This paper presents a joint trajectory smoothing and tracking framework for a specific class of targets with smooth motion. We model the target trajectory by a continuous function of time (FoT), which leads to a curve fitting approach that finds a trajectory FoT fitting the sensor data in a sliding time-window. A simulation study is conducted to demonstrate the effectiveness of our approach in tracking a maneuvering target, in comparison with the conventional filters and smoothers. Note to Practitioners —Estimation, such as automatically tracking and predicting the movement of an aircraft, a train, or a bus, plays a key role in our daily life. In this paper, we provide a new approach for the online estimation of the target trajectory function by means of fitting the time-series observation, which accommodates the lack of quantifiable knowledge about the target motion and of the statistical property of the sensor observation noise. The resulting trajectory function can be used to infer either the past or the present state of the target. Engineering-friendly strategies are provided for computationally efficient implementation. The proposed approach is particularly appealing to a broad range of real-world targets that move in smooth courses, such as passenger aircraft and ships.
However, most existing works work in batch manners based on either MLE @cite_3 or Bayesian inference @cite_81 @cite_1 or as an extra scheme to a recursive filtering algorithm @cite_52 @cite_20 @cite_35 . In @cite_3 , directional bearing data from one or multiple sensors are investigated, where Cardinal splines (i.e., splines with equally spaced knots) of different dimensions are fit to the data in the MLE manner; this is one of the earliest and few attempts that assume a spatio-temporal trajectory for tracking. In @cite_81 , the trajectory is approximated by a cubic spline with an unknown number of knots in 2D state plane, and the function estimate is determined from positional measurements which are assumed to be received in batches at irregular time intervals. For the data drawn from an exponential family, the spline knot con-figurations (number and locations) are changed by reversible-jump Markov chain Monte Carlo @cite_89 . Much more complicated, artificial neural networks were considered as a parametric non-linear model in @cite_11 , which is unaffordable in computation for online estimation.
@cite_35 @cite_1 @cite_52 @cite_3 @cite_89 @cite_81 @cite_20 @cite_11
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We describe a Bayesian method, for fitting curves to data drawn from an exponential family, that uses splines for which the number and locations of knots are free parameters. The method uses reversible jump Markov chain Monte Carlo to change the knot configurations and a locality heuristic to speed up mixing. For nonnormal models, we approximate the integrated likelihood ratios needed to compute acceptance probabilities by using the Bayesian information criterion, BIC, under priors that make this approximation accurate. Our technique is based on a marginalised chain on the knot number and locations, but we provide methods for inference about the regression coefficients, and functions of them, in both normal and nonnormal models. Simulation results suggest that the method performs well, and we illustrate the method in two neuroscience applications. A curve fitting algorithm for batch ship trajectory estimation that employs Bayesian statistical inference for non-parametric regression is presented. It assumes no knowledge about the ship motion model while only assuming standard ship maneuvers. The trajectory is thought to be well represented by a cubic spline with an unknown number of knots in two-dimensional Euclidean plane. The function estimate is determined from positional measurements which are assumed to be received in batches at irregular time intervals. As the measurements are often delivered by different sensors the measurement errors are assumed to be heteroscedastic and correlated. A fully Bayesian approach is adopted by defining the prior distributions on all unknown parameters: the spline coefficients as well as the number and the locations of knots. The quality of the estimator algorithm is evaluated statistically using several simulated scenarios. The results suggest that the algorithm represents efficient methodology for trajectory estimation in maritime surveillance, especially in the absence of prior knowledge of the motion model. We introduce and solve some new problems of efficient reconstruction of an unknown sequence from its versions distorted by errors of a certain type. These erroneous versions are considered as outputs of repeated transmissions over a channel, either a combinatorial channel defined by the maximum number of permissible errors of a given type, or a discrete memoryless channel. We are interested in the smallest N such that N erroneous versions always suffice to reconstruct a sequence of length n, either exactly or with a preset accuracy and or with a given probability. We are also interested in simple reconstruction algorithms. Complete solutions for combinatorial channels with some types of errors of interest in coding theory, namely, substitutions, transpositions, deletions, and insertions of symbols are given. For these cases, simple reconstruction algorithms based on majority and threshold principles and their nontrivial combination are found. In general, for combinatorial channels the considered problem is reduced to a new problem of reconstructing a vertex of an arbitrary graph with the help of the minimum number of vertices in its metrical ball of a given radius. A certain sufficient condition for solution of this problem is presented. For a discrete memoryless channel, the asymptotic behavior of the minimum number of repeated transmissions which are sufficient to reconstruct any sequence of length n within Hamming distance d with error probability spl epsiv is found when d n and spl epsiv tend to 0 as n spl rarr spl infin . A similar result for the continuous channel with discrete time and additive Gaussian noise is also obtained. We propose a new method for creating computationally efficient convolutional neural networks (CNNs) by using low-rank representations of convolutional filters. Rather than approximating filters in previously-trained networks with more efficient versions, we learn a set of small basis filters from scratch; during training, the network learns to combine these basis filters into more complex filters that are discriminative for image classification. To train such networks, a novel weight initialization scheme is used. This allows effective initialization of connection weights in convolutional layers composed of groups of differently-shaped filters. We validate our approach by applying it to several existing CNN architectures and training these networks from scratch using the CIFAR, ILSVRC and MIT Places datasets. Our results show similar or higher accuracy than conventional CNNs with much less compute. Applying our method to an improved version of VGG-11 network using global max-pooling, we achieve comparable validation accuracy using 41 less compute and only 24 of the original VGG-11 model parameters; another variant of our method gives a 1 percentage point increase in accuracy over our improved VGG-11 model, giving a top-5 center-crop validation accuracy of 89.7 while reducing computation by 16 relative to the original VGG-11 model. Applying our method to the GoogLeNet architecture for ILSVRC, we achieved comparable accuracy with 26 less compute and 41 fewer model parameters. Applying our method to a near state-of-the-art network for CIFAR, we achieved comparable accuracy with 46 less compute and 55 fewer parameters.