# Efficient Gamma-Based Zero-Reference Deep Curve Estimation for Low-Light Image Enhancement
## 3 Methodology
### Gamma-Based Light-Enhancement (GLE) Curve  
Our proposed formula is designed to meet the following critical requirements: (1) Enhanced images should be confined to the [0, 1] normalized range, avoiding information loss from overflow truncation. (2) The curve should possess both brightness enhancement and overexposure suppression capabilities. (3) Akin to the LE curve, our curve must maintain monotonicity within the [0, 1] range to preserve the differences between adjacent pixels. In the subsequent sections, we will detail the step-by-step derivation of our final formula, starting from the standard Gamma correction. Here, we first introduce the classic Gamma correction formula, which is defined as follows:  
\[y=cx^{\gamma},\quad x\in[0,1] \tag{2}\]  
where \(x\) represents the input intensity, \(c\) is the scaling coefficient, \(y\) represents the output intensity, and \(\gamma\) is the correction parameter. First, we explore the constraint issue of normalized range of enhanced pixel values. In our work, the parameter \(c\) is set to 1, ensuring that, when input pixel values \(x\) are within the [0, 1] interval, the corrected pixel values \(y\) remain naturally limited to the [0, 1] interval. Based on this property, we decide to retain the design using the original pixel \(x\) as the base such that the formula becomes  
\[y=x^{\gamma},\quad x\in[0,1]. \tag{3}\]  
Next, we analyze how the curve can simultaneously possess the dual functions of brightness enhancement and overexposure suppression. As illustrated in Figure 1, it is evident that, when the \(\gamma\) value is between 0 and 1, the pixel values after Gamma transformation are increased relative to the original values, thereby achieving image brightness enhancement. Conversely, when the \(\gamma\) value exceeds 1, the transformed pixel values decrease, resulting in reduced image brightness and effective suppression of overexposure. To achieve both enhancement of low-illumination regions and suppression of overexposed regions, we modify the formula as follows:  
\[y=x^{1+g},\quad x\in[0,1]. \tag{4}\]  
In this formula, \(x\) still represents the normalized original pixel value, \(y\) is the adjusted pixel value, and \(g\) represents the trainable curve parameter. Through the Tanh activation function, we strictly limit the range of \(g\) to the [\(-1\), 1] interval, with different pixel values \(x\) corresponding to different parameters \(g\). This design enables the curve function to simultaneously meet the requirements of enhancing low-light regions and suppressing high-light regions. However, in practical applications, the above formula poses potential problems. As shown in Figure 2a, when the pixel value \(x\) is small and \(g\) is negative, the adjusted pixel value \(y\) undergoes dramatic changes compared to the original value \(x\), leading to unstable model training. Particularly when parameter \(g\) approaches \(-1\), the function curve becomes abnormally steep, potentially triggering a gradient explosion issue that severely affects the model's convergence performance. To resolve the stability issue, we need to introduce an adaptive regulation factor into the formula to limit the magnitude of changes during brightness adjustment and avoid excessive enhancement. Based on this idea, we attempt to incorporate the original pixel value \(x\) as a regulation factor into the calculation of \(\gamma\), forming the following formula:  
\[y=x^{1+gx},\quad x\in[0,1]. \tag{5}\]Unfortunately, experimental results indicate that directly employing \(x\) as a regulation factor remains suboptimal. This direct application results in excessive regulation, thereby overly suppressing the model's ability to enhance dark regions. This can be observed in Figure 2b, where, when the parameter \(g\) is set to \(-1\), low-pixel-value regions exhibit almost no noticeable changes, which contradicts our original intention of enhancing dark areas.