A time sequence of counts of the number of sensor detections is the sum of the number of detections of objects and the number of false alarms. We model the count sequence as an insertion-deletion process, and model the time-varying number of objects as a birth-death process. Under these modeling assumptions, we derive the optimal recursive Bayesian posterior distribution for the number of objects conditioned only on the count sequence. The method is potentially applicable in management science to detect changes in demand in decisionindependent observed data streams and in social media to estimate the number of users who abuse hashtags. A maximum a posteriori (MAP) algorithm for estimating the parameters of the birth-death and insertion-deletion processes is presented.

Multi-object filtering is a generalisation of stochastic filtering to deal with an unknown and time-varying number of targets, largely based on modelling with point processes. Some early works on this topic from the Soviet Union from 1960s-l980s appeared prior to well known results in the contemporary liter-ature. This article reviews some of these historical contributions.

A multiple target tracking filter is developed for merged measurement problems that arise with finite resolution sensors. The resulting combinatorial problem is incorporated directly in the joint likelihood function using analytic combinatorics techniques. The Bayesian filter is a sum of several terms that correspond one-to-one to the set of all feasible hypotheses about measurements and targets, i.e., resolved/unresolved, detected/undetected. Performance is demonstrated for two targets in both crossing and parallel target motion scenarios.

Object superposition is a way to derive Bayesian estimators for multiple object tracking using point processes. A low computational complexity Bayesian multiple target tracking filter, based on target superposition, is presented. The concept of superposition is introduced and applied to the well-known Joint Probabilistic Data Association (JPDA) filter to derive the JPDA with superposition (JPDAS) filter. The JPDAS intensity function is evaluated to machine precision “for free” by computing the generating functional of the posterior process using complex arithmetic. A simulated example with eight targets is presented.

The method of analytic combinatorics and labeling is shown to be a unifying framework in which to pose both high and low level data fusion problems. The method uses labeled generating functions. Several examples from high level fusion and multiple target tracking are given. Examples from high level fusion include natural language processing and noisy graph association problems. Examples from multitarget tracking include multidimensional assignment problems, unlabeled and labeled JPDA, labeled multiBernoulli filters, and multihypothesis tracking.

The single-object Bayesian filter for an interval, or batch, of data is extended to the multiple object case using the method of analytic combinatorics. The exact expression for the probability generating functional of the Bayes posterior process is derived. It is a nested composition of functions and functionals that is evaluated via a backward recursion. Branching and immigration processes are used to model the initial multiple object process and new object arrival processes, respectively. The exact Bayes posterior distribution and various summary statistics of the interval filter are derivatives of the generating functional. These derivatives are written in equivalent Cauchy integral form and approximated using the saddle point method.

Probabilistic multi-hypothesis tracking (PMHT) is an algorithm for tracking multiple targets when measurement-to- target assignments are unknown and must be jointly estimated with the target tracks. Multi-frame assignment PMHT (MF- PMHT) is an algorithm designed to mitigate some performance problems associated with PMHT. In MF-PMHT, the PMHT algorithm is applied to multi-frame sequences in the last L frames of data and considers the set of all possible measurement sequences. While effective in improving tracking performance compared to PMHT, performance of the original MF-PMHT degrades when the target single-frame detection probability is non-unity. This is because missed detections are not considered in the multi-frame sequences. A new MF-PMHT implementation is derived in this paper which explicitly considers missed detections in the multi-frame sequences. Performance of this MF-PMHT is compared to the original MF-PMHT algorithm as well as to a Homothetic PMHT. Simulation results indicate that the new MF- PMHT algorithm performs the same as the original algorithm when there are no missed detections and also performs better than the alternative algorithms considered when there are missed detections.

An alternating directions method is presented for joint maximum a posteriori estimation of target track and sensor field using bistatic range data. The algorithm cycles over two sub-algorithms: one improves the target state estimate conditioned on sensor field state, and the other improves the sensor field state estimate conditioned on target state. Nonlinearities in the sub-algorithms are mitigated by decomposing their likelihood functions using integral representations. The kernels of these integrals are linear-Gaussian densities in the states to be estimated, a fact that facilitates the use of missing data methods. The resulting sub-algorithms are equivalent to linear-Gaussian Kalman smoothers. The alternating directions algorithm is guaranteed to converge to (at least) a local maximum of the joint target-field likelihood function.

