## 1 Introduction
To gain a faster convergence speed while updating the parameter estimation, the steepest descent algorithm is adopted. Due to its success in data mining and machine learning, the Karhunen-Loeve (KL) transform [23] is utilized to convert several registered adjacent frames into a set of values of linearly uncorrelated variables called principal components. Then, the first principal component is extracted to construct the desired target value. Compared with the existing methods [6; 13], the proposed PCA-NUC shows a better noise elimination capability and an apparent visual quality improvement of the corrected images, as well as a faster convergence speed, with barely any ghosting artifacts. The flow chart of the proposed method is shown in Figure 1.  
Here, the responses of all the detectors are assumed to be linear, and gain and bias are employed to model the nonuniformity. The observed output of the detectors \(Y_{i,j}(n)\) can be written as follows:  
\[ \begin{array}{c}Y_{i,j}(n)=a_{i,j}(n)\cdot X_{i,j}(n)+b_{i,j}(n)\\ (i\in[1,M];j\in[1,N])\end{array}  \tag{1}\]  
where \(a_{i,j}(n)\) and \(b_{i,j}(n)\) represent the real gain and bias of the \((i,j)th\) detector at a given frame \(n\) with a size of \(M\times N\), respectively. \(X_{i,j}(n)\) represents the real incident infrared photon flux, which is collected by the respective detector. By applying linear mapping to the observed pixel values, NUC is performed to provide an estimate of the true scene value  
Figure 1: The flow chart of the proposed PCA-NUC. obtained from \(U\). Therefore, we can extract the first principal component (\(U_{1}\)) to construct the desired target value \(T(n)\), which is the following:  
\[T(n)=U_{1}\cdot U_{1}^{\mathrm{T}}\cdot\left[Y_{i,j}(n)-\overset{-}{\mathbf{I}} _{n,K}\right]+\overset{-}{\mathbf{I}}_{n,K} \tag{9}\]  
By employing the steepest descent algorithm [4], \(g_{i,j}(n)\) and \(o_{i,j}(n)\) can be adaptively updated via the following equations:  
\[g_{i,j}(n+1)=g_{i,j}(n)+\lambda\cdot E_{i,j}(n)\cdot X_{i,j}(n) \tag{10}\]  
\[o_{i,j}(n+1)=o_{i,j}(n)+\lambda\cdot E_{i,j}(n) \tag{11}\]  
where \(\lambda\) denotes the iteration step size coefficient. When evaluating the estimation of nonuniformity parameters in this study, PCA-NUC is studied using a clean IR video sequence with synthetic nonuniformity and is compared with the original NN-NUC (\(3\times 3\) spatial average filter is used as suggested in [6]) and IRLMS methods by applying a set of real infrared image data. Since the stochastic steepest decent technique is utilized by all three methods to achieve an optimization of the correction coefficients, the same global step constant \(\lambda\) (\(\lambda=0.05\)) is adopted. The clean infrared data were captured with a properly calibrated \(320\times 256\) HgCdTe FPA camera operating in the 3-5 \(\upmu\)m range and working at 25 FPS from a tall building. Using a synthetic gain which has a unit-mean Gaussian distribution with a standard deviation of 0.1 and a synthetic offset with a zero-mean Gaussian distribution with a standard deviation of 30, the corrupted video sequence is obtained. To measure the difference between the true infrared image and the corrected result, we use the metric named root mean square error, which is defined as follows:  
\[RMSE=\frac{1}{M\times N}\cdot\sqrt{\sum_{i=1}^{M}\sum_{j=1}^{N}\left[\hat{Y}_{ i,j}-Y_{i,j}\right]^{2}} \tag{12}\]  
where \(Y_{i,j}\) and \(\hat{Y}_{i,j}\) are the true value and the corrected value of the \((i,j)th\) pixel, respectively. \(M\times N\) represents the size of the infrared image. As shown in Figure 2, the RMSE evolution of the three tested algorithms is displayed. The three curves decrease rapidly in the first 50 frames. The red curve is the RMSE result of the PCA-NUC method, in which the result reduced to below 20 within 20 frames, whereas more than 50 frames are needed for the other two methods. In the first 40 frames, the RMSE for the NN-NUC method drops almost as fast as for the IRLMS, but it falls  
Figure 2: RMSE versus frames using different NUC methods.