Mixture reduction techniques and probabilistic intensity models for multiple hypothesis tracking of targets in clutter

https://doi.org/10.1016/j.compeleceng.2013.07.023Get rights and content

Highlights

  • Two Gaussian mixture reduction algorithms (MRAs) for a Multiple Hypothesis Tracker (MHT) are described.

  • The preferred MRA yields a mixture with no more than two components.

  • Both algorithms outperform Probabilistic Data Association (PDA) and an alternative MHT MRA described in the literature.

  • A scheme for incorporating intensity information into a Bayesian tracker in an infrared (IR) sensor is also presented.

Abstract

A linear combination of Gaussian components, i.e. a Gaussian ‘mixture’, is used to represent the target probability density function (pdf) in Multiple Hypothesis Tracking (MHT) systems. The complexity of MHT is typically managed by ‘reducing’ the number of mixture components. Two complementary MHT mixture reduction algorithms are proposed and assessed using a simulation involving a cluttered infrared (IR) seeker scene. A simple means of incorporating intensity information is also derived and used by both methods to provide well balanced peak-to-track association weights. The first algorithm (MHT-2) uses the Integral Squared Error (ISE) criterion, evaluated over the entire posterior MHT pdf, in a guided optimization procedure, to quickly fit at most two components. The second algorithm (MHT-PE) uses many more components and a simple strategy, involving Pruning and Elimination of replicas, to maximize hypothesis diversity while keeping computational complexity under control.

Introduction

Multiple Hypothesis Tracking (MHT) is a complete Bayesian framework for tracking targets in clutter [1]. In theory, it is an ideal data association approach in surveillance sensor systems using detect-before-track architectures where all target and clutter peaks are considered to be point-like with no distinguishing features. Tracks on targets are maintained primarily using kinematic models which define the relationship between spatial and temporal variables, using a Kalman filter. In some cases, intensity measurements may also be used to probabilistically adjust the weights of peak-to-track assignment hypotheses. By comparison, clutter is assumed, on average, to be less intense and spatiotemporally uncorrelated.

After local maxima in the digitized sensor data have been identified, a threshold is applied to extract high-intensity peaks that may be due to targets. This simple but crude peak declaration process reduces the number of possible data association hypotheses by several orders of magnitude. MHT is then applied over time to further reduce the hypotheses to a number that provides a tolerably low probability of track divergence at an acceptable computational cost.

The digitized sensor data are first pre-processed by a whitening algorithm to ensure that the aforementioned assumptions are reasonable. An ideal whitener transforms structured backgrounds into spatiotemporally de-correlated white noise. The target tracking problem is then reduced to a form that is readily solved using MHT, regardless of the sensor modality – acoustic, electro-optic, radio-frequency or hyper-spectral. An infrared (IR) seeker application is considered in this paper [2], [3]. While perfect whitening is unlikely in real electro-optic systems, the assumption of de-correlated clutter is a convenient theoretical starting point.

In practice, MHT implementation in an embedded processor of modest capability is a challenge. MHT is perhaps best regarded as an unrealizable abstraction. MHT instantiations in real systems require simplifications, of one kind or another, to ensure that the number of competing data association hypotheses is manageable. Mixture reduction and sub-problem formation are particularly effective strategies for controlling MHT complexity [4]. Probabilistic Data Association (PDA) is an extreme case where each track is a single sub-problem and each sub-problem is reduced to a single Gaussian component [5], [6]. Joint PDA (JPDA) also maintains a single component for each track but sub-problems with multiple tracks are permitted [6]. Mixture reduction strategies for tracking multiple closely-spaced targets using MHT has more recently been examined in [7], where the minimum mean optimal subpattern assignment (MMOSPA) metric [8], is used to produce “smooth, coalescence-free state estimates”. Reduction algorithms are also required to manage complexity in the newer Gaussian-mixture probability hypothesis density (PHD) trackers [9], [10], [11].

A number of generic reduction strategies have been proposed to allow the use of Gaussian mixtures of arbitrary size. Replacement of pairs and clusters by a single component in proposed in [12]. Candidates for replacement are identified using a statistical measure of inter-component separation. This approach has been incorporated in a multi-target tracker in [13]. The Integral Squared Error (ISE) is used in [14], [15], [16], [17] to quantify the error associated with a given reduction. Unlike the Kullback–Leibler measure used in [18], [11], the ISE has a closed analytical form, allowing it to be quickly evaluated without numerical approximation. Expressions for partial derivatives of the ISE [14], and the normalized ISE [17], have also been derived, which in principal, may be used to define a system of ordinary differential equations (ODEs) [14], [17]; however, in practice, iterative rule-based approaches are found to strike a more sensible balance between tracking performance and computational cost. The error surface of the optimization problem is difficult to negotiate; thus mathematical rigor gives way to a somewhat more pragmatic series of: moment-preserving merge operations [18]; merge-or-prune operations [14], [15], [16]; or successive split operations, guided by an adaptive solver [17]. More recently, in [11] a generalized likelihood ratio is used in a statistical hypothesis test to determine which Gaussian components should be merged; in [19] sparse modelling is used within a L1/2 regularization framework, while a computationally expensive ‘brute-force’ combinatorial approach is adopted in [4].

