A fourth-order PDE denoising model with an adaptive relaxation method

In this paper, an adaptive relaxation method and a discontinuity treatment of edges are proposed to improve the digital image denoising process by using the fourth-order partial differential equation (known as the YK model) first proposed by You and Kaveh. Since the YK model would generate some speckles into the denoised image, a relaxation method is incorporated into the model to reduce the formation of isolated speckles. An additional improvement is employed to handle the discontinuity on the edges of the image. In order to stop the iteration automatically, a control of the iteration is integrated into the denoising process. Numerical results demonstrate that such modifications not only make the denoised image look more natural, but also achieve a higher value of PSNR

A Spatially Adaptive Edge-Preserving Denoising Method Based on Fractional-Order Variational PDEs

Image denoising is a basic problem in image processing. An important task of image denoising is to preserve the significant geometric features such as edges and textures while filtering out noise. So far, this is still a problem to be further studied. In this paper, we firstly introduce an edge detection function based on the Gaussian filtering operator and then analyze the filtering characteristic of the fractional derivative operator. On the basis, we establish the spatially adaptive fractional edge-preserving denoising model in the variational framework, discuss the existence and uniqueness of our proposed model solution and derive the nonlinear fractional Euler-Lagrange equation for solving our proposed model. This forms a fractional order extension of the first and second order variational approaches. Finally, we apply the proposed method to the synthetic images and real seismic data denoising to verify the effectiveness of our method and compare the experimental results of our method with the related state-of-the-art methods. Experimental results illustrate that our proposed method can not only improve the signal to noise ratio (SNR) but also adaptively preserve the structural information of an image compared with other contrastive methods. Our proposed method can also be applied to remote sensing imaging, medical imaging and so onThe work of Dehua Wang was supported in part by the Science and Technology Planning Project of Shaanxi Province under Grant 2020JM-561, in part by the Postdoctoral Foundation of China under Grant 2019M663462, in part by the Innovative Talents Cultivate Program of Shaanxi Province under Grant 2019KJXX-032, in part by the President Fund of Xi’an Technological University under Grant XAGDXJJ17026, and in part by the Teaching Reform Project of Xi’an Technological University under Grant 18JGY08. The work of Juan J. Nieto was supported in part by the Agencia Estatal de Investigacion (AEI) of Spain under Grant MTM2016-75140-P, and in part by the European Community Fund FEDER. The work of Xiaoping Li was supported in part by the NSFC under Grant 61701086, and in part by the Fundamental Research Funds for the Central Universities under Grant ZYGX2016KYQD143S

Properties of higher order nonlinear diffusion filtering

This paper provides a mathematical analysis of higher order variational methods and nonlinear diffusion filtering for image denoising. Besides the average grey value, it is shown that higher order diffusion filters preserve higher moments of the initial data. While a maximum-minimum principle in general does not hold for higher order filters, we derive stability in the 2-norm in the continuous and discrete setting. Considering the filters in terms of forward and backward diffusion, one can explain how not only the preservation, but also the enhancement of certain features in the given data is possible. Numerical results show the improved denoising capabilities of higher order filtering compared to the classical methods