scholarly journals Parameter learning and fractional differential operators: Applications in regularized image denoising and decomposition problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sören Bartels ◽  
Nico Weber
Keyword(s):  
Image Denoising ◽  
Numerical Experiments ◽  
Fractional Laplacian ◽  
Differential Operators ◽  
Riesz Potential ◽  
Regularization Parameter ◽  
Computing Time ◽  
Bilevel Optimization ◽  
Learning Approaches ◽  
Decomposition Models

<p style='text-indent:20px;'>In this paper, we focus on learning optimal parameters for PDE-based image denoising and decomposition models. First, we learn the regularization parameter and the differential operator for gray-scale image denoising using the fractional Laplacian in combination with a bilevel optimization problem. In our setting the fractional Laplacian allows the use of Fourier transform, which enables the optimization of the denoising operator. We prove stable and explainable results as an advantage in comparison to machine learning approaches. The numerical experiments correlate with our theoretical model settings and show a reduction of computing time in contrast to the Rudin-Osher-Fatemi model. Second, we introduce a new regularized image decomposition model with the fractional Laplacian and the Riesz potential. We provide an explicit formula for the unique solution and the numerical experiments illustrate the efficiency.</p>

New Results from Old Investigation: A Note on Fractional M-Dimensional Differential Operators. The Fractional Laplacian

2015 ◽  
Vol 18 (2) ◽  
Author(s):  
Humberto Prado ◽  
Margarita Rivero ◽  
Juan J. Trujillo ◽  
M. Pilar Velasco
Keyword(s):  
Fractional Integral ◽  
Fractional Laplacian ◽  
Singular Integrals ◽  
Differential Operators ◽  
Riesz Potential ◽  
Fractional Power ◽  
Potential Operators ◽  
Relevant Role ◽  
Non Local ◽  
Fractional Power Of Operators

AbstractThe non local fractional Laplacian plays a relevant role when modeling the dynamics of many processes through complex media. From 1933 to 1949, within the framework of potential theory, the Hungarian mathematician Marcel Riesz discovered the well known Riesz potential operators, a generalization of the Riemann-Liouville fractional integral to dimension higher than one. The scope of this note is to highlight that in the above mentioned works, Riesz also gave the necessary tools to introduce several new definitions of the generalized coupled fractional Laplacian which can be applied to much wider domains of functions than those given in the literature, which are based in both the theory of fractional power of operators or in certain hyper-singular integrals. Moreover, we will introduce the corresponding fractional hyperbolic differential operator also called fractional Lorentzian Laplacian.

scholarly journals New Regularization Models for Image Denoising with a Spatially Dependent Regularization Parameter

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
Tian-Hui Ma ◽  
Ting-Zhu Huang ◽  
Xi-Le Zhao
Keyword(s):  
Image Denoising ◽  
Hybrid Model ◽  
Numerical Experiments ◽  
Numerical Scheme ◽  
Regularization Parameter ◽  
Alternating Minimization ◽  
Convergence Properties ◽  
Strictly Convex ◽  
Nonconvex Problem ◽  
Minimization Scheme

We consider simultaneously estimating the restored image and the spatially dependent regularization parameter which mutually benefit from each other. Based on this idea, we refresh two well-known image denoising models: the LLT model proposed by Lysaker et al. (2003) and the hybrid model proposed by Li et al. (2007). The resulting models have the advantage of better preserving image regions containing textures and fine details while still sufficiently smoothing homogeneous features. To efficiently solve the proposed models, we consider an alternating minimization scheme to resolve the original nonconvex problem into two strictly convex ones. Preliminary convergence properties are also presented. Numerical experiments are reported to demonstrate the effectiveness of the proposed models and the efficiency of our numerical scheme.

scholarly journals A well-posed multiscale regularization scheme for digital image denoising

2011 ◽  
Vol 21 (4) ◽  
pp. 769-777 ◽  
Author(s):  
V. Prasath
Keyword(s):  
Image Denoising ◽  
Digital Image ◽  
Regularization Parameter ◽  
Input Image ◽  
Smooth Transition ◽  
Functions Of Bounded Variation ◽  
Edge Preserving ◽  
Regularization Scheme ◽  
Adaptive Parameter ◽  
Well Posed

A well-posed multiscale regularization scheme for digital image denoisingWe propose an edge adaptive digital image denoising and restoration scheme based on space dependent regularization. Traditional gradient based schemes use an edge map computed from gradients alone to drive the regularization. This may lead to the oversmoothing of the input image, and noise along edges can be amplified. To avoid these drawbacks, we make use of a multiscale descriptor given by a contextual edge detector obtained from local variances. Using a smooth transition from the computed edges, the proposed scheme removes noise in flat regions and preserves edges without oscillations. By incorporating a space dependent adaptive regularization parameter, image smoothing is driven along probable edges and not across them. The well-posedness of the corresponding minimization problem is proved in the space of functions of bounded variation. The corresponding gradient descent scheme is implemented and further numerical results illustrate the advantages of using the adaptive parameter in the regularization scheme. Compared with similar edge preserving regularization schemes, the proposed adaptive weight based scheme provides a better multiscale edge map, which in turn produces better restoration.

