# Infrared Stripe Correction Algorithm Based on Wavelet Analysis and Gradient Equalization
## 3 Stripe Correction New Algorithm Based on Wavelet Analysis and Gradient Equalization
### Wavelet-Based Image Decomposition  
Wavelet analysis is a rapidly developing area in mathematics and engineering. It has been successfully applied in several fields. Originating in Fourier analysis, the wavelet transform has the characteristics of multi-scale analysis and the ability to represent the local characteristics of signals in both time and frequency domains. Thus, it can effectively extract information from signals [1; 24]. In image denoising algorithms, the wavelet transform can generate multi-scale representation of the input image. One of the most important properties of the wavelet transform is that the information in different directions of the image can be decomposed into corresponding components. [PERSON] et al. [25; 26] proposed an image cartoon-texture decomposition and sparse representation algorithm, which decomposes the image into cartoon parts and texture parts, and uses sparse representation in the texture part of the image, which can preserve the texture information of the image well. But the noise of the infrared stripes mainly exists in the vertical direction, and so therefore, we choose wavelet decomposition to represent the information of different directions of the image.  
In image denoising, continuous small waves and their wavelet transform should be discretized [8; 24]. Mathematically, this is defined as:  
\[\psi_{m,n}=\frac{1}{\sqrt{2^{m}}}\psi(\frac{1}{2^{m}}-nT_{s}).\;m,n\subset Z \tag{1}\]  
where \(nT_{s}\) represents the sampling interval, \(m\) represents the sampling scale. \(m\) and \(n\) take integers. Formula (1) is the discrete wavelet function. Then the discrete wavelet transform [1; 8] of \(g(t)\) is:  
\[W_{f}(m,n)=\int\limits_{R}g(t)\psi_{j,k}^{*}(t)dt \tag{2}\]  
In order for the wavelet transform to capture the variability of spatial and frequency resolution, a dynamic sampling grid is adopted, and a binary wavelet is commonly used to realize the function of signal zoom analysis. In Formula (3), \(n\) corresponds to \(nT_{s}\) in Formula (1):  
\[Y_{m,n(t)}=\frac{1}{\sqrt{2^{m}}}\psi(\frac{1}{2^{m}}-n) \tag{3}\]  
The wavelet transform extracts high frequency components \(U(i)\) from the original image \(Y(i)\), and the formula is:  
\[Y(i)=X(i)+U(i)=X(i)+N(i)+V(i) \tag{4}\]  
According to Formula (4), the high-frequency components not only contain stripe noise \(N(i)\), but also vertical texture information \(V(i)\). After wavelet decomposition, the signal \(U(i)\) is expressed as:  
\[U(i)=\{A_{i},H_{i},V_{i},D_{i}\} \tag{5}\]  
Figure 2: Workflow of our proposed algorithm. Please zoom in to check details.  
where \(A_{i}\) represents the approximate component, \(H_{i}\) the horizontal component, \(V_{i}\) the vertical component, and \(D_{i}\) the diagonal component. It can be clearly observed from Figure 3 that stripe noise is concentrated in the vertical component. According to this experimental conclusion, stripe noise can be eliminated directly in the vertical component. We tested three different wavelet bases: db1 wavelet, haar wavelet, and sym8 wavelet. Our proposed method chooses three wavelet basis functions for experiments, and the results are shown in Figure 4. It can be seen from the highlighting in the figure that the haar wavelet and the sym8 wavelet cause \"ghosting\" problems. The db1 wavelet preserves the vertical texture information of the image well.