaluations are repeated with misassociations present in the data.
KEYWORDS: Sensors, Filtering (signal processing), Optical filters, Signal to noise ratio, Digital filtering, Infrared search and track, Signal detection, Interference (communication), Target detection, Signal processing
Track features help reduce the number of false associations during tracking. It is expected that the spectral signature vector from a target would tend to be consistent over short periods of time. Therefore, the spectral signature vector direction is a potential track feature candidate. The target's spectral signature and covariance are measured from the data at the output of the spatial-spectral match filter. The spectral vector feature is not independent from the local signal to clutter plus noise ratio (S(C+N)R), at the output of the anomaly detector which is often also used as a track feature. A correction term is introduced in this paper to account for the correlation between these two features. Results from field data collections using a multi-spectral Infra-Red Search and Track System (IRST) are summarized with ROC curves showing the performance improvements achieved by using the unit spectral vector as a feature for both moving and stationary targets.
KEYWORDS: Error analysis, Monte Carlo methods, Signal to noise ratio, Electroluminescence, Signal processing, Data processing, Sensors, Analytical research
Association of observations and tracks is a fundamental component of most solutions to the tracking problem. Association is frequently formulated as a multiple hypothesis test. Typically, the test statistic, called the track score, is the likelihood or likelihood ratio of the observations conditioned upon the association hypotheses. Assuming that the test is reasonably efficient, further reduction in the association error probability necessitates the introduction of additional information into the track score. This additional information is embodied in quantities called track features which are to be included in the track score. In practice, the necessary conditional probabilities of the track features are unknown. The class of non-parametric hypothesis tests is designed to provide such a test in the absence of any probabilistic information about the data. However, the test statistics used in non-parametric tests cannot be used directly in the track score. The one probabilistic quantity generally available with non-parametric tests is the Type I error probability, the probability of failing to accept a true hypothesis. If the non-parametric test is distribution free then the Type I error probability is independent of the distribution of the track features. This paper presents a distribution free, non-parametric test of the track features that can be used to test the association hypotheses and a quantity that can be included in the track score is derived from the Type I error probability of the test.
KEYWORDS: Signal detection, Signal to noise ratio, Sensors, Electronic filtering, Interference (communication), Linear filtering, Signal processing, Gaussian filters, Target detection, Sensor performance
This paper shows that increasing SNR is not necessarily sufficient to improve ROC performance, that is, increase the probability of detection for a given false alarm probability. The general optimal detector is derived with no assumptions on the stationarity of the input noise process or the form of the input signal model.
KEYWORDS: Nonlinear filtering, Signal to noise ratio, Linear filtering, Signal detection, Electronic filtering, Optimal filtering, Receivers, Nonlinear optics, Digital filtering, Interference (communication)
It is well know that the matched filter is the optimal linear detection filter but, this allows for the possible existence of a nonlinear detection filter with better performance. This paper considers the class of nonlinear detection filters that are composed of a linear filter followed by an arbitrary point process. The result is general enough to include detection paradigm in which the signal model is not additive.
The ROC (receiver operating characteristic) curve of a general point process is analyzed. This analysis reveals that nonreversibility and not nonlinearity of the point process is responsible for the improvement of the ROC curve. That is, an reversible point process, either linear or nonlinear, leaves the ROC curve unchanged. However, a nonreversible point process will alter the ROC curve. This result is used to define a canonical ROC curve which is then utilized to derive the optimal point process. Several simple forms of the point process are considered first then the general optimal point process is derived. The technique is illustrated with several examples. Results for the special case of unimodal signal densities receive particular attention.
An extended Kalman filter was developed by Maybeck and Mercier [1] to track unresolved or point targets whose spatial signature in the point spread function of the sensor. An extension of this filter which takes into account the target shape and its variations with aspect angle will be developed and should offer improvement in several areas: performance against structured clutter; Maybeck considered only a white noise background; the errors and computations associated with segmentation are eliminated.
