## 2 Image Alignment  
### Matching Feature Points  
In the image acquisition process, due to the inconsistency of the position, attitude, and acquisition parameters of infrared and visible cameras, it is necessary to align the images captured by each camera in pixel-level image fusion. The alignment process includes three steps: feature point extraction, feature point matching, and coordinate mapping. In this paper, the features from the accelerated segment test (FAST) algorithm [28] are used to extract the key points of the image, and the DAISY descriptor [29] is used to quantify the features of the key points. The geometric structure of the DAISY descriptor is a multilayer concentric circle. In each layer, eight sampling points are evenly distributed at 45\({}^{\circ}\) intervals, and a Gaussian convolution is applied to compute the relationship between each point and its neighboring pixels, as shown in Figure 1.  
Figure 1: DAISY descriptor. Where “+” is the position of the sampling point and the size of the circle is positively correlated with the Gaussian convolution kernel scale  
[PERSON] [30] found that the DAISY descriptor performs optimally in different datasets by comparing several feature aggregation strategies for chunking in Cartesian and polar coordinate systems. The computation process of the DAISY descriptor is specified as follows:  
1. The gradient of the input image \(\mathbf{I}\) in direction \(d\) is calculated. Only those values with gradients greater than zero are retained in the result. The gradient matrix is denoted as \(\mathbf{G}_{d}=\left(\frac{\partial\mathbf{I}}{\partial d}\right)^{+}\), where the operator \((\cdot)^{+}=\text{max}(\cdot,0)\). The gradient calculation needs to be decomposed in the \(x\) direction and \(y\) direction separately. Let the angle between the gradient direction and the positive \(x\) direction be \(\theta\). The formula for the gradient matrix is \[\mathbf{G}_{d}=\left(\frac{\partial\mathbf{I}}{\partial x}\cos\theta+\frac{ \partial\mathbf{I}}{\partial y}\sin\theta\right)^{+}\] (1) where the convolutions in the \(x\) and \(y\) directions are applied using one-dimensional convolution kernels [1, \(-1\)] and [1, \(-1\)]T, respectively. 2. We use Gaussian convolution kernels with different scales to perform separate convolution operations on the gradient matrix \(\mathbf{G}_{d}\) to form different scaled convolution matrices \(\mathbf{G}_{d}^{\Sigma}\), which are calculated as follows: \[\mathbf{G}_{d}^{\Sigma}=\mathbf{G}_{\Sigma}*\mathbf{G}_{d}\] (2) where \(\mathbf{G}_{\Sigma}\) denotes a Gaussian convolution kernel with \(\Sigma\) standard deviation. 3. Let \(\mathbf{h}_{\Sigma}(u,v)=\left[\mathbf{G}_{d_{1}}^{\Sigma}(u,v),\mathbf{G}_{ d_{2}}^{\Sigma}(u,v),\ldots,\mathbf{G}_{d_{l}}^{\Sigma}(u,v)\right]^{T}\) denote the feature vector of location \((u,v)\) after convolution by a Gaussian kernel of standard deviation \(\Sigma\). \(H\) denotes the number of gradient directions in the feature vector. Let \(\widetilde{\mathbf{h}}_{\Sigma}^{\Sigma}(u,v)\) be the normalized feature vector. The DAISY descriptor \(D\) of the feature point location \((u,v)\) is denoted as \[ \begin{array}{ll}D(u,v)&=\left[\widetilde{\mathbf{h}}_{\Sigma_{1}}^{T}(u,v),\right.\\ &\widetilde{\mathbf{h}}_{\Sigma_{1}}^{T}(l_{1}(u,v,R_{1})),\ldots,\widetilde{ \mathbf{h}}_{\Sigma_{1}}^{T}(l_{T}(u,v,R_{1})),\\ &\widetilde{\mathbf{h}}_{\Sigma_{2}}^{T}(l_{1}(u,v,R_{2})),\ldots,\widetilde{ \mathbf{h}}_{\Sigma_{2}}^{T}(l_{T}(u,v,R_{2})),\\ &\cdots,\\ &\widetilde{\mathbf{h}}_{\Sigma_{Q}}^{T}(l_{1}(u,v,R_{Q})),\ldots,\widetilde{ \mathbf{h}}_{\Sigma_{Q}}^{T}(l_{T}(u,v,R_{Q}))]^{T}\end{array} \] (3) where \(Q\) is the number of layers, and \(T\) is the number of sampling points in each layer. \(l_{i}(u,v,R_{i})\) denotes the location of the \(i\)th sampling point above the \(j\)th concentric ring centered at the point \((u,v)\). \(R_{j}\) is the distance between the sampling point and the center point \((u,v)\).