# Infrared Dim and Small Target Detection Based on the Improved Tensor Nuclear Norm
## 4 Construction of the a Priori Model in This Paper  
In scenes with large edge contours in the background, the simple maximum cannot fully utilize the local feature information in the background, and the background contour region is not well suppressed. This study addresses the deficiencies in the literature [17] in terms of addressing situations with more edge contour backgrounds by establishing a new anisotropic background feature weight function to characterize the background space difference features as a way to achieve the suppression of edge contours in backgrounds of complex scenes and enhance the significant target features in the structure tensor by improving high-order accumulation. A new anisotropic background feature weight function is calculated as follows: for an image \(D\in R^{m\times n}\), the eigenvalue is calculated as \(K_{1},K_{2}\). First, each anisotropy filter is applied to \(K_{1},K_{2}\)[7], and the calculation procedure is as follows:  
\[\left\{ \begin{array}{l}up=|K(x,y)-K(x,y-step)|\\ down=|K(x,y)-K(x,y+step)|\\ right=|K(x,y)-K(x-step,y)|\\ left=|K(x,y)-K(x+step,y)|\end{array} \right. \tag{11}\]  
\[\left\{ \begin{array}{l}cn=\left(1-e^{-(up/k)^{2}}\right)\times up\\ cs=\left(1-e^{-(down/k)^{2}}\right)\times down\\ ce=\left(1-e^{-(right/k)^{2}}\right)\times right\\ cw=\left(1-e^{-(left/k)^{2}}\right)\times left\end{array} \right. \tag{12}\]  
\(K\) represents the feature value \(K_{1},K_{2}\) obtained by the calculation, \(K(x,y)\) is the gray value of the image element at position \((x,y)\) in the image, step is the step between image elements, and \(k\) is a constant. The mean value of the two largest numbers in \(cn,cs,ce,cw\) is \(t\). According to the above idea, we can obtain the values of \(K_{1},K_{2}\) after anisotropic filtering as \(t_{1},t_{2}\), respectively, and the final weight function of the new background features is as follows:  
\[W_{g}(x,y)=\min(t_{1},t_{2}) \tag{13}\]  
\(W_{g}\) is the minimum value of \(t_{1},t_{2}\). This weight function is shown in Figure 4b; the target feature still uses the corner point intensity function \(W_{t}\) to describe it, and the final improved a priori weight function is obtained as:  
\[W=W_{t}\times W_{g} \tag{14}\]  
This weight function is shown in Figure 4d. Figure 4: The local a priori extraction method proposed by the algorithm in this paper. (**a**) Infrared image containing dim and small targets; (**b**) weighted image describing corner point information; (**c**) weighted image describing background information; (**d**) final weighted image obtained. Finally, the weights W are normalized, and the normalization formula is [17]:  
\[W(x,y)=\frac{W(x,y)-w_{\text{min}}}{w_{\text{max}}-w_{\text{min}}} \tag{15}\]  
where \(W(x,y)\) denotes the gray value of the image element at position \((x,y)\) of the weighted image and \(w_{\text{max}},w_{\text{min}}\) are the maximum and minimum gray values of the weighted image \(W(x,y)\), respectively. Comparing Figure 3 above with Figure 4, we can see that the a priori model proposed by [PERSON] et al. to describe the background feature information is not ideal for strong edge contour suppression in complex scenes due to the simple operation of taking the maximum value [17], and there is still a large amount of edge information in the final weight map. In contrast, the algorithm proposed in this paper combines the traditional spatiotemporal filtering idea with anisotropic filtering to describe the background information and it achieves a better suppression effect on the edge contour in the original image. However, because the background image description is obtained as the minimum value of \(K_{1},K_{2}\) after anisotropic filtering, this leads to a loss of energy for the target point in the final weight image.