## 2 Proposed Approach
### Nonuniformity Correction Model Based on \(1d\) Guided Image Filtering  
The guided filter denoising model was first proposed by [PERSON] (2012) in 2012. In 2015, [PERSON] (2015) applied the guided filter denoising model to the nonuniformity correction of infrared uncooled array detectors. They obtained good results. In this subsection, we propose a novel non-uniform correction model based on two 1-D guided filtering for infrared focal plane linear array scan detectors. The implementation of guided filtering [PERSON] (2012) is based on the assumption that the guided picture and the output image captured in a limited window have a linear connection, as follows:  
\[q_{i}=a_{k}v_{i}+b_{k},\forall i\in w_{k} \tag{1}\]  
where \(q_{i}\) is the pixel value of the output image, \((a_{k},b_{k})\) is the linear coefficients in the small window, and \(v_{i}\) is the pixel value of the guided image. \(w_{k}\) is the window, and noise characteristics must be taken into account when designing the window form. The guided filtering is completed by constraining the minimum difference between the output image \(q_{i}\) and the input image \(u_{i}\):  
\[E(a_{k},b_{k})=\sum_{i\in w_{k}}((a_{k}v_{i}+b_{k}-u_{i})^{2}+ea_{k}) \tag{2}\]  
where \(E(a_{k},b_{k})\) is the cost function, \(\varepsilon\) is a regularization parameter penalizing large \(a_{k}\). The results of coefficients \((a_{k},b_{k})\)He (2012) is shown as  
Figure 1: Non-uniform correction effect of real infrared scanning image. (**a**) Raw infrared scanning image. (**b**) Non-uniform correction results of our proposed algorithm. \[a_{k}=\frac{\frac{1}{|w|}\sum_{i\in w_{k}}v_{i}u_{i}-\mu_{k}\bar{a}_{k}}{\frac{1} {|w|}\sum_{i\in w_{k}}v_{i}^{2}-\mu_{k}^{2}+\varepsilon} \tag{3}\]  
\[b_{k}=\bar{u}_{k}-a_{k}u_{k} \tag{4}\]  
where \(\mu_{k}\) is the mean of the guide image \(v\) in the window, \(\bar{u}_{k}\) is the mean of the input image \(u_{i}\) in the window \(w_{k}\), and \(|w|\) is the number of pixels in the window \(w_{k}\). \(\varepsilon\) is a manually set parameter, which determines the strength of local averaging. Its effect is similar to the range variance parameter \(\sigma_{r}^{2}\) in a bilateral filter [28]. However, a pixel \(i\) is included in all overlapping windows \(w_{k}\) that cover \(i\), so the value of \(q_{i}\) in (4) is not the same when computed in various windows. A basic technique is to calculate the mean of all conceivable values of \(q_{i}\). Thus, after computing \((a_{k},b_{k})\) for each window \(w_{k}\) in the image, we compute the output of the filtering by  
\[q_{i}=\frac{1}{|w|}\sum_{k|i\in w_{k}}(a_{k}v_{i}+b_{k})=\bar{a}_{i}v_{i}+\bar {b}_{i} \tag{5}\]  
where \(k|i\in w_{k}\) defines a number of overlapping windows \(w_{k}\) covering the pixel \(i\), \(\bar{a}_{i}=\frac{1}{|w|}\sum_{k\in w_{k}}a_{k}\), and \(\bar{b}_{i}=\frac{1}{|w|}\sum_{k\in w_{k}}b_{k}\) are the coefficient averages of all windows overlapping pixel \(i\). For the nonuniformity of the infrared focal plane linear array scan detector, this paper designed a guided filtering correction model, as shown in Figure 2.  
Since the nonuniformity of the focal plane linear array scan image appears as horizontal strips, the algorithm first takes the infrared raw image \(l_{p}\) as a guided image and an input image. It smooths the image through a 1D column guided filter window, and at this point, Equations (3) and (4) become Equations (6) and (7). \[a_{k}^{{}^{\prime}}=\frac{\sigma_{k}^{2}}{\sigma_{k}^{2}+\varepsilon} \tag{6}\]  
\[b_{k}=(1-a_{k}^{{}^{\prime}})\mu_{k} \tag{7}\]  
where \(\mu_{k}\) and \(\sigma_{k}^{2}\) are the mean and variance of the image \(I_{p}\) in the window \(w_{k}\). Equation (1) is used to conduct guided filtering operations, and the output image \(I_{u}\) is the smoothed image in the vertical direction.