## 4 Experimental Verification
PSNR is used to measure the pixel quality of an image, and the larger the value, the better the quality of the image, as represented by the calculation formula shown in Equation (17):  
\[PSNR=10\log\frac{M\times N\times 255^{2}}{\sum\limits_{i=1}^{M}\sum\limits_{j=1}^{ N}\left[W(i,j)-W^{\prime}(i,j)\right]^{2}} \tag{17}\]  
where \(M\times N\) denotes the image size, and \(W(i,j)\) and \(W^{\prime}(i,j)\) denote the greyscale values of the image before and after enhancement. SSIM was used to measure the similarity of an image, and its value is in the range of 0-1. The higher the value, the lower the distortion; standard deviation is used to measure the contrast of an image. The higher the value, the higher the contrast; the mean gradient is used to measure an image's sharpness and texture variation. The higher the value, the more precise the image; the mean value is used to measure the brightness of an image. The higher the value, the higher the brightness, and the information entropy of an image contains the amount of information contained in the entropy of an image. The higher the entropy value, the more informative the image. It is calculated as shown in Equation (18):  
\[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{xy}+c_{2})}{(\mu_{x}{}^{2}+\mu _{y}{}^{2}+c_{1})(\sigma_{x}{}^{2}+\sigma_{y}{}^{2}+c_{2})} \tag{18}\]  
where \(x\) and \(y\) are images of the same size, \(\mu_{x}\) is the mean of \(x\), \(\sigma_{x}\) is the variance of \(x\), \(\sigma_{xy}\) is the covariance of \(x\) and \(y\), and \(c_{1}\) and \(c_{2}\) are constants. Each calculation takes an \(N\times N\) window from the image, and then keeps sliding the window for calculation, and finally takes the average value as the global SSIM. The validation process used the MSR algorithm, the MSRCR algorithm and the BF-MSRCR algorithm for comparative validation. It measures the effectiveness of the enhanced images according to the evaluation criteria mentioned above, and the validation results are shown in the following Tables 1 and 2 (using downhole image 2 as an example). \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm Categories** & **PSNR** & **SSIM** & **Information Entropy** \\ \hline homomorphic filtering & 13.45 & 0.75 & 12.44 \\ Laplace enhancement & 14.98 & 0.71 & 12.05 \\ MSR & 11.62 & 0.66 & 11.66 \\ MSRCR & 15.93 & 0.79 & 12.37 \\ BF-MSRCR & 19.90 & 0.81 & 13.70 \\ \hline \end{tabular}
\end{table} 
Table 1: Evaluation criteria for downhole image enhancement effect. As can be seen in Table 1, the BF-MSRCR algorithm improves the peak signal-to-noise ratio, structural similarity and image information entropy by 8.28, 0.15 and 2.04, respectively, compared to the MSR algorithm; by 3.97, 0.02 and 1.33, respectively, compared to the MSRCR algorithm; by 6.45, 0.06, 0.93, respectively, compared to the homomorphic filtering algorithm; and by 4.92, 0.1, 1.65, respectively, compared to the Laplace enhancement algorithm. As can be seen in Table 2, the BF-MSRCR algorithm improved the standard deviation, mean gradient and mean value by 40.39, 7.38 and 52.92, respectively, compared to the MSR algorithm; by 34.33, 6.46 and 26.21, respectively, compared to the MSRCR algorithm; by 24.76, 4.72 and 42.34, respectively, compared to the homomorphic filtering algorithm; and by 22.18, 7.0 and 33.56, respectively, compared to the Laplace enhancement algorithm. All evaluation criteria of the enhanced images improved, indicating that the quality of the images enhanced by the BF-MSRCR algorithm was optimised in all aspects. Through objective verification of the indicators in Tables 1 and 2, the image quality, brightness and contrast are shown to be enhanced by the BF-MSRCR algorithm are significantly improved. The amount of information in the image also increased, while the anti-distortion performance of the image improved in the presence of disturbing factors, such as low contrast, low brightness, uneven light distribution and high dust influence in the underground mine image.