# Small Infrared Target Detection via a Mexican-Hat Distribution
## 1 Introduction  
Small infrared (IR) target detection has been widely used in airborne early warning, IR guidance, surveillance and tracking, and others. Usually, the earlier we detect a small IR target, the more time we get for dealing with it, and thus the more suitable decision can be made. However, factors such as low signal-to-noise ratio (SNR), variable target sizes, variable target intensity, less shape and texture information, blurred edges, and serious background clutters, cause small IR target detection to be a challenging task. So far, many IR target detection algorithms have been developed ranging from recursive estimation techniques [1] to partial sum of the tensor nuclear norm [2]. Among them, morphology filtering such as the top-hat transform and its variants plays an important role [3, 4]. Top-hat transforms are also combined with other techniques such as genetic algorithm [5] to improve the detection performance. Though these methods can detect targets to a certain degree, their performance is greatly degraded under complex background clutters. Over the last few years, it is widely shared that algorithms derived from visual attention mechanics work well for detecting a small target. The early work in 1998 made intellectuals pour attentioninto visual attention distribution [6], and then saliency estimation methods corresponding to visual attention became popular in small IR target detection [7]. [PERSON] et al. are the early researchers to use the Difference of Gaussian (DoG) filters to get a saliency map for detecting real targets [8]. However, due to problems like signal interference, a small target does not strictly obey the 2D Gaussian distribution, and thus cannot be successfully detected by merely adopting DoG filters. Further studies on visual attention focus on the relationship between a target and its neighborhoods. The local contrast method [9; 10], the multiscale patch-based contrast measure [11], weighted local difference measure [12], human visual mechanism detectors [13; 14], variance difference detector [15], entropy-based contrast measure [16], weighted local contrast detector [17], local intensity and gradient detector [18], visual saliency guided detector [19], local difference adaptive measure [20], the adaptive local measurement contrast, and salient region extraction and gradient vector processing [21] are reported successively, demonstrating that the contrast between a target and its neighbor is helpful for small IR target detection. Though many improved algorithms have been reported, it is still challenging to detect a small IR target in images with low SNR, low contrast, and serious background clutters. This paper presents an effective detection method based on the Mexican-hat distribution. A raw IR image is first processed by the modified top-hat transformation [22] and the DoG filter, getting a filtered IR image. Then, the adjacent region around a pixel of the filtered image is radially divided into three sub-regions. Next, the pixels whose adjacent regions have the Mexican-hat distribution are determined as candidate targets. Finally, a small target is segmented out by locating the brightest pixel. Experimental results show that the adoption of the Mexican-hat distribution benefits our method with higher detection rate, lower false alarm rate, and faster detection speed than existing detectors. Though the existing detector in [15] is resemble to our method, it performs on raw IR images and only compares the relationship between two regions. On contrast, we operate on the filtered image and detect an IR target based on the Mexican-hat distribution, and our experimental results show that such strategy can effectively enhance the small IR target and improve the detection rate. The rest of the paper is organized as follows. Section 2 analyzes the characteristics of the adjacent region of a target in a DoG-filtered image. Section 3 describes the details of the proposed method. Section 4 presents experimental results along with some analysis.