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

Abstract

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.

1. Introduction

Fractional calculus has been an important branch of mathematical analysis over the last 300 years [1–4]; however, it is still little known by many mathematicians and physical scientists in both the domestic and overseas engineering fields. Fractional calculus of the Hausdorff measure is not well established after more than 90 years studies [5, 6], whereas fractional calculus in the Euclidean measure seems more completed. So, Euclidean measure is commonly required in mathematics [5, 6]. In general, fractional calculus in the Euclidean measure extends the integer step to a fractional step. Random variable of physical process in the Euclidean measure can be deemed to be the displacement of particles by random movement; thus, fractional calculus can be used for the analysis and processing of the physical states and processes in principle [7–15]. Fractional calculus has one obvious feature, that is, that most fractional calculus is based on a power function and the rest is based on the addition or production of a certain function and a power function [1–6]. It is possible that this feature indicates some changing law of nature. Scientific research has proved that the fractional-order or dimensional mathematical approach provides the best description for many natural phenomena [16–19]. Fractional calculus in the Euclidean measure has been used in many fields, including diffusion process, viscoelasticity theory, and random fractal dynamics. Methods to apply fractional calculus to modern signal analysis and processing [18–30], especially to digital image processing [31–38], are an emerging branch to study, which has been seldom explored.

Integer-order partial differential equation-based image processing is an important branch in the field of image processing. By exploring the essence of image and image processing, people tend to reconstruct the traditional image processing approaches through strictly mathematical theories, and it will be a great challenge to practical-oriented traditional image processing. Image denoising is a significant research branch of integer-order partial differential equation-based image processing, with two kinds of denoising approach: the nonlinear diffusion-based method and the minimum energy norm-based variational method [39–42]. They have two corresponding models, which are the anisotropic diffusion proposed by Perona and Malik [43] (Perona-Malik or PM) and the total variation model proposed by Rudin et al. [44] (Rudin-Osher-Fatemi or ROF). The PM model simulates the denoising process by a thermal energy diffusion process and the denoising result is the balanced state of thermal diffusion, while the ROF model describes the same thermal energy by a total variation. In further study, some researchers have applied the PM model and the ROF model to color images [45, 46], discussed the selection of the parameters for the models [47–51], and found the optimal stopping point in iteration process [52, 53]. Rudin and his team proposed a variable time step method to solve the Euler-Lagrange equation [44]. Vogel and Oman proposed improving the stability of ROF model by a fixed point iteration approach [54]. Darbon and Sigelle decomposed the original problems into independent optimal Markov random fields by using level set methods and obtained globally optimal solution by reconstruction [55–57]. Wohlberg and Rodriguez proposed to solve the total variation by using an iterative weighted norm to improve the computing efficiency [58]. Meanwhile, Catté et al. proposed to perform a Gaussian smoothing process in the initial stage to improve the suitability of the PM model [59]. However, PM model and ROF model have some obvious defects in image denoising; that is, they can easily lose the contrast and texture details and can produce staircase effects [39, 60, 61]. Some improved models have been proposed to solve these problems. To maintain the contrast and texture details, some scholars have proposed to replace the 𝐿² norm with the 𝐿¹ norm [62–65], while Osher et al. proposed an iterative regularization method [66]. Gilboa et al. proposed a denoising method using a numerical adaptive fidelity term that can change with the space [67]. Esedoglu and Osher proposed to decompose images using the anisotropic ROF model and retaining certain edge directional information [68]. To remove the staircase effects, Blomgren et al. proposed to extend the total variation denoising model by changing it with gradients [69, 70]. Some scholars introduced high-order derivative to energy norm [71–76]. Lysaker et al. integrated high-order deductive to original ROF model [77, 78], while other scholars proposed a two-stage denoising algorithm, which smoothes the corresponding vector field first and then fits it by using the curve surface [79, 80]. The above methods have provided some improvements in maintaining contrast and texture details and removing the staircase effect, but they still have some drawbacks. First, the improved algorithms have greatly increased calculation complexity for real-time processing and excessive storage and computational requirements will lead them to be impractical. Second, the above algorithms are essentially integer-order differential based algorithm, and thus they may cause the edge field to be somewhat fuzzy and the texture-preservation effect to be less effective than expected.

