# Low-Light Image Enhancement Method Based on Retinex Theory by Improving Illumination Map
## 3 The Proposed Algorithm
### Illumination Map Enhancement Based on Gaussian Laplacian Pyramid Fusion
#### 3.2.1 Brightness Enhancement of Illuminance Map  
This section designs the BEF function by analyzing several brightness enhancement functions given in Section 2.2. Figure 6 shows two graphs for the different transformation functions, and the values of Equations (2) and (4) at any point are higher than the identity line, indicating that the pixel value at any point of the enhanced image is higher than that of the original image.  
Figure 7 displays one raw low-light image and its counterparts enhanced by different transform functions. Both GC (2) and truncated normalized sigmoid (4) can enhance the brightness of the image. However, Equation (4) has limited enhancement ability for dark regions marked by red boxes, as shown in Figure 7c. Observing the curves of each formula in Figure 6a, it is found that GC (2) has the greatest enhancement in the region of low pixel value. Therefore, we can hopefully enhance the brightness details simultaneously for dark regions by cascading GC (2) and sigmoid (4) as follows:  
\[y_{1}=\sqrt[2]{E_{1}}. \tag{11}\]  
Figure 5: Experimental results of several methods in the brighter areas of the image. (**a**) Original image; (**b**) BIMEF; (**c**) LIME; (**d**) LECARM; (**e**) SDD.  
Figure 6: Different transformation function graphs. (**a**) The graphs of Equations (2), (4), (11) and (13); (**b**) Graph of the \(SF\) function.  
Figure 7d shows the image enhanced by Equation (11), which is better than Equations (2) and (4). However, for the window area in the red box of the image, Equation (11) produces overexposure, resulting in complete loss of image content. This is due to the fact that in the high-pixel value region, the curve of Equation (11) is much higher than the identity line in the high-pixel value region, as shown in Figure 6a. Inspired by the characteristics of the sine function (\(SF\)), in which the values are greater than 0 between [0, 0.5] and less than 0 between [0.5, 1], as shown in Figure 6b, we accordingly considered adding the \(SF\) function to the original image to suppress the bright areas in the image. We designed the following transform function:  
\[s_{i}=S_{i}+SF(S_{i}), \tag{12}\]  
where  
\[SF(S_{i})=\sin(S_{i}\times 2\pi)/2\pi\]  
It is obvious that we have \(SF(S_{i})\geq 0\) for \(0\leq S_{i}\leq 0.5\), while \(SF(S_{i})\leq 0\) for \(0.5\leq S_{i}\leq 1\).  
Obviously, the enhancement result of Equation (12) can retain more details as shown in Figure 8a, but the whole image is dark. In order to further improve the visual quality, we combine Equations (11) and (12) to obtain the first image for fusion:  
\[I_{1}=0.5\times(y_{1}+s). \tag{13}\]  
Figure 8: Enhanced results of the raw image shown in Figure 7a. (**a**) Image enhanced by Equation (12); (**b**) image enhanced by Equation (13).  
Figure 7: Enhanced results for several transformation functions. (**a**) Original image; (**b**) image enhanced by Equation (2); (**c**) image enhanced by Equation (4); (**d**) image enhanced by Equation (11).  
So far, we have completed the luminance improvement of the illuminance map through Equation (13).  
The corresponding plot of Equation (13) is shown in Figure 5(a), where the entire curve is almost always above the identity line with the first part of the curve being significantly higher than the identity line, due to the need for greater enhancement in the low pixel value region. However, it almost coincides with the line near 1, indicating that there is almost no enhancement here, so that over-enhancement can be avoided. Figure 7(b) shows the enhanced result with Equation (13). As can be seen, the resulting image not only improved the brightness, but also was not overexposed. It is clearly better than that obtained with Equation (12).