Probabilistic multi-hypothesis tracking (PMHT) is an algorithm for tracking multiple targets when measurement-to-target assignments are unknown and must be estimated jointly with the target tracks. PMHT is linear in the number of targets and the number of measurements; moreover, it is guaranteed to converge to locally optimal state estimates. However, it violates the rule that no target can be assigned more than one measurement. This hereby leads to a plethora of local maxima that cause performance problems. These problems are greatly reduced by applying the PMHT method to multi-frame data sequences, that is, to the set of all possible measurement sequences in the last L scans. The blend of PMHT and limited enumeration reduces the mismatch induced by violating the "at most one measurement per target" rule. Two new PMHT algorithms are presented. Both are linear in the number of targets and the number of enumerated sequences

Detecting signals that are long, weak, and narrowband is a well known and important problem in acoustic signal processing. In this paper an ad hoc scheme is developed: its stages include the DFT, a multiresolution decomposition in the frequency domain, and a GLRT. The computational load is light, and the performance is remarkably good. This is so not just in the original narrowband situation, but also, due to an inherent adaptivity to the data, in the detection of signals that are relatively broadband in nature. Generalizations are given to CFAR operation in both prewhitened and unwhitened cases, and to the detection of multi-band signals. As regards the last, it is discovered that there is little loss from over-estimating the number of bands.

A simple yet effective statistic is proposed for detecting a transient buried in partially unknown ambient noise. The transient model is frequency scattered increased variance observations. We pose the transient detection problem as a homogeneity test and the statistic is derived as the (generalized) likelihood ratio test of overdispersion when the underlying observation sequence follows a double exponential distribution. Numerical testing focuses on the comparison of this scheme with the CFAR power-law detector.

Here we analyze the tracking characteristics of a new data-association/tracking algorithm proposed by Streit and Luginbuhl, the probabilistic multi-hypothesis tracker (PMHT). The algorithm uses a recursive method (known amongst statisticians as the expectation-maximization or EM method) to compute in an optimal way the associations between the measurements and targets. Until now, no comparative performance analysis has been done. We compare the performance of this new scheme to that of a commonly used tracking algorithm, the joint probabilistic data association filter (JPDAF).

A maximum likelihood method is presented for training probabilistic neural networks (PNN's) using a Gaussian kernel, or Parzen window. The proposed training algorithm enables general nonlinear discrimination and is a generalization of Fisher's method for linear discrimination. Important features of maximum likelihood training for PNN's are: 1) it economizes the well known Parzen window estimator while preserving feedforward NN architecture, 2) it utilizes class pooling to generalize classes represented by small training sets, 3) it gives smooth discriminant boundaries that often are "piece-wise flat" for statistical robustness, 4) it is very fast computationally compared to backpropagation, and 5) it is numerically stable. The effectiveness of the proposed maximum likelihood training algorithm is assessed using nonparametric statistical methods to define tolerance intervals on PNN classification performance.

A three-layer feedforward neural network (NN) that implements the optimum Neyman-Pearson (N-P) classifier is described. This NN is useful whenever it is appropriate to characterize (1) input classes as multivariate random variables, and (2) input data vectors as realizations of one of the multivariate random variables. The purpose of the NN is thus simply to compute the conditional likelihoods necessary for the N-P classifier. Because the N-P classifier is optimal, the classification performance of the NN is optimal too. Therefore, three-layer feedforward NN classifiers can equal but not exceed the performance of the N-P classifier. The optimal N-P classifier requires multivariate probability density functions (PDFs) characterizing the input classes. Class PDFs are approximated (arbitrarily closely) by mixtures of multivariate Gaussian PDFs. Supervised training of the class PDFs from input data vectors is, thus, equivalent to training the NN. Maximum likelihood training of the PDFs is performed by the EM algorithm (or by any other suitable optimization method)

article Free Access Share on ALGORITHM 635: an algorithm for the solution of systems of complex linear equations in the L∞ norm with constraints on the unknowns Author: Roy L. Streit Naval Underwater Systems Center, New London, CT Naval Underwater Systems Center, New London, CTView Profile Authors Info & Claims ACM Transactions on Mathematical SoftwareVolume 11Issue 3Sept. 1985 pp 242–249https://doi.org/10.1145/214408.214415Published:01 September 1985Publication History 16citation338DownloadsMetricsTotal Citations16Total Downloads338Last 12 Months15Last 6 weeks0 Get Citation AlertsNew Citation Alert added!This alert has been successfully added and will be sent to:You will be notified whenever a record that you have chosen has been cited.To manage your alert preferences, click on the button below.Manage my AlertsNew Citation Alert!Please log in to your account Save to BinderSave to BinderCreate a New BinderNameCancelCreateExport CitationPublisher SiteeReaderPDF