# A Low-Brightness Image Enhancement Algorithm Based on Multi-Scale Fusion
## 3 Proposed Method
### Brightness Transformation Function  
The main characteristic of a low-light image is that the image brightness and contrast are low, resulting in poor detail recognition. To address this problem, researchers have explored various methods to improve image brightness. The simplest and most direct approach is to use linear functions. However, this method neglects the spatial distribution of image brightness, resulting in over-saturation in high-brightness areas and significant loss of detail.  
To mitigate these problems, researchers have turned to various brightness transformation function (BTF) models. The most representative of these BTF models are nonlinear functions, which are better suited for improving image brightness. By adjusting parameters, these nonlinear functions modify the enhancement amplitude of the image brightness, avoiding the distortion caused by uniform enhancement across varying brightness levels. This ensures the naturalness of the enhancement results. Among the nonlinear functions used, the gamma function [8] is widely used and can be expressed as:  
\[s(I(x))=c\cdot I(x)^{\alpha} \tag{1}\]  
where \(c\) is the constant, \(I(x)\) is the input image, and \(\alpha\) is the model adjustment parameter. However, due to unreasonable assumptions of model parameters, the enhancement result often looks unreal.  
[PERSON] [25; 26] noted that the histogram of the underexposed image is mainly concentrated in the low-brightness region. Consequently, by linearly increasing the pixel values before conventional gamma correction, the resulting image will closely resemble a well-exposed image. Therefore, he proposed a new model: the camera response model of beta-gamma, which was modified on the basis of gamma correction. Before gamma correction, the pixels of the image were enlarged to obtain the brightness transformation function \(g(I(x),k)\):  
\[\begin{split} g(I(x),k)&=\beta\cdot I(x)^{\gamma} \\ \beta&=e^{b(1-k^{a})}\\ \gamma&=k^{a}\end{split} \tag{2}\]  
where \(I(x)\) is the input image, \(a\) and \(b\) are the parameters of the camera, \(k\) is the exposure rate, and \(\beta\) and \(\gamma\) are calculated by Equation (2). Finally, the brightness transformation function is:  
\[g(I(x),k)=e^{b(1-k^{a})}\cdot I(x)^{k^{a}}. \tag{3}\]  
However, in this model, the image exposure ratio needs to be estimated in advance, which makes the algorithm complex. Therefore, we propose to use a new and simple brightness transformation function, which can be defined as  
\[L(I(x),a)=(a+1)I(x)-aI(x)^{2} \tag{4}\]  
where \(I(x)\) is the input image, and \(a\in(0,1)\) is the parameter, which controls the exposure level and adjusts the magnitude of the curve. This model can be iterative as follows:  
\[L_{n}(x)=(a+1)L_{n-1}(x)-aL_{n-1}(x)^{2} \tag{5}\]where \(n\) is the number of iterations. The graphs of Equation (5) are as follows:  
As can be seen in Figure 2b, \(n\) controls the curvature, and higher-order curves have a stronger fitting ability (i.e., larger curvature). Compared to the model proposed by [PERSON], our model is more concise. However, this model is still a global adjustment. A global mapping to adjust image brightness can easily cause image brightness distortion. Therefore, it is also necessary to process the image after the brightness transformation.