# An Image Denoising Method for a Visible Light Camera in a Complex Sky-Based Background
## 2 Materials and Methods
### Wavelet Transform and Curvelet Transform
#### 2.1.2 Limitations of the Wavelet Transform and Introduction of the Ridgelet Transform  
Although the wavelet transform, known as the \"mathematical microscope\", can express the details of image waveforms and has significant application value and advantages, it has a limitation due to the lack of anisotropy in wavelet bases. The wavelet transform can only reflect discontinuities or points where the derivative is zero in 1D images, and it cannot capture the edge features of 2D spatial images. Therefore, at the end of the 20 th century, [PERSON] developed the ridgelet transform algorithm [PERSON] (1980), which expresses line singularities and can provide a good approximation of line singularities in multivariable functions. The expression of the ridgelet transform is given by:  
\[\Psi_{\gamma}(x)=a^{-\frac{1}{2}}\Psi\bigg{(}\frac{[u,x]-b}{a}\bigg{)} \tag{6}\]  
where \(\Psi_{\gamma}\) is called a ridgelet generated from \(\Psi\) under specific admissibility conditions. \(a\) represents the scale parameter, \"\(u\)\" represents the direction parameter, and \(b\) represents the position parameter. \(\Psi_{\gamma}\) has rapid decay properties that satisfy:  
\[\int\Psi(t)dt=0 \tag{7}\]  
For functions with several variables, the ridgelet transform is defined as:  
\[R(f)(r)=\langle f,\Psi_{r}\rangle \tag{8}\]  
The ridgelet transform represents the direction by adding an additional parameter to the wavelet basis function. In the cross-section, the basis of the ridge is the wavelet. The continuous ridgelet transform in 2D space is expressed in Equation (9).  
\[CRT_{f}\big{(}a,b_{x},b_{y}\big{)}=\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}f(x,y)\Psi a,b_{x},b_{y}(x,y)dxdy \tag{9}\]  
When comparing the expressions of the 2D wavelet transform, as shown in Equation (10), with the ridgelet transform, we can observe the following contrast: in the wavelet transform formula, the parameter is a point, whereas in the ridgelet transform, the parameter is a line.  
\[w_{f}\big{(}a,b_{x},b_{y}\big{)}=\int_{-\infty}^{\infty}\int_{-\infty}^{ \infty}f(x,y)\Psi a,b_{x},b_{y}(x,y)dxdy \tag{10}\]  
The introduction of the ridgelet transform solves the problem of the wavelet transform's inability to describe 2D edge singularities. However, the ridgelet transform still has limitations when dealing with functions with curve singularities. Moreover, in practical images, edges are rarely straight lines, which limits the widespread application of the ridgelet transform. Additionally, the ridgelet transform suffers from high computational redundancy. Therefore, in the last year of the 20 th century, [PERSON] and [PERSON] developed the curvelet transform and formulated its theory. The curvelet transform effectively captures the singular points and characteristics of curves in images. The basis of the curvelet transform exhibits directionality and anisotropy, allowing for the optimal approximation and description of nonlinear parts of 2D images. Thus, curvelet analysis combines the properties of wavelet multiresolution and locality with directionality, providing an optimal approach for representing images with these three features.