We therefore propose to introduce a new mathematical method—fractional calculus to the denoising field for texture image and implementing a fractional partial differential equation to solve the above problems by the integer-order partial differential equation-based denoising algorithms [23, 33–38]. Guidotti and Lambers [81] and Bai and Feng [82] have pushed the classic anisotropic diffusion model to the fractional field, extended gradient operator of the energy norm from first-order to fractional-order, and numerically implemented the fractional developmental equation in the frequency domain, which has some effects on image denoising. However, the algorithm still has certain drawbacks. First, it simply took the gradient operator of the energy norm from first order to fractional order and still cannot essentially solve the problem of how to nonlinearly maintain the texture details via the anisotropic diffusion. Therefore, the texture information is not retained well after denoising. Second, the algorithm does not include the effects of the fractional power of the energy norm and the fractional extreme value on nonlinearly maintaining texture details. Third, the method does not deduce the corresponding fractional Euler-Lagrange formula according to fractional calculus features and directly replace it according to the complex conjugate transpose features of the Hilbert adjoint operator. It greatly increased the complex of the numerical implementation of the fractional developmental equation in frequency field. Finally, the transition function of fractional calculus in Fourier transform domain is (𝑖𝜔)^𝑣. Its form looks simple, but the Fourier’s inverse transform of (𝑖𝜔)^𝑣 belongs to the first kind of Euler integral, which is difficult to calculate theoretically. The algorithm simply converted the first-order difference into the fractional-order difference in the frequency domain form and replaced the fractional differential operator, which has not solved the computing problem of the Euler integral of the first kind.

The properties of fractional differential are as follows [23, 24, 38]. First, the fractional differential of a constant is non-zero, whereas it must be zero under integer-order differential. Fractional calculus varies from a maximum at a singular leaping point to zero in the smooth areas where the signal is unchanged or not changed greatly; note that, by default, any integer-order differential in a smooth area is approximated to zero, which is the remarkable difference between the fractional differential and integer-order differential. Second, the fractional differential at the starting point of a gradient of a signal phase or slope is nonzero, which nonlinearly enhances the singularity of high-frequency components. With the increasing fractional order, the strengthening of the singularity of high-frequency components is also greater. For example, when 0<𝑣<1, the strengthening is less than when 𝑣=1. The integral differential is a special case of the fractional calculus. Finally, the fractional calculus along the slope is neither zero nor constant but is a nonlinear curve, while integer-order differential along slope is the constant. From this discussion, we can see that fractional calculus can nonlinearly enhance the complex texture details during the digital image processing. Fractional calculus can nonlinearly maintain the low-frequency contour features in smooth area to the furthest degree, nonlinearly enhance the high-frequency edge and texture details in those areas where gray level changes frequently, and nonlinearly enhance high-frequency texture details in those areas where gray level does not change obviously [23, 33–38].

A fractional partial differential equation-based denoising algorithm is proposed. The experimental results prove that it can not only preserve the low-frequency contour feature in the smooth area but also nonlinearly maintain the high-frequency edge and texture details both in the areas where gray level did not change obviously or change frequently. As for texture-rich images, the abilities for preserving the high-frequency edge and complex texture details of the proposed fractional based denoising model are obviously superior to the traditional integral based algorithms. The outline of the paper is as follows. First, it introduces three common-used definitions of fractional calculus, that is, Grümwald-Letnikov, Riemann-Liouville, and Caputo, which are the premise of the fractional developmental equation-based model. Second, we obtain fractional Green’s formula for two-dimensional image by extending the classical integer-order to a fractional-order and also fractional Euler-Lagrange formula. On the basis, a fractional partial differential equation is proposed. Finally, we show the denoising capabilities of the proposed model by comparing with Gaussian denoising, fourth-order TV denoising, bilateral filtering denoising, contourlet denoising, wavelet denoising, nonlocal means noise filtering (NLMF) denoising, and fractional-order anisotropic diffusion denoising.

2. Related Work

The common-used definitions of fractional calculus in the Euclidean measure are Grümwald-Letnikov definition, Riemann-Liouville definition, and Caputo definition [1–6].