KEYWORDS: Target detection, Optical filters, Sensors, Signal processing, Signal detection, Point spread functions, Interference (communication), Linear filtering, Reflectivity, Signal to noise ratio
Typically, a tracker receives the position coordinates of the threshold exceedances from the detection process. The threshold nonlinearity serves to prevent superfluous data from entering the tracker; it also prevents other information about a detection from being used by the tracker. Track features were developed to provide a shunt for useful information around the detection threshold. Track features such as the measured C(C+N)R of a detection or local measures of clutter severity have been shown to significantly reduce track confirmation times and the probability of confirming a false track. This paper considers the development of track features for multispectral data. The multispectral track features are used in conjunction with available spatial and temporal track features. Ideally, multispectral track features would provide the tracker with information about how target-like the spectral signature of a detection is. Unfortunately, the spectral signature of the target is unknown a priori because of its dependence upon unmeasured environmental variables, uncertainties in factors effecting the emissivity and reflectivity of the target's surface and unknown operating history. This prevents the general development of a multispectral track features that provide target- likeness. The alternative, which is developed in this paper, is to use the consistency of the spectral signatures of the detections that form a track as a track feature. This multispectral track feature helps suppress the formation of tracks from random detections. It also inhibits a true track from branching to a false detection. Finally, it reduces the true track confirmation time.
The spectral signature of a target is typically unknown apriori because of its dependence upon environmental conditions (e.g., sun angle, atmospheric attenuation and scattering), factors effecting the reflectivity and emissivity of the target's surface (dirt, dust, water, paint, etc) and recent operating history (hot or cold engine, exhaust parts, wheels or tracks, etc.). Because of the high variability of the spectral signature of a target, multispectral detection typically detects spectral anomalies. For example, the canopy of a helicopter hovering in front of tree clutter may glint in the midwave infrared band while the reststrahlen spectral feature of the fuselage paint occurs in the longwave infrared band. Both of these are spectral anomalies relative to the tree clutter. If the target is slightly extended so that it subtends more than one pixel, the spectral anomalies by which the target may be detected will not be spatially collocated. This effectively lowers the ROC (receiver operating characteristic) curve of the detection process. This paper derives the ROC curves for several alternative solutions to this problem. One solution considers all possible spectral n-tuples within a small region. One of these n-tuples would likely contain all of the spectral anomalies of the target. Another solution is to apply a spatial maximum operator to each spectral band prior to the anomaly detector. This also combines all the spectral anomalies form the target into a single n-tuple. These methods have the potential to increase PD but an increase in PFA will also occur. The ROC curves of these solutions to the problem of detecting slightly extended targets are derived and compared to establish relative levels of performance.
This paper considers the problem of tracking dim unresolved ground targets and helicopters in heavy clutter with a ground based sensor. To detect dim targets the threshold must be set low which result in a large number of false alarms. The tracker typically uses the target dynamics to prevent the false tracks. The interesting aspect of this problem is that the targets may be or may become stationary. The tracks of stationary targets are difficult to discriminate from tracks formed by persistent false alarms.
The ability to detect and track dim unresolved targets in heavy clutter can be improved by the inclusion of the spectral dimension. Because of the great variation in targets, operating conditions and environments factors the spectral signature of the target is typically unknown. This paper present a fully adaptive matched filter and tracking paradigm which assumes no a priori information about the spectral signature of the target. It is shown that the full SCR gain can be realized in the absence of the spectral signature of the target. The ROC curve of the detector is used to show that performance loss due to the absence of spectral information is entirely due to an increase in the false alarm probability. This increase in PFA adversely effects tracker performance. The SCR track feature is developed to mitigate these effects. Track features provide an information shunt around the detection threshold nonlinearity that would otherwise block the flow of useful information to the tracker.
KEYWORDS: Sensors, Target detection, Signal processing, Infrared search and track, Filtering (signal processing), Electronic filtering, Data modeling, Infrared sensors, Atmospheric modeling, Signal to noise ratio
This paper describes an analytic model which generates a synthetic list of detection observations from an IRST. The observation list contains both false detects and target detections. The false detects are generated from a statistical model of the clutter and noise. The user is able to select from a menu of clutter types. This selection determines the values of the statistical parameters. The target type and trajectory are user specified. The target type is selected from a menu and determines the signature of the target. Both the target signature and clutter are propagated through the atmosphere and the sensor. The sensor is modeled as the cascade of transfer functions. The sensor model includes optics, detectors, electronics and noise sources. The signal processing which is part of the sensor model assumes a matched filter is used to increase the S(C + N)R prior to detection. The detection threshold is set to provide the user specified probability of false alarm. Each entry in the observation list includes the observation list includes the observation time, the angular position of the observation, the estimated S(C + N)R of the observation and the number of degrees of freedom which is a measure of clutter severity in the region of the observation. The model is intended to be used as part of a larger simulation for example in a sensor fusion study or to provide tracker test sequences for performance comparison and evaluation.
