# Labeled Multi-Bernoulli Filter Joint Detection and Tracking of Radar Targets
## 3 The S-LMB-GOM Filter
### A Method to Overcome the Cardinality Estimate Bias
#### 3.3.2 The One-to-One Correspondence Method  
Motivated by the standard LMB filter, we propose a one-to-one correspondence method between measurements and tracks to overcome the problem of the cardinality estimate bias. The idea of our method is similar with the Nearest Neighbor (NN) method [29]. The difference is that our method matches the maximum likelihood but the NN method matches the smallest distance.  
The method can be summarized as Algorithm 1. We update the predicted track in order of existence probability \(r_{+}^{\ell}\) to prevent the new birth predicted tracks from interfering with the surviving predicted tracks. Given that the predicted track \(\ell_{1}\) update first and after update the existence probability \(r_{+}^{\ell_{1}}\) is greater than the detection threshold \(\lambda\) (e.g., \(\lambda\) = 0.5), the hypothesized track \(\ell_{1}\) is considered a true track and has tracked a target \(\mathbf{x}\). Still as shown in the Figure 1, the target \(\mathbf{x}\) can be approximated by the particle \(\mathbf{\hat{x}}\) with the maximum likelihood in the particle group of track \(\ell_{1}\). The associated measurements by \(T(\mathbf{x})\) are reset to noise, where \(T(\mathbf{x})\) is approximated by \(T(\mathbf{\hat{x}})\). This operation makes the track \(\ell_{1}\) only correspond to the associated measurements by \(T\left(\mathbf{x}\right)\). The setting of \(T\left(\mathbf{x}\right)\) is detailed in Section 3.4. Then, another predicted track \(\ell_{2}\) is updated and cannot track the same target \(\mathbf{x}\) again due to the resetting of the associated measurements. Therefore, the overlapped particle groups will no longer lead to a repeated cardinality estimate.  
Figure 1: A sketch diagram to show the measurements affected by a target are associated with two tracks. Solid dots denote the particles of track \(\ell_{1}\), Hollow dots denote the particles of track \(\ell_{2}\) and triangles denote the associated measurements of \(\mathbf{x}\), \(\mathbf{\hat{x}}\) and \(\mathbf{\hat{x}}\).  
```
1. Descending sort: \(r_{+}^{\ell}=\underset{\ell\in\mathbb{L}_{+}}{\text{sort}}(r_{+}^{\ell}\), 'descend').
2: Update in order of \(r_{+}^{\ell}\) to get \(r_{+}^{\ell}\) and \(\bar{p}_{+}^{\ell}(\mathbf{x})\). If \(\bar{r}_{+}^{\ell}\geq\lambda\) \(\mathbf{\hat{x}}=\underset{\mathbf{x}\in\mathbf{X}}{\text{max}}(\text{compute-likelihood}(Z,\mathbf{x}))\). \(\mathbf{x}\approx\mathbf{\hat{x}}\). \(Z=\text{reset-noise}(Z,T\left(\mathbf{x}\right))\). end
```  
**Algorithm 1** One-to-one correspondence method  
When the two targets \(\mathbf{x}_{1}\) and \(\mathbf{x}_{2}\) are closely spaced and the templates \(T\left(\mathbf{x}_{1}\right)\) and \(T\left(\mathbf{x}_{2}\right)\) are partially overlapped as shown in the Figure 2, the one-to-one correspondence method can work properly. After update of the track \(\ell_{1}\), the target \(\mathbf{x}_{1}\) is associated with track \(\ell_{1}\) and the associated measurements by \(T\left(\mathbf{x}_{1}\right)\) are reset to noise where \(T(\mathbf{x}_{1})\approx T(\mathbf{\hat{x}}_{1})\). If the setting range of \(T\left(\mathbf{x}_{1}\right)\) is small enough and \(T\left(\mathbf{x}_{1}\right)\) do not include the peak region of the likelihood function of the target \(\mathbf{x}_{2}\), then \(\mathbf{x}_{2}\) can be distinguished at the update step, because the peak region of likelihood function roughly corresponds to a target position and determines the existence probability of the target.