# Research on Image Denoising in Edge Detection Based on Wavelet Transform
## 3 Materials and Methods  
### Wavelet Transform  
The wavelet transform replaces the infinitely trigonometric basis of the Fourier transform with a finite decaying wavelet basis. The wavelet basis has a finite energy, usually concentrates around a single point, and the integral value is zero. In the Fourier transform,the variable is only \(\omega\), while the wavelet transform contains two variables, namely the scale \(a\) and the translation \(b\). The scale \(a\) corresponds to frequency, and the translation \(b\) corresponds to time, so the wavelet transform can be used for time-frequency analysis to obtain the time-frequency spectrum of the signal. The wavelet sequence can be derived using the scaling and translation of the mother wavelet function, and the general form of the wavelet sequence is given below [14]. \[\psi_{a,b}(t)=\frac{1}{\sqrt{a}}\psi(\frac{t-b}{a})\;a,b\in R \tag{1}\]  
In the process of performing the wavelet transform, the scale factor \(a\) and the time shift \(b\) are theoretically continuously varying, but this is a calculation that a computer cannot complete in a finite amount of time. In the wavelet transform, the scale factor \(a\) and the time shift \(b\) are taken to be discrete according to certain rules, also known as the discrete wavelet transform (DWT). If the scale factor \(a\) and the time shift \(b\) are chosen according to a power of 2, the analysis of the signal will become more accurate and efficient [15; 16]. The wavelet function can be written as [14]. \[\psi_{m,n}(k)=2^{-\frac{\omega}{2}}\psi(2^{-m}k-n)\;m,n\in Z \tag{2}\]  
The wavelet transform can decompose the original image information into approximate and detailed components that mainly display the noise in the image. After the wavelet reconstruction is then performed on the thresholding detailed components, we can obtain smoother image information. The general process of wavelet transform denoising is shown in Figure 1.  
Based on the theory of function spaces, the wavelet transform is suitable for multi-scale analysis. Under different scales and space conditions, a set of scaling function vectors V and a set of wavelet function vectors V are created. At a particular level, the convolution of the signal in the scale space V obtains the approximation of the signal, which is a low-frequency signal. Meanwhile, the details of the signal obtained by convolving the signal in the wavelet space W are the high-frequency signals. During the decomposition process, the signal data are passed through a low-pass filter and a high-pass filter. The low-pass filter produces a low-frequency component of the signal, which is the approximation coefficient, while the high-pass filter produces the high-frequency component, which is the detail coefficient. The advantage of the wavelet transform is that it can effectively avoid the disadvantages of the Fourier transform. It develops the localization idea of the short-time Fourier transform, and it can provide a \"time-frequency\" window that changes with frequency, which is an ideal tool for time-frequency analysis and signal processing. It can highlight the characteristics of certain aspects of the problem by the transformation, localize the analysis of time (space) frequency and automatically adapt to the requirements of time-frequency signal analysis. Thus, it can focus on any details of the  
Figure 1: General process of wavelet transform denoising. signal to solve the difficult problems faced by the Fourier transform, making it a major breakthrough in the scientific method since the Fourier transform. In the wavelet transform, the functions are crucial and should be selected in a comprehensive manner in terms of support length, symmetry, regularity and similarity. The functions, such as db, sym, coif and fk [11; 17; 18; 19; 20; 21; 22; 23], are the most commonly used. db is short for Daubechies. The wavelet function was constructed by Daubechies and is generally abbreviated as db\(N\) (\(N\) being the order of the wavelet).