# Edge-Preserving Image Denoising Based on Lipschitz Estimation
###### Abstract
proposed DWT- and DT-CWT-based image denoising methods and employed statistical goodness of fit tests on wavelet coefficients at multi-scales to identify noise at different levels [22]. [PERSON] proposed a method based on an unsupervised learning model. They presented adaptive dictionary learning-based denoising of approximation. The wavelet coefficients were denoised by using an adaptive dictionary learned over the set of extracted patches from the wavelet representation of the corrupted image [23]. Several other methods have also been proposed in the past, but there is limited study regarding the preservation of edges. Recently, [PERSON] and [PERSON] presented a brief evolution of the research on edge-preserving denoising methods in [24], and [PERSON] presented an overview of image denoising algorithms ranging from wavelet shrinkage to patch-based non-local processing with the main focus on the suppression of additive Gaussian noise [25]. [PERSON] et al. reviewed the benchmark edge-preserving smoothing algorithms and keeping the main focus on anisotropic diffusion and bilateral filtering [26]. Many other filtering algorithms exists but, as mentioned before, very few of them discussed the edges and the spurious oscillations or artifacts introduced after denoising. Gibbs phenomena explain these oscillations in the neighborhood of discontinuities. An edge separates two different areas with limited or no connections between them e.g., no common characteristics: texture, noise, etc. Figure 0(a), presents an edge part of the House image. Figure 0(b),c represents the same image with the addition of white Gaussian noise and denoised image [19]. It can be observed from the figure that, while denoising, some additional artifacts have been produced. This is because most linear or nonlinear methods use neighborhood pixels while denoising. It is, therefore, logical to separate edges and transitions from the rest of the background. Most of the wavelet-based methods use Discrete Wavelet Transform (DWT) for denoising; however, DWT suffers from three main issues, lack of shift-invariant, poor directional, and lack of phase information. Stationary wavelet transform reduced the problem of partial translation in variance and considerably improved the denoising results; however, it suffersfrom the cost of very high redundancy and is ultimately computationally expensive. Many algorithms have been proposed to solve the shortcomings of DWT by using different forms of Complex Wavelet Transforms (CWT) [27]. CWT is nearly translation invariant with directional selectivity but at the cost of high redundancy, [27]. Similarly, through continuous wavelet transform analysis, a set of wavelet coefficients is obtained, indicating how close the signal is to a particular basis function. Since the continuous wavelet transform behaves like orthonormal basis decomposition, it can be shown that it is isometric, i.e., it preserves energy [28]. Hence, the signal can be recovered from its transform. The wavelet transform in its continuous form can accurately represent minor variations/edges present in signal. In most of the denoising methods, these minor variations that correspond to the edges of different levels in images are not considered while denoising. In this work, we classify this separation as classes for edges and background using continuous wavelet transform. This classification is based on the information obtained from the Lipschitz estimation. The rest of the paper is organized as follows. Section 2 introduces the method and the principle of method. Section 3 gives a detailed explanation of the multi-scale analysis based edge detection. In Section 4, we present the Lipschitz regularity in the context of images, followed by the explanation of the reconstruction method. Section 5 explains the obtained results and evaluates the performance of the proposed method, and Section 6 finally concludes the paper.