The detection of dim targets in heavy clutter requires large gains in the SCR. Gains of the required magnitude have been obtained with space-temporal processing. However, in many cases these gains are either difficult or expensive to realize. If the range to the clutter is small relative to the clutter velocity, the temporal processing will need to include scene registration and optical flow correction. Scene registration is computationally expensive especially for large search volumes. The correction of optical flow is both expensive and typically less than satisfactory. The spectral dimension provides an alternative to the temporal dimension. Since the data in each of the spectral bands is collected simultaneously or nearly so, the problems of registration and optical flow are eliminated. This paper considers the performance of the multi-spectral IR bands. Dual band performance results comparing space spectral processing with space temporal will be shown. An analytic model of the probability of false alarm as a function of the number of spectral bands is presented. A comparison of this model to experimental result using multi-spectral IRST data is given.
KEYWORDS: Detection and tracking algorithms, Target detection, Sensors, Signal to noise ratio, Filtering (signal processing), Performance modeling, Electronic filtering, Missiles, Signal processing, Digital filtering
Advanced track-after-detect (TAD) trackers are able to operate with detection thresholds as low as 9.5dB with the use of track features. At lower threshold the increased number of false alarms inhibits track confirmation. In order to track weaker targets, the target SNR must be increased prior to detection. Assuming that the SNR has been increased as much as possible through signal processing, further increase in SNR can be obtained by preceding the detection threshold with a track-before-detect algorithms. This paper analyzes the performance of the cascade of a TBD and a TAD tracking algorithm.
Fully adaptive matched filters typically can suppress clutter to the level of the sensor fixed pattern noise. A fully adaptive filter assumes that the clutter is a wide-sense stationary process which can be modeled by a constant means and unknown covariance function. Fixed pattern noise within a data sequence is unknown and tends to be a non-stationary process. As a result fixed pattern noise is minimally affected by fully adaptive filters. The signal processing philosophy for detecting unresolved targets is to enhance the target signal based on the sensor point spread function. When sensor fixed pattern noise exists, the signal from a point target can be significantly different from the sensor point spread function and can result in a loss in SCR. This SCR loss can make weak targets undetectable. This paper describes the effect of a fully adaptive filter on fixed pattern noise manifested as channel dependent bias and gain errors. Spectral analysis which quantifies the impact of these errors is presented. Experimental results on synthetic data and on real data from an infrared scanning sensor with channel dependent fixed pattern noise are given.
KEYWORDS: Image filtering, 3D acquisition, Electronic filtering, Point spread functions, Digital filtering, Linear filtering, Signal attenuation, Sensors, Gaussian filters, Optical filters
The velocity filter follows naturally from the derivation of the 3D (spatio-temporal) matched filter. It assumes constant velocity targets and that the spatial distribution of the targets remains unchanged. Typically the target velocity is unknown and a bank of velocity filters is implemented which covers the range of possible target velocities. Classically it is assumed that the gain of the velocity filter is proportional to the square root of the number of image frames filtered. As the number of frames is increased to enhance the filter gain, the width of the filter response in velocity space decreases. Consequently, more gain requires more filters to cover the same range of velocities. Each of the filters produces a set of false alarms. To maintain a constant false alarm rate, the detection threshold must be increased. The higher detection threshold reduces the probability of detection and offsets some of the gain achieved by the velocity filters. This performance trade is quantitatively analyzed.
