# Zero-TCE: Zero Reference Tri-Curve Enhancement for Low-Light Images
## 3 Materials and Methods
### Tri-Curve Enhancement for Light (TCE-L)  
Images captured under low-light or backlight conditions often suffer from insufficient brightness and a lack of detail in dark areas due to inadequate lighting. To address this, we propose the use of TCE-L to adjust the brightness information about the images. The TCE-L framework is illustrated in Figure 2.  
The input of TCE-L is a low-light image, and its output is a set of pixelated curve parameter maps corresponding to high-order curves. We employ a 7-layer convolutional neural network (CNN) [42] architecture. Each layer consists of 32 convolution kernels with a size of 3 \(\times\) 3 and a stride of 1, followed by an ReLU activation function. The final layer uses a Tanh function to map the parameter maps into the range of \(-1\) to 1, and a Split function is utilized to decompose the parameter map into 8 parameter maps with 3 channels each, which are then utilized by the enhancement module.  
Figure 1: The architecture of the proposed Zero-TCE.  
Figure 2: The architecture of the proposed TCE-L module.  
To enable automatic mapping of low-light images to their brightness-enhanced versions, we adopt a second-order light enhancement curve (LE-curve). The formula can be expressed as follows:  
\[LE(I(\mathbf{x});\alpha)=I(\mathbf{x})+\alpha I(\mathbf{x})[1-I(\mathbf{x})], \tag{3}\]  
where \(I\) denotes the input low-light image, \(\mathbf{x}\) represents the pixel coordinates, and the function \(LE(I(\mathbf{x});\alpha)\) signifies the enhanced image output at \(\mathbf{x}\). \(\alpha\in[-1,1]\) is a trainable curve parameter that adjusts the magnitude of the LE curve and controls the exposure level concurrently. Each pixel is normalized to the range \([0,1]\), and all operations are performed on a per-pixel basis. To achieve more versatile adjustments and tackle challenging low-light environments, we iteratively apply the LE curve defined in Equation (3). Specifically, it can be expressed as follows:  
\[LE_{n}(\mathbf{x})=LE_{n-1}(\mathbf{x})+\alpha_{n}LE_{n-1}(\mathbf{x})[1-LE_{n-1}(\mathbf{x})], \tag{4}\]  
where \(n\) is the number of iterations, controlling the curvature of the curve. In this paper, we set \(n\) to 8, which suffices for most scenarios. When \(n\) equals 1, Equation (4) is reduced to Equation (3). Figure 3 displays the LE curves with different parameters. From this figure, it can be observed that each pixel value of the enhanced image falls within a certain range, and the curve is monotonic and differentiable during gradient backpropagation. As seen in Figure 3b, after iterations, the curve gains stronger adjustment capabilities, enabling it to more effectively enhance image brightness. Global mapping often results in the over- or under-processing of local regions, leading to the amplification of noise along with the enhancement of dark detail. To address this, we define \(\alpha\) as a pixel-wise parameter, meaning each pixel of the input image is assigned an optimal fitting curve in order to adjust its dynamic range. This approach avoids drastic adjustments to image pixels directly, as shown in Equation (2).