Grümwald-Letnikov defined that 𝑣-order differential of signal 𝑠(𝑥) can be expressed by 𝐷^𝑣_𝐺-𝐿𝑠(𝑥)=𝑑^𝑣[𝑑(𝑥−𝑎)]^𝑣𝑠(𝑥)_𝐺-𝐿=lim_𝑁→∞((𝑥−𝑎)/𝑁)^−𝑣Γ()×−𝑣_{𝑁−1𝑘=0}Γ(𝑘−𝑣)Γ()𝑠𝑘+1𝑥−𝑘𝑥−𝑎𝑁, (1) where the duration of 𝑠(𝑥) is [𝑎,𝑥] and 𝑣 is any real number (fraction included). 𝐷^𝑣_𝐺-𝐿 denotes Grümwald-Letnikov defined fractional-order differential operator, and Γ is Gamma function. Equation (1) shows that Grümwald-Letnikov definition in the Euclidean measure extends the step from integer to fractional, and thus it extends the order from integer differential to fractional differential. Grümwald-Letnikov defined fractional calculus is easily calculated, which only relates to the discrete sampling value of 𝑠(𝑥−𝑘((𝑥−𝑎)/𝑁)) that correlates to 𝑠(𝑥) and irrelates to the derivative or the integral value.

Riemann-Liouville defined the 𝑣-order integral when 𝑣<0 is shown as 𝐷^𝑣_𝑅-𝐿𝑠(𝑥)=𝑑^𝑣[𝑑(𝑥−𝑎)]^𝑣𝑠(𝑥)_𝑅-𝐿=1Γ(−𝑣)^𝑥_𝑎𝑥−𝜂^−𝑣−1𝑠𝜂=𝑑𝜂−1Γ(−𝑣)^𝑥_𝑎𝑠𝜂𝑑𝑥−𝜂^−𝑣,𝑣≺0, (2) where 𝐷^𝑣_𝑅-𝐿 represents the Riemann-Liouville defined fractional differential operator. As for 𝑣-order differential when 𝑣≥0, 𝑛 satisfies 𝑛−1<𝑣≤𝑛. Riemann-Liouville defined 𝑣-order differential can be given by 𝐷^𝑣_𝑅-𝐿𝑠(𝑥)=𝑑^𝑣[𝑑(𝑥−𝑎)]^𝑣𝑠(𝑥)_𝑅-𝐿=𝑑^𝑛𝑑𝑥^𝑛𝑑^𝑣−𝑛[𝑑(𝑥−𝑎]^𝑣−𝑛𝑠(𝑥)_𝑅-𝐿=^𝑛−1_𝑘=0(𝑥−𝑎)^𝑘−𝑣𝑠^(𝑘)(𝑎)Γ()+1𝑘−𝑣+1Γ(𝑛−𝑣)^𝑥_𝑎𝑠^(𝑛)𝜂𝑥−𝜂^{𝑣−𝑛+1}𝑑𝜂,0≤𝑣≺𝑛. (3) Fourier transform of the 𝑠(𝑥) is expressed as FT𝐷^𝑣𝑠(𝑥)=(𝑖𝜔)^𝑣FT[𝑠(𝑥)]−^𝑛−1_𝑘=0(𝑖𝜔)^𝑘𝑑^{𝑣−1−𝑘}𝑑𝑥^{𝑣−1−𝑘}𝑠(0), (4) where 𝑖 denotes imaginary unit and 𝜔 represents digital frequency. If 𝑠(𝑥) is causal signal, (4) can be simplified to read FT𝐷^𝑣𝑠(𝑥)=(𝑖𝜔)^𝑣FT[𝑠(𝑥)]. (5)

3. Theoretical Analysis for Fractional Partial Differential Equation: Fractional Total Variation and Fractional Steepest Descent Approach Based Multiscale Denoising Model for Texture Image

3.1. The Fractional Green Formula for Two-Dimensional Image

The premise of implementing Euler-Lagrange formula is to obtain the proper Green’s formula [83]. We therefore extend the order of Green’s formula from the integer to a fractional first in order to implement fractional Euler-Lagrange formula of two-dimensional image.