KEYWORDS: Signal to noise ratio, Signal attenuation, Electronic filtering, Digital filtering, Interference (communication), Statistical analysis, Signal detection, Optimal filtering, Linear filtering, Signal analyzers
The matched filter is a common solution to the problem of detecting a known signal in noise. The matched filter is composed of the signal template to enhance the signal response and second order noise statistics to suppress the noise. The second order statistics of the noise are typically unknown. Fully adaptive implementations estimate these statistics from the noise present in the data to be filtered. If the signal is present, then it will be included in the estimate of the noise statistics used in the matched filter. Since these statistics are used by the matched filter to suppress noise, the signal will act to suppress itself, this is referred to as signal capture loss. In this paper an analytic model for signal capture loss is developed and experimentally verified. The use of the sample statistics to suppress the noise from which they are derived alters the noise rejection performance of the filter. Unlike the analysis of Reed et. al. which considers the use of the sample covariance to filter data which is independent of the sample covariance, the case of filtering the same data which was used to calculate the sample covariance is explicitly analyzed. This form of noise suppression is called self- whitening. The effect of self-whitening upon the noise rejection performance of the filter is analyzed and the results are verified experimentally. Signal capture loss and self-whitening are competing effects in terms of the number of samples used to form the sample covariance matrix. The output SNR includes both of these effects and is used to measure filter performance as a function of the number of samples. The output SNR performance is obtained by combining the results for signal capture loss with the self-whitening results. To obtain the performance of a fully adaptive filter relative to the optimal matched filter designed with the true population covariance, the results derived in this paper are combined with those of Reed et. al.
KEYWORDS: Signal processing, Target detection, Sensors, Filtering (signal processing), Data processing, Signal detection, 3D acquisition, Logic, Clouds, Electronic filtering
Long range detection and tracking of moving targets against clutter requires advanced signal and track processing techniques in order to exploit the ultimate capabilities of modern electro-optical sensors. These include three- dimensional filtering and multiple hypothesis tracking. Unfortunately, features present in real backgrounds can lead to false alarms which must be recognized in order to achieve a low false track rate. This paper describes one approach which was successful at mitigating clutter-induced false tracks while maintaining the low thresholds necessary for the detection of weak targets. This technique uses information derived in the signal processor describing the local background as additional discriminants in the track processor to identify false tracks caused by clutter leakage. We present an overview of the 3D signal track/processor, the false track mitigation methodology, and experimental results against real background data.
KEYWORDS: Signal to noise ratio, Sensors, Target detection, Statistical analysis, Infrared search and track, Filtering (signal processing), Signal processing, Data analysis, Detection and tracking algorithms, Logic
The heavy tailed false alarm density function of a common CFAR detector was previously derived. That density function was shown to be well approximated by a t-distribution with a reduced number of degrees of freedom. The number of degrees of freedom used in the approximate probability density function must be estimated form the data at the output of the clutter filter. Three estimators for the number of degrees of freedom have been developed. Their relative advantages and disadvantages are discussed. The most practical one is presented and its performance is analyzed. Experimental results on synthetic and real data are provided. The synthetic data is used to empirically test the bias of the estimator and to qualitatively evaluate its efficiency. The effectiveness of this estimator has been quantitatively demonstrated on ocean scenes with glint. The interest in knowing the false alarm density goes beyond just setting a CFAR threshold. This estimator is incorporated into an IRST signal processing and tracking algorithm suite containing a constant threshold. This density function together with the estimated number of degrees of freedom is used to adaptively estimate the probability of false alarm. Regions with a small number of degrees of freedom have a higher false alarm probability and consequently the tracker is more conservative in initiating tracks. The tracker uses the adaptive PFA to improve the logic which initiates, confirms and deletes tracks.
KEYWORDS: Signal to noise ratio, Statistical analysis, Sensors, Target detection, Gaussian filters, Linear filtering, Electronic filtering, Point spread functions, Detector development, Chemical elements
A CFAR detector commonly used for the detection of unresolved targets normalizes the background variance by dividing the detection filter output by the local sample standard deviation. A number of researchers have measured the experimental false alarm probability of this detector and found it to be higher than the probability predicted by a Gaussian density function. This is the case even when the filter output statistics are known to be Gaussian distributed. A number of attempts have been made to heuristically construct distributions which exhibit the heavy tails associated with the measured false alarm probability (e.g. sum of two Gaussian densities or the modified gamma density). This paper presents a first principle derivation of the detector false alarm density function based upon the assumption that the filter output is Gaussian distributed. The resulting false alarm density function is very nearly Gaussian out to about 3.5 standard deviations. Past 3.5 standard deviations the tails of the derived density function are markedly heavier than the corresponding Gaussian tails. The parameters of this new density function are easily estimated from the filter outputs. The analytic results are validated using a Chi-Square goodness-of-fit test and experimental measurements of the false alarm density.
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