Raman spectroscopy has received a great deal of attention in recent years in the chemical and biological detection
research community because of its unique ability to determine the chemical composition of substances. This has led to
development of fast and numerically efficient algorithms for Raman spectra estimation. There are two types of
algorithms for Raman spectra estimation, namely supervised and unsupervised. In the supervised approach, a number of
reference spectra for known chemicals is used. It is also assumed that the measured spectra of one or more unknown
substances belong to one of the individual substances in the reference library, or that they originate from a linear
combination of a number of reference spectra. The mixing coefficients for a measured spectrum are often estimated
using the nonnegative least squares (NNLS) or nonnegative weighted least squares (NNWLS) algorithms. This is a
constrained parameter estimation problem due to the inherent nonnegativity of the mixing coefficients.
Some previous researchers have used the NNLS method, in which no weight matrix is used, or all measurement error
variances are treated as equal. In our Fusion 2009 paper, we found that the measurement error variances or weights
vary significantly with the wavenumber and that it is therefore necessary to use non-uniform weights in parameter
estimation. Previously we used the true weights and have done limited study using estimated weights. In this paper, we
perform extensive study for Raman spectra estimation using WLS and NNWLS for one, two, and three chemicals, using
simulated data and Monte Carlo simulations.
Raman spectroscopy is a powerful technique for determining the chemical composition of a substance. Our objective
is to determine the chemical composition of an unknown substance given a reference library of Raman spectra. The
unknown spectrum is expressed as a linear combination of the reference library spectra and the non-zero mixing
coefficients represent the presence of individual substances, which are not known. This approach is known as the
supervised learning method. The mixing coefficients are usually estimated using the nonnegative least squares (NNLS)
or nonnegative weighted least squares (NNWLS). This problem is a constrained estimation problem due to the presence
of the nonnegativity constraint. In this paper, we present a swarm based algorithm, the particle swarm optimization
(PSO), to estimate the mixing coefficients and Raman spectra. The PSO is used to determine the mixing coefficients.
PSO efficiently finds an optimum solution. Results are presented for simulated data obtained from the Jennifer Kelly
Raman spectra library. The reference library consists of Raman spectra for nine minerals and the measured spectrum is
simulated by using spectrum/spectra of single/multiple minerals. We compare the root mean square error (RMSE) for
parameter estimation and measurement residual and computational time of the NNWLS and nonnegative weighted PSO
(NNWPSO) algorithms.
KEYWORDS: Video, Unmanned aerial vehicles, Detection and tracking algorithms, Information fusion, Cameras, Video surveillance, Video acceleration, Video processing, Sensors, Monte Carlo methods
Surveillance and ground target tracking using multiple electro-optical and infrared video sensors onboard
unmanned aerial vehicles (UAVs) has drawn a great deal of interest in recent years. We compare a number of track-to-track
fusion algorithms using a single target with the nearly constant velocity dynamic model and two UAVs. A local
tracker is associated with each UAV and processes video measurements to produce local tracks. The video measurement
is the centroid pixel location in the digital image corresponding to the target positions on the ground. In order to handle
arbitrary height variations, we use the perspective transformation for the video measurement model. In addition, the
video measurement model also includes radial and tangential lens distortions, scale, and offset. Since the video
measurement model is a nonlinear function of the target position, the tracking filter uses a nonlinear filtering algorithm.
A fusion center fuses track data received from two local trackers. The track-to-track fusion algorithms employed by the
fusion center include the simple convex combination fusion, Bhattacharya fusion, Bar-Shalom-Campo fusion, and
extended information filter based fusion algorithms. We compare the fusion accuracy, covariance consistency, bias in
the fused estimate, communication load requirements, and scalability. Numerical results are presented using simulated
data.
Video cameras onboard multiple unmanned aerial vehicles (UAVs) can provide effective and inexpensive tracking
and surveillance functions for ground targets. In our previous work, we quantified the degree of nonlinearity (DoN) of
the video filtering problem by considering the perspective transformation for the video measurement model and
constant velocity motion for the target dynamic model. In this paper, we generalize the formulation by using a more
realistic video measurement model which is based on the perspective transformation, radial and tangential lens
distortions, scale, offset, and skew. The centroid pixel coordinates of a target in the digital image represent the sensor
measurement for this model. This measurement model is commonly used in photogrammetry, computer vision, and
video tracking, where significant height variation can occur.
