A Nonlinear Fractional Partial Differential Equation for Image Denoising

We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

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Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach-Based Multiscale Denoising Model for Texture Image

Abstract and Applied Analysis ◽

10.1155/2013/483791 ◽

2013 ◽

Vol 2013 ◽

pp. 1-19 ◽

Cited By ~ 10

Author(s):

Yi-Fei Pu ◽

Ji-Liu Zhou ◽

Patrick Siarry ◽

Ni Zhang ◽

Yi-Guang Liu

Keyword(s):

Differential Equation ◽

Image Processing ◽

Partial Differential Equation ◽

Fractional Order ◽

Differential Calculus ◽

Order Partial Differential Equation ◽

Texture Image ◽

Fractional Partial Differential Equation ◽

Partial Differential ◽

Integer Order

The traditional integer-order partial differential equation-based image denoising approaches often blur the edge and complex texture detail; thus, their denoising effects for texture image are not very good. To solve the problem, a fractional partial differential equation-based denoising model for texture image is proposed, which applies a novel mathematical method—fractional calculus to image processing from the view of system evolution. We know from previous studies that fractional-order calculus has some unique properties comparing to integer-order differential calculus that it can nonlinearly enhance complex texture detail during the digital image processing. The goal of the proposed model is to overcome the problems mentioned above by using the properties of fractional differential calculus. It extended traditional integer-order equation to a fractional order and proposed the fractional Green’s formula and the fractional Euler-Lagrange formula for two-dimensional image processing, and then a fractional partial differential equation based denoising model was proposed. The experimental results prove that the abilities of the proposed denoising model to preserve the high-frequency edge and complex texture information are obviously superior to those of traditional integral based algorithms, especially for texture detail rich images.

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Patch Similarity Modulus and Difference Curvature Based Fourth-Order Partial Differential Equation for Image Denoising

Mathematical Problems in Engineering ◽

10.1155/2015/636295 ◽

2015 ◽

Vol 2015 ◽

pp. 1-12

Author(s):

Yunjiao Bai ◽

Quan Zhang ◽

Hong Shangguan ◽

Zhiguo Gui ◽

Yi Liu ◽

...

Keyword(s):

Differential Equation ◽

Partial Differential Equation ◽

Image Denoising ◽

Fourth Order ◽

Objective Evaluation ◽

Order Partial Differential Equation ◽

Partial Differential ◽

The Difference ◽

Adaptively Adjusting ◽

Difference Curvature

The traditional fourth-order nonlinear diffusion denoising model suffers the isolated speckles and the loss of fine details in the processed image. For this reason, a new fourth-order partial differential equation based on the patch similarity modulus and the difference curvature is proposed for image denoising. First, based on the intensity similarity of neighbor pixels, this paper presents a new edge indicator called patch similarity modulus, which is strongly robust to noise. Furthermore, the difference curvature which can effectively distinguish between edges and noise is incorporated into the denoising algorithm to determine the diffusion process by adaptively adjusting the size of the diffusion coefficient. The experimental results show that the proposed algorithm can not only preserve edges and texture details, but also avoid isolated speckles and staircase effect while filtering out noise. And the proposed algorithm has a better performance for the images with abundant details. Additionally, the subjective visual quality and objective evaluation index of the denoised image obtained by the proposed algorithm are higher than the ones from the related methods.

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