# Edge-Preserving Image Denoising Based on Lipschitz Estimation
## 5 Results and Analysis  
This section presents the performance evaluation of the proposed algorithm. The set of set of images used for experimentation consisted of standard test images, including House, Lena, Circuit, Pepper, and Hand, and were tested with various noise levels. The images were corrupted with the addition of Gaussian noise of zero mean and standard deviation ranging from 15 to 35. The peak signal to noise ratio (PSNR) was employed as the measure of quantitative performance as explained in Section 2. We computed Lipschitz exponents to identify the edges or image structure. These exponents were estimated using the modulus maxima lines. We analyzed these estimations with the addition of Gaussian white noise (28 dB). In order to illustrate more detailed analysis, we show in Figure 9, a single row (125) of the image as a curve without and with the addition of noise. It can be seen from the results that sharp transitions preserved their Lipschitz estimation even in the presence of noise. To illustrate the robustness of the estimation (\(\alpha\)) in the case of natural images, we analyzed our test samples in the case of a constant affine transformation, as shown in Figure 10. The original image has been deformed by the \(10^{\circ}\) of rotation. Lipschitz exponents were estimated, and, from the histogram, we observed that approximately 92% of the Lipschitz regularity values lay in between the same range. The regularity (\(\alpha\)) was preserved even when a constant affine deformation was applied to an image. Therefore, we concluded from our findings that the Lipschitz exponents can be used as a significant measure to study natural images. At the same time, the problem of preserving edges in any linear or nonlinear denoising algorithm can be handled with this estimation. The transitions identified based on their Lipschitz exponents represent the main structure of the image; therefore, we preserved these points and performed smoothing on the rest of the image, as explained in the Section 4. Figure 9: (**a**) Row No. 125 of image shown in Figure 2a, (**b**) Row No. 125 of the image with the addition if Gaussian noise (28 dB) modulus maxima are marked as a circle in both images, and the corresponding Lipschitz value is written in numerical form. We defined an adaptive restoration method and performed data samples based non-linear functioning to estimate the best fit. The method restores therest of the image by utilizing noisy sample and their smoothness. As an example, Figure 11 illustrates the focused edges of the image overlaid with the results of edge extraction. Noise level of 24.59 dB PSNR were added and by performing Lipschitz analysis and restoration process as explained in previous sections, the PSNR is improved to 29.14 dB. The sharpness of the edges were significantly preserved, and the smoothing process on the rest of the image results in increasing PSNR of the image. The obtained results in Figure 11c shows that the noise level has reduced and preserved the image structures without introducing spurious oscillations. To illustrate the results of denoising more precisely, we show in Figures 12 and 13, a single row (200) of House and Lena image with noise (22.11 dB white Gaussian noise), plotted as a curve. It can be seen from the figure that the method has filtered most of the noise elements while preserving the edge points. The sharpness and magnitude of these edges were not altered during denoising and the noise elements were removed from the homogeneous regions. The results in Figure 13 also illustrate that the sharp transition or edges (near to step function or discontinuity), are equally preserved and hence the main information about the structure of the image remain unchanged. We performed the multi scale analysis based splitting and heuristic approach for the reconstruction of House, Lena, circuit, Pepper and Hand images.