Since the measurement model is a nonlinear function of the target state, the filtering problem is nonlinear. We
quantify the DoN of the video filtering problem by calculating the differential geometry based parameter-effects
curvature and intrinsic curvature. These measures help a filter designer to select an appropriate nonlinear filtering
algorithm for the video filtering problem so that tracking accuracy and computational load requirements are satisfied.
Our results show that the DoN of the video filtering problem is quite low and hence a computationally simple filter such
as the extended Kalman filter (EKF) is a better choice than the particle filter (PF) which has a much higher
computational cost. The state estimation accuracies of the EKF and PF are nearly the same.
KEYWORDS: Video, Unmanned aerial vehicles, Cameras, Sensors, Detection and tracking algorithms, Video surveillance, Video acceleration, Video processing, Infrared sensors, Monte Carlo methods
Surveillance and ground target tracking using multiple
electro-optical and infrared video sensors onboard
unmanned aerial vehicles (UAVs) have drawn a great deal of interest in recent years due to inexpensive video sensors
and sensor platforms. In this paper, we compare the convex combination fusion algorithm with the centralized fusion
algorithm using a single target and two UAVs. The local tracker for each UAV processes pixel location measurements
in the digital image corresponding to the target location on the ground. The video measurement model is based on the
perspective transformation and therefore is a nonlinear function of the target position. The measurement model also
includes the radial and tangential lens distortions. Each local tracker and the central tracker use an extended Kalman
filter with the nearly constant velocity dynamic model.
We present numerical results using simulated data from two UAVs with varying levels of process noise power
spectral density and pixel location standard deviations. Our results show that the two fusion algorithms are unbiased and
the mean square error (MSE) of the convex combination fusion algorithm is close to the MSE of the centralized fusion
algorithm. The covariance calculated by the centralized fusion algorithm is close to the MSE and is consistent for most
measurement times. However, the covariance calculated by the convex combination fusion algorithm is lower than the
MSE due to neglect of the common process noise and is not consistent with the estimation errors.
In our previous work, we presented an algorithm to quantify the degree of nonlinearity of nonlinear filtering problems
with linear dynamic models and nonlinear measurement models. A quantitative measure of the degree of nonlinearity was
formulated using differential geometry measures of nonlinearity, the parameter-effects curvature and intrinsic curvature.
We presented numerical results for a number of practical nonlinear filtering problems of interest such as the bearing-only
filtering, ground moving target indicator filtering, and video filtering problems. In this paper, we present an algorithm to
compute the degree of nonlinearity of a nonlinear filtering problem with a nonlinear dynamic model and a linear measurement
model. This situation arises for the bearing-only filtering problem with modified polar coordinates and log polar
coordinates. We present numerical results using simulated data.
The GMTI radar sensor plays an important role in surveillance and precision tracking of ground moving targets. A class of GMTI sensors which employs a linear antenna measures the range, path difference between the received beams, and range-rate. The path difference is equivalent to the cone angle between the axis of the antenna and the radar line-of-sight. The measurement errors for the range, cone angle, and range-rate are independent. The measurements for the conventional GMTI measurement model are range, azimuth, and range-rate. The azimuth is a derived measurement obtained from the range and cone angle measurements. Therefore, the errors in the range and azimuth are correlated. However, the conventional GMTI measurement model ignores this correlation. We derive an analytic expression for the cross-covariance between the range and azimuth errors and show that the cross-covariance is inversely proportional to the ground-range. Thus for a stand-off GMTI sensor, the approximation used in neglecting the cross-covariance is reasonable.
We present a new algorithm for the geolocation of the target using the maximum likelihood estimator and range, cone angle, and surface height measurements. Along-track, cross-track, and vertical errors in the sensor position and errors in the antenna orientation are taken into account. We use a flat Earth approximation. An initial estimate of the target state and associated covariance are required in a tracking filter using the first GMTI report. We present an extended Kalman filter based algorithm for GMTI track initialization using the GMTI geolocation results. Numerical results are presented using simulated data.
