**1. Introduction**

Signals with strength higher than the detection threshold of the sensor are used as measurements for track initiation and track state update of target tracking. These measurements include not only the target measurements, but also clutter measurements due to environmental factors. Since the source of the measurements in the tracking system is not known in advance, target tracking performance may be significantly degraded if measurements generated by clutter are used when the track state is updated. It is essential to use a tracking algorithm based on data association that can statistically distinguish target and clutter measurements in a cluttered environment [1–4].

Since the number of targets existing in the surveillance region and information on the appearance and disappearance of the target cannot be known in advance, it is important to have a means for determining whether the target is being tracked by a tracking algorithm. For target tracking with track management, integrated probabilistic data association (IPDA) [5,6] and integrated track splitting (ITS) [7–9] have been proposed as data association algorithms for single target tracking, which include a track management method that utilizes the target existence probability of each track for controlling the track status and track number or track label. Linear multitarget-IPDA (LM-IPDA) [10], joint IPDA (JIPDA) [11], and iterative JIPDA (iJIPDA) [12] have been proposed for multiple target tracking by extending IPDA and ITS.

In the aforementioned data association algorithms, it is assumed that the number of clutter measurements is Poisson distributed with a parameter called the clutter measurement density, and the clutter measurements are assumed to be uniformly distributed in the surveillance space. The clutter measurement density is defined as the mean number of clutter measurements per unit volume of the surveillance space. The clutter measurement density is an important parameter used to calculate the data association probability and the target existence probability in the data association algorithms.

If the clutter measurement density is fixed to a design value for target tracking in heterogeneous clutter environments, the error in the clutter measurement density deteriorates not only the target state estimation performance, but also the false track discrimination (FTD) performance because prior information about the clutter measurement is unknown in actual target tracking environments. For accurate and robust target tracking in these environments, it is required to estimate the clutter measurement density adaptively. Clutter measurement density estimation methods are divided into track based estimation methods and measurement based estimation methods. In addition, they are divided into single scan estimation methods and multiple scan estimation methods depending on whether the memory is used in the calculation.

The clutter map method [13,14] is a multi-scan estimation method that uses the measurements from previous scans to calculate the clutter measurement density in the current scan. It divides the surveillance region into a finite number of cells and then estimates the clutter measurement density in each cell by statistically counting the number of existing measurements in the cell during a pre-determined multiple scan period. The clutter map can reduce the influence of bias caused by the target measurements, but estimation performance is sensitive to the parameters such as the cell size and the length of multiple scan period. It is difficult to apply the clutter map when the number of measurements and the spatial probability distribution are time varying.

In [15,16], the clutter measurement density estimation method based on the probability hypothesis density (PHD) filter [17] was handled in conjunction with a target tracking algorithm based on data association. It was designed as a feedback structure that used the intensity of clutter estimated through PHD. However, since the clutter generator is assumed to be a Gaussian function with unknown mean and unknown covariance, it is difficult to use in practical implementations due to heavy computational loads. The work in [18] proposed an interactive clutter measurement density estimator (ICMDE) based on a Gaussian mixture PHD (GM-PHD) filter [19] to estimate the clutter measurement density adaptively in environments where the clutter measurement densities are nonuniform and time varying. In [18], the Gaussian model for the clutter generator was assumed to have a known covariance for reducing the computational loads required to calculate the updated state PHD. By dividing the entire surveillance area, the clutter generator for each partition is represented as a component with the Gaussian model. These processes are performed for multiple scans to generate a reliable clutter map of the surveillance area. In [20], a method of forming a clutter map as proposed by using the histogram probabilistic multi-hypothesis tracker (H-PMHT) based on expectation maximization for image target tracking with each scene composed of millions of pixels. This method forms a clutter map through many iterations until local convergence is guaranteed.

The track based and the measurement based clutter measurement density estimation methods are classified as single scan estimation methods in which the clutter measurement density of the previous scan does not affect the clutter measurement density of the current scan. The track based clutter measurement density estimation method uses the validation gate of the track and the validated measurements existing in this gate. There exist several methods such as the conditional mean estimator based on the target perceivability [21] and the maximum likelihood estimator based on the assumption of unknown, but non-random clutter measurement density [22]. The conditional mean estimator [22] requires prior knowledge of clutter measurement density so that the maximum likelihood estimator may be used as an auxiliary estimator. For the track based clutter measurement density estimation methods, different clutter measurement densities are produced for the same measurement shared by the two tracks as the size of the validation gate of each track is different. This is a drawback of the track based clutter measurement density estimators.

The spatial clutter measurement density estimator (SCMDE) [23] is a measurement based clutter measurement density estimation method that calculates the sparsity as the reciprocal of the clutter measurement density by evaluating the volume of the hyper-sphere centered at the measurement of interest and counting the number of measurements inside the volume. The number of measurements and the hyper-sphere volume are determined by the sparsity order. Unlike the track based clutter measurement density estimation methods, it produces a unique sparsity for each measurement regardless of the validation gate size of the track involved.

It was pointed out in [23] that the existing SCMDE generates the unbiased estimates of clutter measurement density when the point of interest is the target detection for single target tracking environments. It produces biased and bigger clutter measurement density estimates than the actual ones when the point of interest is a clutter detection, which results in improved target tracking performance as the data association probabilities become smaller for the clutter detection. However, when the existing SCMDE is used for multi-target tracking environments, biased clutter measurement density estimation is expected from the nature of SCMDE that all the adjacent measurements to the point of interest are considered to be clutter detections. Through the study in this paper, a major source of biased clutter measurement density estimation of the existing SCMDE for multi-target tracking environments is analyzed, and remedies to reduce the biases are proposed. The new adaptive SCMDE for multi-target tracking (MTT-SCMDE) utilizes the clutter measurement probability to take into account only the clutter measurements for improved accuracy by reducing the biases in the clutter measurement density estimation. Through the analysis, an expansion of the volume of the hyper-sphere corresponding to each sparsity order from that of the existing SCMDE is proposed for more accurate clutter measurement density estimation.

A method that takes into account clutter-originated measurements in the clutter measurement density calculation was proposed in [24]. The performance of the SCMDE algorithm for multi-target tracking was presented in [24]. In this paper, we elaborated the theoretical development by analyzing the source of biases in the MTT-SCMDE algorithm for multi-target tracking, and refined its performance by increasing the hyper-sphere volume for the measurement of interest. The improvement was based on strict analysis presented in this paper. To verify the performance of the proposed clutter measurement density estimation method, a series of simulation runs was executed in heterogeneous clutter environments, and the results were analyzed by performance comparison to check how closely the estimated clutter measurement densities followed the true clutter measurement densities for multiple targets. In addition, the clutter measurement density estimation performance and the target tracking performance were tested for various sparsity orders and various numbers of targets involved. The proposed MTT-SCMDE was also applied to a set of real radar data for performance evaluation.

The remainder of this paper is organized as follows. The stochastic models in the target tracking algorithm are described in Section 2. Section 3 derives the LM-IPDA algorithm for multi-target tracking in a cluttered environment. The SCMDE method is briefly described in Section 4. Section 5 describes the proposed clutter measurement density estimation method in detail. The clutter measurement density estimation performance and multiple target tracking performance of the proposed method are analyzed through a series of Monte Carlo simulation runs in various tracking environments, as well as a set of real radar data in Section 6, followed by the Conclusions. Performance analysis of the existing SCMDE used in multi-target tracking environments is presented in the Appendix A.
