# An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization
## 4 Experimental Results and Analysis
### Face Denoising with Illumination Variation  
Face images of the same subject under different illumination conditions generally lie in a low-dimensional subspace, while the outliers resulting from lighting variations can be assumed to be sparse [52]. RPCA can therefore balance the uneven brightness of the image surface and retain important facial features while removing noise. Moreover, given enough facial images of the same person, it is possible to reconstruct real facial images. We experimented with 64 images from the Extended Yale B database for each participant. All images were converted to 32,256-dimensional column vectors, hence \(\mathbf{Y}\in\mathbb{R}^{32256\times 1}\) for each subject. Since the images were well aligned, L should have a rank of one. In order to demonstrate the effectiveness of the proposed method, we selected facial images of three subjects from the Extended Yale B database and added different noises to each subject to demonstrate the method's performance, as shown in Figures 13-15. By observing these three sets of photos, it can be shown that in comparison to the experimental outcomes of the comparison methods, the proposed method could more effectively remove the occlusion shadow of the image and keep the important features of the face image on the basis of successfully removing the noise, thereby restoring the true face image. The denoising results of RPCA and WNNM-RPCA were similar, and all of them eliminated a significant amount of noise when removing impulse noise (Figure 13). However, the brightness of the image surface was unbalanced, making it difficult to clearly distinguish facial characteristics. The AMlogtv method could not remove noise in facial images. The TDV method removed noise, but the shadow in the image was not removed, which made the face difficult to identify. Although NonRPCA removed most of the shadows of the face, the denoised image was not smooth enough to cause the detail structure to be very blurred. Similarly, even when removing Poisson and Gaussian noise, respectively, these comparison methods' denoising effects remained poor. For instance, when removing Poisson noise (Figure 14), WNNM-RPCA (Figure 14f) had the worst impact, not only not entirely removing the noise but also leaving half of the facial image in the shadow. Although NonRPCA (Figure 14e) preserved facial features well, it produced some artifacts in the denoising process. When removing Gaussian noise (Figure 15), RPCA (Figure 15c) and AMlogtv (Figure 15d) still had a small amount of noise, and the uneven brightness of the image surface was not balanced, which blocked the facial features of the image. It can be seen that the experimental effect of our method was the best. analysis, the quantitative results based on PSNR, SSIM, and FSIM showed that our method had higher scores, and the objective index was obviously improved compared with some methods. In addition, in the experiment of removing facial image noise, our method could highlight facial features more clearly, without artifacts, while removing noise. At the same time, our method had a lower computational cost than NonRPCA and TDV. Although its cost was higher than the other three methods, our denoising effect was better than theirs. To summarize, the proposed RPCA framework based on a nonconvex low-rank approximation and TV norm showed good performance in noise removal and in the retention of important features of images, which provides a reference for the future application of nonconvex functions in the field of image denoising. However, our method can only process grayscale images at present, and it needs to be optimized in terms of computational cost. Therefore, in the future, we will continue to investigate new iterative optimization methods for solving nonconvex optimization problems, as well as extend our research to a tensor framework for solving the problem of color image denoising in complex noise backgrounds. Conceptualization, T.C., Q.X. and D.Z. ; methodology, T.C., Q.X. and D.Z. ; formal analysis, Q.X.