Tracking people and vehicles in an urban environment using video cameras onboard unmanned aerial vehicles has drawn
a great deal of interest in recent years due to their low cost compared with expensive radar systems. Video cameras
onboard a number of small UAVs can provide inexpensive, effective, and highly flexible airborne intelligence,
surveillance and reconnaissance as well as situational awareness functions. The perspective transformation is a
commonly used general measurement model for the video camera when the variation in terrain height in the object scene
is not negligible and the distance between the camera and the scene is not large. The perspective transformation is a
nonlinear function of the object position. Most video tracking applications use a nearly constant velocity model
(NCVM) of the target in the local horizontal plane. The filtering problem is nonlinear due to nonlinearity in the
measurement model.
In this paper, we present algorithms for quantifying the degree of nonlinearity (DoN) by calculating the differential
geometry based parameter-effects curvature and intrinsic curvature measures of nonlinearity for the video tracking
problem. We use the constant velocity model (CVM) of a target in 2D with simulated video measurements in the image
plane. We have presented preliminary results using 200 Monte Carlo simulations and future work will focus on detailed
numerical results. Our results for the chosen video tracking problem indicate that the DoN is low and therefore, we
expect the extended Kalman filter to be reasonable choice.
KEYWORDS: Error analysis, Monte Carlo methods, Time metrology, Linear filtering, Gaussian filters, Digital filtering, Velocity measurements, Smoothing, Computer simulations, Optimal filtering
Out-of-sequence measurement (OOSM) filtering algorithms have drawn a great deal of attention during the last few years. A number of multiple-lag OOSM filtering algorithms exists in the research literature. Only one of the OOSM filtering algorithms is optimal and remaining algorithms are suboptimal even for the linear dynamics and linear measurement models with additive Gaussian noises. A general feature of each OOSM filtering algorithm is that the algorithm calculates optimally or sub-optimally, the smoothed or retrodicted state estimate, associated covariance, and cross-covariance between the state and the measurement at the OOSM time. The existing optimal OOSM algorithm calculates these three quantities using a forward recursive algorithm. In this paper, we show that the OOSM filtering problem can be solved optimally using a generalized smoothing or retrodiction framework for the linear dynamics and linear measurement models with additive Gaussian noises. We develop a new optimal smoothing based OOSM filtering algorithm which uses the Rauch-Tung-Streibel (RTS) fixed-interval optimal backward smoother. We present numerical results using simulated data which includes two-dimensional position and velocity measurements and analyze the performance of the algorithm using Monte Carlo simulations.
The Multiple Hypotheses Tracking (MHT) algorithm has been shown to have the best tracking performance among existing multi-target tracking algorithms using real world sensors with probability of detection less than unity and in the presence of false alarms. The improved performance of the Multiple Hypotheses Tracking comes at the cost of signicantly higher computational complexity. Most Multiple Hypotheses Tracking implementations only form the best global hypothesis. This paper compares the Linear Multitarget Integrated Track Splitting (LMITS) tracking algorithm with the Multiple Hypotheses Tracking algorithm. LMITS has a simpler structure than Multiple Hypotheses Tracking as it decouples local hypotheses and avoids the measurement to multi-track allocation entirely. The number of LMITS hypotheses equals the sum of the number of local hypotheses added to the number of initiation hypotheses. Thus LMITS can retain a deeper hypotheses subtree which can result in better performance. We compare tracking performances of LMITS and MHT algorithms using simulated data for multiple maneuvering targets in heavy and non-uniform clutter.
The bearing-only tracking problem arises in many radar and sonar tracking applications. Since the bearing measurement model is a nonlinear function of the target state, the filtering problem is nonlinear in nature. A great deal of attention has been focused on this problem due to the difficulty posed by the so-called high degree of nonlinearity (DoN) in the problem. However, a quantitative measure of the DoN is not calculated in previous works. It has been observed that the extended Kalman filter (EKF) in which the state vector consists of the Cartesian components of position and velocity is unstable and diverges in some cases. The range parametrized EKF (RPEKF) and particle filter (PF) have been shown to produce improved estimates for the bearing-only tracking problem. In this paper, we calculate two measures of nonlinearity, (1) the parameter-effects curvature and (2) intrinsic curvature for the bearing-only tracking problem using the differential geometry measures of nonlinearity. We present numerical results using simulated data for the constant velocity motion of a target in 2D with bearing-only measurements where the sensor platform uses a higher order motion than the target to achieve observability. We analyze the DoN by varying the distance between the target and sensor.