scholarly journals Performance of Restarted Homotopy Perturbation Method for TV-Based Image Denoising Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-18 ◽  
Author(s):  
Yu Du Han ◽  
Jae Heon Yun
Keyword(s):  
Perturbation Method ◽  
Image Denoising ◽  
Numerical Experiments ◽  
Homotopy Perturbation Method ◽  
Nonlinear Pde ◽  
Finite Difference Schemes ◽  
Numerical Implementation ◽  
Homotopy Perturbation ◽  
Efficiency And Reliability ◽  
Nonlinear Denoising

We first propose a restarted homotopy perturbation method (RHPM) for solving a nonlinear PDE problem which repeats HPM process by computing only the first few terms instead of computing infinite terms, and then we present an application of RHPM to TV- (Total Variation-) based image denoising problem. The main difficulty in applying RHPM to the nonlinear denoising problem is settled by using binomial series techniques. We also provide finite difference schemes for numerical implementation of RHPM. Lastly, numerical experiments for several test images are carried out to demonstrate the feasibility, efficiency, and reliability of RHPM by comparing the performance of RHPM with that of existing TM and recently proposed RHAM methods.

Image Denoising with Contour-Based Directionlet Transform

2014 ◽  
Vol 513-517 ◽  
pp. 3607-3611
Author(s):  
Huan An Xu ◽  
Guo Hua Peng ◽  
Zhe Liu
Keyword(s):  
Image Processing ◽  
Image Denoising ◽  
Numerical Experiments ◽  
Adaptive Threshold ◽  
Perceptual Quality ◽  
Laplacian Pyramid ◽  
Low Frequencies ◽  
Threshold Algorithm ◽  
Image Transform

A novel mutiscale and directionally adaptive image transform called contour based directionlet tansform is presented. Directionlet transform (DT) has shown its charming performance in image processing, but it has scrambled frequencies. Laplacian Pyramid is employed here to separate the low frequencies before applying DT for avoiding the drawback. And an adaptive threshold algorithm is proposed for denoising. Numerical experiments are performed to assess the applicability of the proposed method. The obtained results show that the proposed scheme outperforms Wavelet and Directionlet transforms in terms of numerical and perceptual quality.

scholarly journals Riesz potential versus fractional Laplacian

2014 ◽  
Vol 2014 (9) ◽  
pp. P09032 ◽  
Author(s):  
Manuel D Ortigueira ◽  
Taous-Meriem Laleg-Kirati ◽  
J A Tenreiro Machado
Keyword(s):  
Fractional Laplacian ◽  
Potential Versus ◽  
Riesz Potential

Computation analysis of regularization methods and parameter selection for acoustics radiation modes source reconstruction of vibrating plates

2021 ◽  
Vol 263 (2) ◽  
pp. 4295-4302
Author(s):  
Luis Corral ◽  
Pablo E. Román
Keyword(s):  
Tikhonov Regularization ◽  
Graphics Processing Unit ◽  
Regularization Parameter ◽  
Computing Time ◽  
Surface Velocity ◽  
Processing Unit ◽  
Vibrating Plates ◽  
Speed Performance ◽  
Multiple Threads ◽  
Graphics Processing

Source localization and power estimation is a topic of great interest in acoustics and vibration. Acoustic source radiation modes reconstruction is a method of particular interest. Solutions leads to determinate sound/vibration power and surface velocity distribution from sparse acoustics samples. It has been shown that the quality of the results depends on Tikhonov regularization parameter. This inverse method is based on the radiation resistance matrix and we show that a single instruction multiple threads computing approach for graphics processing unit device exhibit better speed performance than common approaches to achieve the solution. We compare four regularization and three estimating methods for regularization parameters. We use a similarity measure to the simulated cases in three frequencies. Tikhonov regularization exhibits best reconstruction results. However, truncated singular vector decomposition also shows good performance with the advantage of not using a regularization parameter. Graphics processing unit implementation reduce reconstruction's computing time at least in a half.

An Adaptive Wavelet Shrinkage and its Application in Image De-Noising

2011 ◽  
Vol 128-129 ◽  
pp. 500-503
Author(s):  
Tian Jie Cao
Keyword(s):  
Linear Function ◽  
Image Denoising ◽  
Signal To Noise Ratio ◽  
Computing Time ◽  
Adaptive Method ◽  
Risk Estimate ◽  
Wavelet Coefficients ◽  
Adaptive Wavelet ◽  
Proportional Relation ◽  
Unbiased Risk

In this paper an adaptive method of shrinkage of the wavelet coefficients is presented. In the method, the wavelet coefficients are divided into two classes by a threshold. One class of them with the smaller absolute values at a scale is transformed with a proportional relation，another class with the larger absolute values at the same scale is transformed with a linear function. The threshold and the coefficient in the proportional relation or in the linear function are determined by the principle of minimizing the Stein’s unbiased risk estimate. In the paper, the method of estimation of the threshold and the coefficient is given and the adaptive method of shrinkage of the wavelet coefficients is applied to image denoising. Examples in the paper show that the presented method has an advantage over SureShrink from the point of view of both the Stein’s unbiased risk estimate and the signal-to-noise ratio. In addition, the method takes almost the same computing time as the SureShrink in image denoising.

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