# Efficient Gamma-Based Zero-Reference Deep Curve Estimation for Low-Light Image Enhancement
## 3 Methodology
### Gamma-Based Light-Enhancement (GLE) Curve
Through in-depth analysis, we discover that the previous regulation factor excessively suppresses the model's ability to enhance dark regions. This necessitates an optimized design of the regulation factor to provide moderate suppression in dark areas without being  
Figure 1: The curves generated by the formula \(y=x^{\gamma}\) for different values of the parameter \(\gamma\). Figure 2: Comparison of the effects of including the regulation factor \(x\) versus excluding it for different values of the parameter \(g\). (**a**): The curves generated by the formula \(y=x^{1+g}\) without a regulation factor. (**b**): The curves generated by the formula \(y=x^{1+gx}\) with a regulation factor. overly restrictive. This requirement calls for a nonlinear transformation of the regulation factor. When \(x\) is directly used as the regulation factor, its gradient remains constant, resulting in poor enhancement effects in dark regions. The ideal regulation factor should be a function that monotonically increases within the \([0,1]\) range but with gradually decreasing derivatives. Based on the above analysis, we decide to apply exponential transformation to the regulation factor, adopting the form \(x^{a}\), where \(a\) is a hyperparameter. Validated through ablation studies, when \(a\) is set to 0.13, it can achieve good enhancement effects in dark regions while maintaining moderate suppression. As such, considering all the above design factors, our GLE curve formula can be defined as  
\[y=x^{1+gx^{a}},\quad x\in[0,1]. \tag{6}\]  
Theoretical analysis demonstrates that our proposed formula exhibits significant adaptive advantages. When the pixel value \(x\) approaches 0, although the regulation factor \(x^{0.13}\) also approaches 0, its gradient changes relatively smoothly, ensuring that the formula maintains stable and effective enhancement capabilities in low-light regions. Conversely, when \(x\) approaches 1, the regulation factor approaches 1, effectively suppressing excessive enhancement of high-light regions and preventing overexposure distortion. To further enhance performance, we observe that, while using the Tanh activation function limits parameter \(g\) to the \([-1,1]\) range, this constrains the formula's ability to suppress overexposure. To address this issue, we implement an asymmetric mapping for parameter \(g\) after Tanh activation, amplifying its value by a factor of 10 when \(g>0\) (used for overexposure suppression) while keeping it unchanged when \(g<0\) (used for brightening dark regions). This asymmetric design significantly enhances the model's control over high-light regions while preserving the fine adjustment capability for dark areas. Figure 3 illustrates the curve characteristics corresponding to different \(g\) values under the fixed hyperparameter \(a=0.13\). As such, the derivation of the proposed GLE formula and its parameter settings is ultimately complete. The proposed GLE formula satisfies the three objectives we previously established. It requires only a single application to accomplish the mapping of pixel brightness levels, thereby enhancing low-illumination regions while suppressing overexposed areas. Figure 3: The curves generated by the formula \(y=x^{1+gx^{a}}\) with asymmetric parameter values of g. (**a**): The curves of enhancement and suppression exposure with g limited to the range of \([-1,1]\). (**b**): The suppression exposure curves after being expanded by 10 times when g is greater than 0.