KEYWORDS: Sensors, Monte Carlo methods, Detection and tracking algorithms, Data fusion, Data communications, Error analysis, Telecommunications, Computer simulations, Data processing, Solids
Most practical multi-platform data fusion systems use the distributed tracking architecture where each sensor platform has its own local tracker. A local tracker performs tracking using measurements from one or more sensors and sends its track data to a central fusion system. When the track data from a local tracker is transmitted to the central fusion system using a communication network, the track data can arrive out-of-sequence due to random delays in the communication network and different processing times at local trackers. Track-to-track fusion using the equivalent decorrelated pseudo-measurement approach is an efficient algorithm for the distributed tracking problem. In this paper, we use an existing multiple-lag out-of-sequence measurement (OOSM) algorithm and the decorrelated pseudo-measurement approach for track-to-track fusion of out-of-sequence track (OOST) data. We present numerical results using simulated data for a scenario where a global tracker processes track data from two local trackers. Each local tracker processes two-dimensional position and velocity measurements from a single sensor. We use Monte Carlo simulations to evaluate the performance of the algorithm.
Out-of-sequence measurements (OOSMs) frequently arise in a multi-platform central tracking system due to delays in communication networks and varying pre-processing times at the sensor platforms. During the last few years, multiple-lag OOSM filtering algorithms have received a great deal of attention. However, a comparative analysis of these algorithms for multiple OOSMs is lacking. This paper analyzes a number of multiple-lag OOSM filtering algorithms in terms of optimality, accuracy, statistical consistency, and computational speed. These factors are important for realistic multi-target multi-sensor tracking systems. We examine the performance of these algorithms using a number of examples with Monte Carlo simulations. We present numerical results using simulated data, which includes two-dimensional position and velocity measurements.
Tracking using the ground moving target indicator (GMTI) sensor measurements plays an important role in situation awareness of the battlefield, surveillance, and precision tracking of ground moving targets. The GMTI sensor measurements range, azimuth, and range-rate are nonlinear functions of the target state. The extended Kalman filter (EKF) is widely used to solve the GMTI filtering problem. Since the GMTI measurement model is nonlinear, the use of an EKF is sub-optimal. The sub-optimality depends on the degree of nonlinearity of the measurement function and GMTI measurement error covariance. We can convert polar measurements range and azimuth to Cartesian measurements and approximately treat the range-rate as a linear function of the target velocity by considering the radar line-of-sight (RLOS) vector as a constant. This allows the use linear Kalman filter (KF) with linearized measurements in an approximate way. The unscented Kalman filter (UKF) and particle filter (PF) have been shown recently as robust alternate algorithms for a wide range of nonlinear estimation problems. This paper compares the performance of the KF with linearized measurements, EKF, iterated EKF (IEKF), UKF, and PF for the GMTI measurement filtering problem using a wide range of operating conditions. Estimation accuracy, statistical consistency, and computational speed and storage are used to evaluate the performance of these estimators. We use Monte-Carlo simulations and calculate the average mean square error (MSE) matrix, normalized estimation error squared (NEES), and normalized innovation squared (NIS) to analyze the accuracy and statistical consistency.
KEYWORDS: Sensors, Nonlinear filtering, Detection and tracking algorithms, Electronic filtering, Linear filtering, Monte Carlo methods, Error analysis, Telecommunications, Radar, Kinematics
Measurements can arrive out-of-sequence at a central tracker due to varying data pre-processing times and communication delays in a multi-sensor target tracking system. A number of single-lag and multiple-lag out-of-sequence measurement (OOSM) filtering algorithms for the linear filtering problem are known in the research literature. In this paper, we present a multiple-lag nonlinear OOSM filtering algorithm based on an extension of the existing multiple-lag linear OOSM filtering algorithm Ground target tracking using multiple airborne ground moving target indicator (GMTI) radar sensors is an important problem in surveillance and precision tracking of ground moving targets. Sensor geometry with two nearly orthogonal GMTI sensors can significantly improve the position measurement accuracy with fast revisit times due to the narrow elliptical nature of the range and cross-range measurement error covariance matrix of a single sensor. We present numerical results for the multiple-lag nonlinear OOSM filtering algorithm using simulated GMTI measurements with nearly constant velocity motion in two dimensions. Our numerical results show that the results from the nonlinear OOSM algorithm are in close agreement with those obtained from the EKF using time-ordered GMTI measurements.
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