# Tracking System for a Coal Mine Drilling Robot for Low-Illumination Environments
## 3 The Proposed Method  
### Discriminative Correlation Filter  
The fast DSST [39] is adopted as the baseline due to its outstanding performance. It utilizes two optional correlation filters for estimating the translation (two-dimensional) and scale (one-dimensional) of the target in the new frame. The goal for a two-dimensional (M \(\times\) N pixels) image is to learn a set of multichannel correlation filters \(\mathbf{f}_{t}^{l}\in\mathbb{R}^{\text{M}\times\text{N}\times\text{D}}\) based on the sample \(\{(\mathbf{x}_{t},\mathbf{y}_{t})\}_{t}^{l}\) at the time \(t\). Each training sample \(\mathbf{x}_{t}\in\mathbb{R}^  
Because Equation (1) is a linear least squares problem, it can be computed efficiently in the Fourier domain by [PERSON]'s formula. Hence, the filter that minimizes Equation (1) is expressed as:  
\[\varepsilon(\mathbf{F})=\left\|\sum_{l=1}^{D}\overline{\mathbf{X}}^{l}\odot \mathbf{F}^{l}-\mathbf{Y}\right\|_{2}^{2}+\lambda\sum_{l=1}^{D}\left\|\mathbf{ F}^{l}\right\|^{2} \tag{2}\]  
where the capital letters denote the discrete Fourier transform (DFT) of the corresponding quantities, the bar \(\overline{\bullet}\) denotes a complex conjugation, and \(\odot\) denotes a Hadamard product. Therefore, in the first frame, Equation (2) can be solved as:  
\[\mathbf{F}^{l}=\frac{\overline{\mathbf{Y}}\odot\mathbf{X}^{l}}{\sum_{k=1}^{D} \overline{\mathbf{X}}^{l}\odot\mathbf{X}^{k}+\lambda},\quad l=1,\dots D. \tag{3}\]  
The numerator \(\mathcal{A}_{t}^{l}\) and the denominator \(\mathcal{B}_{t}\) are defined for the \(t\)-th frame. An optimal update strategy for the filter \(\mathbf{F}_{t}^{l}\) in the new sample \(\mathbf{x}_{t}\) is as follows:  
\[\mathcal{A}_{t}^{l}=(1-\eta)\mathcal{A}_{t-1}^{l}+\eta\overline{\mathbf{Y}} \odot\mathbf{X}_{t}^{l}, \tag{4}\]  
\[\mathcal{B}_{t}=(1-\eta)\mathcal{B}_{t-1}+\eta{\sum_{k=1}^{D} \overline{\mathbf{X}}_{t}^{l}\odot\mathbf{X}_{t}^{l}}, \tag{5}\]  
where the scalar \(\eta\) is a parameter of the learning rate. The correlation scores \(\mathbf{y}_{t}\) for a new test sample \(\mathbf{z}_{t}\) can be computed in the Fourier domain to detect the change in the position in the new frame \(t\):  
\[\mathbf{y}_{t}=\mathcal{F}^{-1}\left\{\frac{\sum_{l=1}^{D}\overline{\mathcal{ A}}_{t-1}^{l}\odot\mathbf{Z}^{l}}{\mathcal{B}_{t-1}+\lambda}\right\}, \tag{6}\]  
where \(\mathbf{Z}^{l}\) denotes the \(l\)-dimensional features extracted from the frame of pending detection. \(\mathcal{F}^{-1}\) is the IFFT. Equation (6) can be used to find the maximum correlation score to determine the position of the current target. The image feature pyramid of the current sample is constructed in a rectangular area behind the learned translation filter to estimate the scale. The scale of the filter is defined as \(S=\left\{\begin{array}{ll}\alpha^{n}\left|n=-\frac{N-1}{2}\right|,\dots, \left|\frac{N-1}{2}\right|\right\}\right\}\). The current target region with a size of \(\text{W}\times\text{H}\) is reconstructed to form a series of scale patches \(I_{n}\) of size \(\alpha^{n}\text{W}\times\alpha^{n}\text{H}\) based on \(N\) scale levels. The scale filter has a one-dimensional Gaussian score \(y_{s}\). The max value \(S_{t}(n)\) of the training sample \(x_{t,scale}\) for \(I_{n}\) is the current scale. Because the computational cost of the fast DSST depends primarily on the FFT, we use PCA for dimensionality reduction to improve the computing speed. To update the target template \(\mu_{t}=(1-\eta)\mu_{t-1}+\eta\mathbf{x}_{t}\), a projection matrix \(\mathbf{P}_{t}\in\mathbb{R}^{d\times D}\) is constructed for \(\mu_{t}\), where \(d\) is the dimension of the compression feature.