Fractional-Order Anisotropic Diffusion for Image Denoising

Abstract

This paper introduces a new class of fractional-order anisotropic diffusion equations for noise removal. These equations are Euler-Lagrange equations of a cost functional which is an increasing function of the absolute value of the fractional derivative of the image intensity function, so the proposed equations can be seen as generalizations of second-order and fourth-order anisotropic diffusion equations. We use the discrete Fourier transform to implement the numerical algorithm and give an iterative scheme in the frequency domain. It is one important aspect of the algorithm that it considers the input image as a periodic image. To overcome this problem, we use a folded algorithm by extending the image symmetrically about its borders. Finally, we list various numerical results on denoising real images. Experiments show that the proposed fractional-order anisotropic diffusion equations yield good visual effects and better signal-to-noise ratio.

Graph fractional-order total variation EEG source reconstruction

“…GRAPH FRACTIONAL-ORDER TOTAL VARIATION Fractional-order TV has been proposed recently in image processing to enhance smoothness with relatively low computational cost [6], [7], [9]. In this paper, we focus on the anisotropic version in which the objective function of the derived minimization problem is separable.…”

High-order fractional partial differential equation transform for molecular surface construction

“…(16). Consider the central finite difference in the Fourier domain defined as [3] $$\widehat{D^{n}u}=false(e^{ihw/2}-e^{-ihw/2}false)\widehat{D^{n-1}u}=\cdots ={false(e^{ihw/2}-e^{-ihw/2}false)}^n\hat{u},$$ where the n th-order finite difference D n u is defined by $$D^{n}ufalse(xfalse)=Dfalse(D^{n-1}ufalse)=\cdots =D^{n-1}false(Dufalse)=D^{n-1}true(utrue(x+\frac{h}{2}true)-utrue(x-\frac{h}{2}true)true).$$ …”

“…A key component of the fractional PDE transform is an arbitrarily high-order fractional PDE, which is defined via the fractional variational principle. Many authors have discussed the fractional variational principle [1, 3]. For any α ∈ ℝ and 0 < α < ∞, denote by ${\nabla }^{\alpha }=true(\frac{{\partial }_l^{\alpha }}{\partial x^{\alpha }},\frac{{\partial }_l^{\alpha }}{\partial y^{\alpha }},\frac{{\partial }_l^{\alpha }}{\partial z^{\alpha }}true)$ a gradient vector in ℝ 3 , where $\frac{{\partial }_l^{\alpha }}{\partial x^{\alpha }}$ is the left Riemann-Liouville fractional derivative of order α in x , according to Eq.…”

“…(53) (see Ref. [3] for a similar treatment) or the amplitude of the term ( i w ) α . In this work, we explore the performance of both approaches.…”

“…Such a power law in the wavenumber corresponds to the fractional derivative of order α in the coordinate space, since the inverse Fourier transform of an exponent gives rise to a derivative of the same order. Fractional derivative has found its success in physical and biological modeling [2, 10, 70], financial analysis [26, 44, 46], and image processing [3]. Currently, most attention in the field is paid to the fractional derivatives of order less than 2.…”