In this paper, an efficient bimodal reduction algorithm is presented (MHT-2) [20]. The use of two components is sufficient to resolve temporary association ambiguity that might otherwise lead to track divergence in a simpler one-component approach. In the simple abstract scenarios investigated here, maintaining a greater number of components, using a more complicated mixture reduction scheme, offers diminishing returns. An alternative reduction algorithm (MHT-PE), capable of maintaining an arbitrary number of components at a reasonable computational cost, is also presented [20]. It is used to highlight the performance/complexity trade-off associated with MHT system design.

In the next section the use of intensity information is discussed. The target tracking problem is then posed, in the context of an IR seeker in a long-range air-to-air engagement. The low-observable target, which is well approximated by a single point, is set against blue sky. It is assumed that the track management functions (initiation and termination) are manually performed by the pilot therefore the formulations do not include an integrated track confidence model. Finally, Monte Carlo (MC) simulations are used to assess the various mixture reduction strategies, using the mean-time-before-failure (mtbf) and the mean-time-to-process-frame (mtpf) metrics as measures of performance.

Section snippets

Intensity ‘Information’

The use of non-kinematic feature attributes has been shown to aid data association in imaging sensors. Peak curvature and volume [21]; color distribution [22]; down-range and cross-range extent [23]; target size, shape and orientation [24]; and of course signal strength or intensity [21], [25], [26]; have all been proposed. The volume and diversity of published research on the use of intensity information in PDA and MHT trackers suggests that the subject is more of an art than a science. In [21]

Formulation

The Nc brightest peaks in a given frame are retained and passed to the tracker. The detection threshold intensity (i.e. the intensity of the dimmest peak) is subtracted from all raw intensity measurements. This adjusted intensity I, is then used in all downstream processing functions. The way in which intensity information is incorporated into the PDA filter is presented in this section, along with a description of the two-component MHT filter. As the focus here is on the maintenance of a

Scenarios

Clutter was randomly generated and distributed over a 64 × 64 pixel (pix) Field-Of-View (FOV), according to (3), (15). Unless otherwise indicated, the ‘baseline’ scenario used 100 peaks per frame (frm) and SNR = 3 dB, where SNR = 20log10(κtθt/κcθc). A single point-target was injected, with pd = 1.0 for the first two frames to facilitate track establishment, and pd = 0.8 thereafter. Unity pd was used in [12]. Unfortunately, pd < 1.0 is an inextricable part of the tracking-in-clutter problem and consecutive

Summary of results

The results show that in the baseline scenario, the mtbf of MHT-2 is approximately twice that of PDA and that the mtbf of MHT-PE is approximately twice that of MHT-2. The mtbf of MHT-P is slightly greater than the mtbf of PDA; however, the mtpf of MHT-P is an order of magnitude greater.

Computational complexity

When an upper limit is placed on the number of MHT components, the removal of duplicates is required because they consume computing resources that could be better utilized elsewhere. However, the complexity of a

Conclusion

The global ISE is a simple and effective measure of the goodness-of-fit for the reduction of the high-order Gaussian mixtures used in MHT; however, putting it to use in an ideal algorithm is not straightforward, due to: the high dimension of the parameter search space, constraints imposed on the pdf, and the ‘dimpled’ error surface. As tracking algorithms are required to run in real-time in mission-critical systems, mathematically sub-optimal implementations are needed. MHT-2 uses the ISE in a

Hugh L. Kennedy (BE & PhD from The University of New South Wales) is a principal engineer in the Defence and Systems Institute at the University of South Australia. Prior to joining the university in late 2010, he worked in industry on the design, development, integration and maintenance of a variety of different sensor systems – electro-optic, radio-frequency and acoustic.

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    Hugh L. Kennedy (BE & PhD from The University of New South Wales) is a principal engineer in the Defence and Systems Institute at the University of South Australia. Prior to joining the university in late 2010, he worked in industry on the design, development, integration and maintenance of a variety of different sensor systems – electro-optic, radio-frequency and acoustic.

    Reviews processed and recommended for publication to Editor-in-Chief by Deputy Editor Dr. Ferat Sahin.

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