## 5 Iterative Maximum Weighted Likelihood Estimation  
The IMWLE method is an improvement of the traditional ToF-based method. It is a statistical method that improves the accuracy of 3-D target localization using the distribution of pseudo-targets. The way to get pseudo-targets is the same as that in Section 3. If the problem of disjoint ellipsoids appears, the curve is defined as a region where the distance to the two disjoint ellipsoids is smaller than a given number. If the problem of disjoint curves appears, the pseudo-target is defined as the midpoint of the line segment whose length is the closest distance between the two disjoint curves.  
IMWLE is only related to the spatial distribution of pseudo-targets, so it is necessary to understand the rules of distribution. ToF measurement errors affect the spatial distribution of pseudo-targets. Some pseudo-targets are located close to the target, while other pseudo-targets are far from the target and may have a negative influence on the accuracy of localization.  
IMWLE takes into account the spatial distribution of pseudo-targets for localization. The flow chart of the method is shown in Figure 7.  
The pseudo-targets, whose coordinates are denoted by vector \(\mathbf{x}^{k}\), are assigned weight factors \(\omega_{j}^{k}\). Superscript \(k\) refers to the pseudo-target location numbers. Subscript \(j\) refers to the iteration number.  
Figure 7: The flow chart of the Iterative Maximum Weighted Likelihood Estimation (IMWLE) method.  
When \(j=1\), \(\omega_{1}^{k}\) is initialized to the same values 1. By assuming that all pseudo-locations satisfied the Gaussian distribution, mean \(\mu_{1}\) and covariance matrix \(\Omega_{1}\) are calculated by the following equations:  
\[\mu_{1}=\frac{1}{L}\sum_{k=1}^{L}x^{k}, \tag{4}\]  
\[\Omega_{1}=\frac{1}{L}\sum_{k=1}^{L}\left(x^{k}-\mu_{1}\right)\left(x^{k}-\mu_ {1}\right)^{T}, \tag{5}\]  
where \(L\) is the number of pseudo-targets.  
In the next iteration, the weight factors of the pseudo-targets are defined as the probability density of the Gaussian distribution.  
\[\omega_{j}^{k}=\frac{1}{\left(2\pi\right)^{3/2}}\frac{1}{\left|\Omega_{j-1} \right|^{1/2}}\exp\left\{-\frac{1}{2}\left(x^{k}-\mu_{j-1}\right)^{T}\left( \Omega_{j-1}\right)^{-1}\left(x^{k}-\mu_{j-1}\right)\right\} \tag{6}\]  
MWLE is used to fit the Gaussian distribution of pseudo-targets.  
\[\mu_{j}=\frac{\sum_{k=1}^{L}\omega_{j}^{k}x^{k}}{\sum_{k=1}^{L}\omega_{j}^{k}} \tag{7}\]  
\[\Omega_{j}=\frac{\sum_{k=1}^{L}\omega_{j}^{k}\left(x^{k}-\mu_{j}\right)\left( x^{k}-\mu_{j}\right)^{T}}{\sum_{k=1}^{L}\omega_{j}^{k}} \tag{8}\]  
The mean \(\mu_{j}\) of the Gaussian distribution can be considered as the estimated target location in the \(j\)-th iteration.  
If \(\frac{\left|\mu_{j}-\mu_{j-1}\right|}{\left|\mu_{2}-\mu_{1}\right|}<\textit{ threshold}\) (in our case, we set \(\textit{threshold}=1\%\)), the stop iteration and \(\mu_{j-1}\) are the final estimated target location obtained by IMWLE; otherwise, set \(j=j+1\) and repeat from 2.  
At each iteration, MWLE assigns new weight factors to pseudo-targets by fitting a Gaussian distribution to their positions, and small weight factors are assigned to the pseudo-targets that are distant from the target so as to minimize their influence on the accuracy of localization. Using the proposed method, the spatial distribution of pseudo-targets were taken into account in order to achieve accurate target localization.