# Edge-Preserving Image Denoising Based on Lipschitz Estimation
###### Abstract  
The information transmitted in the form of signals or images is often corrupted with noise. These noise elements can occur due to the relative motion, noisy channels, error in measurements, and environmental conditions (rain, fog, change in illumination, etc.) and result in the degradation of images acquired by a camera. In this paper, we address these issues, focusing mainly on the edges that correspond to the abrupt changes in the signal or images. Preserving these important structures, such as edges or transitions and textures, has significant theoretical importance. These image structures are important, more specifically, for visual perception. The most significant information about the structure of the image or type of the signal is often hidden inside these transitions. Therefore it is necessary to preserve them. This paper introduces a method to reduce noise and to preserve edges while performing Non-Destructive Testing (NDT). The method computes Lipschitz exponents of transitions to identify the level of discontinuity. Continuous wavelet transform-based multi-scale analysis highlights the modulus maxima of the respective transitions. Lipschitz values estimated from these maxima are used as a measure to preserve edges in the presence of noise. Experimental results show that the noisy data sample and smoothness-based heuristic approach in the spatial domain restored noise-free images while preserving edges. denoising edge detection; Lipschitz estimation; continuous wavelet transform +
Footnote †: journal: Applied  
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**Subject**: 31 May 2021  
**Publicisher's Note**: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. and [PERSON] used a neighborhood filtering method [6; 7]. Image denoising in the spatial domain is complex and more time-consuming especially for real-time problems. Some researchers worked in the frequency domain to denoise images, e.g., Wiener filters [6]. [PERSON] and [PERSON] improved the performance of denoising by taking advantage of patch redundancy from the Weiner filter-based method [8]. The scope of these linear techniques is limited to stationary data, as the Fourier transform is not suitable to perform analysis on non-stationary and nonlinear data. As a result, denoising methods based on multi-scale denoising employing nonlinear operations in the transform domain came up as an alternative. These multiple scales provide sparse representation of the signal in the transform domain. Several methods based on wavelet transform have been proposed in the past [9; 10; 11; 12; 13; 14; 15]. A brief survey of some of these methods is given by [PERSON] et al. in [16; 17]. [PERSON] et al. presented a comparison and discussion on the kernel based methods [18]. The method proposed by [PERSON] and [PERSON] minimized the mean square error estimate termed as --[PERSON]'s unbiased risk estimate (SURE) [19]. In another method, dual tree complex wavelets were transformed into ridgelet transforms by [PERSON] and [PERSON] [20]. [PERSON] et al. proposed image denoising based on the family of transforms--the ridgelet and curvelet transforms--as alternatives to the wavelet representation of image data [12]. Similarly, [PERSON] et al. used curvelet coefficients to distinguish two classes of coefficients, e.g., noise-free components, which they termed as \"signal of interest,\" and other [10]. More recently, [PERSON] et al. adopted a hybrid approach that combined frequency and multi-scale analysis [13]. They formulated the restoration problem as a nonlinear estimation problem leading to the minimization of a criterion derived from [PERSON]'s unbiased quadratic risk estimate. [PERSON] et al. investigated patch-based denoising methods for additive noise reduction [21]. Recently, [PERSON] et al.