# Labeled Multi-Bernoulli Filter Joint Detection and Tracking of Radar Targets
## 1 Introduction  
Multi-target tracking based on multi-target Bayes filter makes an important contribution to capturing and maintaining the awareness of the environment [1; 2]. Multi-target Bayes filter is usually computationally intractable [3] by simply extending the classic single-target Bayes filters [4] for multi-target case. With the development of the emerging random finite set (RFS) theory, tremendous efforts have been devoted to investigating various approximate multi-target Bayes filters [5; 6; 7; 8], including probability hypothesis density (PHD) filter, cardinalized PHD (CPHD) filter and multi-Bernoulli (MB) filter, and their various revised versions. In order to integrate track management into the filtering scheme, the so-called generalized labeled multi-Bernoulli (GLMB) filter and labeled multi-Bernoulli (LMB) filter were proposed [9; 10; 11]. Most of these filters are based on the standard multi-target observation model which assumes that one target generates at most one measurement and so any measurement corresponds to at most one target [12]. However, in many scenarios, the observation models are nonstandard and cannot be modeled by standard multi-target likelihood function, for example, track before detect (TBD) in video [13; 14], superpositional sensors [15; 16; 17], unresolved target tracking [18] and extended target tracking [19]. As a typical application of the nonstandard observation model, the TBD algorithm of radar targets [20; 21; 22; 23; 24; 25] works with the raw radar data directly, and detects and tracks the targets jointly, so it is suitable for application in a low signal-to-noise ratio (SNR) scenario. The observation model for radar sensor is called a generic observation model (GOM) [20; 26]. This is because the measurements affected by different target point spread functions may overlap. The joint likelihood function for the GOM cannot be denoted as the product of each target likelihood function and so it is typically referred to as inseparable likelihood [20]. However, for computational efficiency, it could be assumed that themeasurements affected by different targets do not overlap and then the separable likelihood could be derived [20; 27]. References [27; 28] proposed the MB filter and the LMB filter for the GOM under the assumption of the separable likelihood which are called the S-MB-GOM filter and the S-LMB-GOM filter, respectively. Both filters are computational efficiently but the cardinality estimate bias easily occurs when the separable likelihood assumption is violated. Firstly, due to the influence of new or close tracks, the measurements generated by a target may be associated with multiple tracks which causes an increase of cardinality estimate. Secondly, the measurements generated by multiple closely spaced targets may only be associated with a track due to merging [12] which causes a decrease of cardinality estimate. To avoid the problem, References [20; 26] proposed the GLMB filter and the group LMB filter for the GOM, which are called the GLMB-GOM filter and the G-LMB-GOM filter, respectively. Both filters adapt to inseparable likelihood and improve the tracking performances for radar. However, the implementations of both filters require high-dimensional particle sampling, so the calculation amounts are intensive. In our paper, we propose a one-to-one correspondence method between measurements and tracks to overcome the cardinality estimate bias of the S-LMB-GOM filter [28]. Further, to improve the computational efficiency of the S-LMB-GOM filter, we propose an approximate method to calculate the point spread function according to its property analysis. The Sequential Monte Carlo (SMC) approximation method has been used to implement our S-LMB-GOM filter. The performance is verified by a scenario with closely spaced targets in low SNR and compared with the G-LMB-GOM filter.