# An Image Denoising Method for a Visible Light Camera in a Complex Sky-Based Background
## 2 Materials and Methods  
### Wavelet Transform and Curvelet Transform  
2.1.1. Wavelet Transform  
In the 1980s, [PERSON] first proposed the concept of wavelets [20]. In the same period, mathematicians [PERSON] and [PERSON] further studied the Mallat algorithm based on previous work, and the wavelet transform was officially created [21; 22]. From a theoretical perspective, wavelet transform theory evolved from Fourier analysis. Fourier analysis can be used to analyze the characteristics of data separately using the time axis and frequency axis transformations, but it cannot reflect the characteristics of signals changing simultaneously along both the time and frequency axes. It can only consider one of the two axes, allowing a global understanding of the signal's characteristics, but is unable to describe the local characteristics of a specific time-frequency region of the signal. In reality, signals are often nonstationary, and it is difficult to maintain stability. Most signals are nonstationary, and the local time-frequency characteristics precisely describe the distinctive features of nonstationary signals. While scaling the signal by shrinking or expanding the variable multiplier and transforming the signal through smooth distance shifts, the wavelet transform can decompose the signal into multiple levels, surpassing the limitations of the Fourier transform in this aspect. The traditional approach of the wavelet transform is to separate the high-frequency and low-frequency signals for processing. The low-frequency signal, which undergoes minimal changes, is expanded to analyze the fine details of the rapidly changing parts in the high-frequency signal. In the time dimension, the signal is further segmented, while in the low-frequency signal, the frequency is divided more finely, adjusting the time-frequency characteristics analysis based on the specific high and low frequencies of the signal. The continuous wavelet transform involves extensive data operations, so it needs to be discretized during implementation to simplify the form. In the expression of continuous wavelets, the binary number scale parameter \(a\) and the translation parameter \(b\) remain continuous, and only the parameters in the semi-discrete continuous wavelet formula are discretized while the variable \(t\) remains unchanged. The discrete form of the wavelet transform is:  
\[(W_{\Psi}f)\left(\frac{1}{2^{i}},b\right)=\int_{-\infty}^{\infty}f(t)\Big{\{}2 ^{j/2}\Psi\left(2^{j}(x-b)\right)\Big{\}}dx \tag{1}\]  
This transformation is also known as the binary wavelet transform, which meets the condition:  
\[A\leq\sum_{j\in z}\left(2^{-j}\omega\right)\Bigg{|}^{2}\leq B \tag{2}\]  
Here, \(0<A<B<\infty\), where \(A\) and \(B\) are constants. For a certain level of decomposition, the binary wavelet transform can be considered \(a\) function of the variable \(b\) in Equation (2), and \(b\) is not discretized. Equation (2) can be transformed into a convolution operation expression, which is shown below:  
\[W_{s}f(x)=f^{*}\Psi_{s}(x) \tag{3}\]  
By scaling the wavelet mother function \(\Psi(x)\) with a factor of \(s\) (where \(s=2\bar{\jmath}\)), we obtain the wavelet function as follows:  
\[\Psi_{s}(x)=\frac{1}{s}\Psi_{s}\Big{(}\frac{x}{s}\Big{)} \tag{4}\]  
Let us assume \(f_{\tau}(x)=f(x-\tau)\). Then,  
\[\left(W_{2\bar{\jmath}}f_{\tau}\right)(x)=W_{2\bar{\jmath}}[f(x)] \tag{5}\]If we first shift the function \(f\) along the scale and then perform the binary wavelet transform, it is equivalent to first applying the binary wavelet transform and then shifting it along the scale. In other words, \(W_{2j}f\) possesses the properties of \(f\), which is the meaning of translation invariance. Although the binary wavelet transform is in a discrete form, the variable \(b\) is not discretized along the scale, and the transformation of the wavelet transform at different scales is not discrete. This translation invariance exhibits a zooming characteristic.