# Infrared Dim and Small Target Detection Based on the Improved Tensor Nuclear Norm
## 1 Introduction
The traditional filtering method only uses the local feature information of the target to predict the background or highlight information by filtering without using the global properties of the image, so the false alarm rate of the detection effect is often high under complex and variable scenes. Numerous scholars have explored the regularity of the distribution of dim and small targets and used the ability of neural networks to approximate any function to apply neural networks to dim and small target detection. [PERSON] [8] constructed GAN models based on U-net networks to learn the features of dim and small targets, predict the target intensity, and obtain better detection results. [PERSON] [9] integrated the guiding image filter and convolutional neural network into particle filtering. The guiding image filter was shown to effectively retain the edge characteristics of the image; the convolutional neural network was able to effectively extract the target features, while the particle filtering algorithm could retain the sample information to produce a good detection effect. However, this type of algorithm requires a large number of samples to train the detection model to obtain good detection results, and because the target does not have a fixed shape and texture characteristics, the actual scene is also complex and variable. Thus, often, models pre-trained for the actual application of the results are not ideal. In recent years, infrared dim and small target detection based on the low-rank and sparse theory has received significant attention. The method considers global information about the target and background and divides the original image into three parts: the low-rank background image, sparse target image, and noise signal. The expression of this theoretical model is:  
\[f_{D}=f_{A}+f_{T}+f_{N} \tag{1}\]  
where \(f_{D}\), \(f_{A}\), \(f_{T}\), \(f_{N}\) represent the original image, the target image, the background image, and the noise signal, respectively. Based on the low-rank and sparse theory, [PERSON] [10] conducted an analysis to obtain the characteristics of the local block image of the background image and the approximate linear correlation of the block image. The original image was transformed into a block matrix through a sliding window, and the IPI model was obtained. The model expression is as follows:  
\[D=A+T+N \tag{2}\]  
where \(D\), \(A\), \(T\), \(N\) denote the original block image, background block image, target block image, and noise block image, respectively. Based on the IPI model, many scholars have improved the model to achieve better results [11; 12; 13; 14; 15]. [PERSON] [12] proposed a full-variance principal component tracking model based on the IPI model by using full variance to constrain the model, which achieved better detection results in scenes with few edge contours. [PERSON] [13] achieved better detection results through the construction of a new dim and small target detection model that combines the L1 and L2 norms to address the problem of leaving background residuals in the target image due to the defects of the nuclear norm and the L1 norm. Based on the IPI model, [PERSON] [14] transformed the NP-hard problem into a convex optimization problem by strengthening the constraints on the target image through the \(L_{p}\) norm while reducing the constraints on the background image to obtain better detection results. Although many scholars have improved the IPI model, in scenes containing many complex edge contours, there are still some contours in the background image that remain in the sparse target image, resulting in a high final false alarm rate. To solve the above problem, some scholars have extended the IPI model to the tensor domain. [PERSON] [16] proposed a robust infrared patch tensor model (RIPT) with a combination of the local prior and nonlocal prior before transforming the target and background image separation problem into a weighted robust tensor recovery problem.