Consider Ω to be simply connected plane region, taking the piecewise smooth curve 𝐶 as a boundary; then the differintegrable functions 𝑃(𝑥,𝑦) and 𝑄(𝑥,𝑦) [1–6] are continuous in Ω and 𝐶, and the fractional continuous partial derivatives for 𝑥 and 𝑦 exist. If we consider 𝐷¹ to represent the first-order differential operator, then 𝐷^𝑣 represents the 𝑣-order fractional differential operator, 𝐼¹=𝐷⁻¹ denotes the first-order integral operator, and 𝐼^𝑣=𝐷^−𝑣 represents 𝑣-order fractional integral operator when 𝑣>0. Note that (𝐼^𝑣_𝑥𝐼^𝑣_𝑦)_Ω represents the 𝑣-order integral operator of curve surface in the Ω plane. 𝐼^𝑣_{𝐶(𝐴𝐶1𝐵)} is the 𝑣-order integral operator in the 𝐴𝐶₁𝐵 section of curve 𝐶 along the direction of −−−−−→𝐴𝐶₁𝐵. 𝐼^𝑣_𝐶− is 𝑣-order fractional integral operator in the closed curve 𝐶 along counter-clockwise direction. Consider that boundary 𝐶 is circled by two curves 𝑦=𝜙₁(𝑥), 𝑦=𝜙₂(𝑥), 𝑎≤𝑥≤𝑏 or 𝑥=𝜓₁(𝑦), 𝑥=𝜓₂(𝑦), 𝑐≤𝑦≤𝑑, as shown in Figure 1.

Figure 1 

Simply connected space Ω and its smooth boundary curve 𝐶.

As for differintegrable function 𝑃(𝑥,𝑦) [1–6], when 𝑃−𝐷^−𝑣1𝐷^𝑣1𝑃≠0, it follows that 𝐷^𝑣1𝐷^𝑣2𝑃  =  𝐷^𝑣1+𝑣2𝑃  −  𝐷^𝑣1+𝑣2(𝑃−𝐷^−𝑣1𝐷^𝑣1𝑃). Thus, when 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦𝐷^𝑣1_𝑦𝑃(𝑥,𝑦)  =  𝐼^𝑣2_𝑥{𝐷^{𝑣1−𝑣2}_𝑦𝑃(𝑥,𝑦)  −  𝐷^{𝑣1−𝑣2}_𝑦[𝑃(𝑥,𝑦)−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃(𝑥,𝑦)]}, it has 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑦𝑃=𝑥,𝑦_𝑏𝑎𝐼^𝑣2_𝑥^𝜙2(𝑥)_𝜙1(𝑥)𝐼^𝑣2_𝑦𝐷^𝑣1_𝑦𝑃=𝑥,𝑦_𝑏𝑎𝐼^𝑣2_𝑥𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦[𝑃(𝑥,𝑦)−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃(𝑥,𝑦)]^𝜙2_𝜙^(𝑥)_1(𝑥)=−𝐼^𝑣2_{𝑥𝐶(𝐵𝐶2𝐴)}𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃𝑥,𝑦−𝐼^𝑣2_{𝑥𝐶(𝐴𝐶1𝐵)}𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃𝑥,𝑦=−𝐼^𝑣2_𝑥𝐶−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃𝑥,𝑦. (6) Similarly, it has 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑥𝑄𝑥,𝑦=𝐼^𝑣2_𝑦𝐶−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^−𝑣1_𝑥𝐷^𝑣1_𝑥𝑄𝑥,𝑦. (7) Fractional Green’s formula of two-dimensional image can be expressed by 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑥𝑄𝑥,𝑦−𝐷^𝑣1_𝑦𝑃𝑥,𝑦=𝐼^𝑣2_𝑥𝐶−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃𝑥,𝑦+𝐼^𝑣2_𝑦𝐶−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^−𝑣1_𝑥𝐷^𝑣1_𝑥𝑄𝑥,𝑦. (8) When 𝐷^𝑣1 and 𝐷^−𝑣1 are reciprocal, that is, 𝜑−𝐷^−𝑣1𝐷^𝑣1𝜑=0, it follows that 𝐷^𝑣1𝐷^𝑣2𝜑=𝐷^𝑣1+𝑣2𝜑. Then, (8) can be simplified to read 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑥𝑄𝑥,𝑦−𝐷^𝑣1_𝑦𝑃𝑥,𝑦=𝐼^𝑣2_𝑥𝐶−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦+𝐼^𝑣2_𝑦𝐶−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦. (9) We know from (9) the following. First, when 𝑣₁=𝑣₂=𝑣, (9) can be simplied as (𝐼^𝑣_𝑥𝐼^𝑣_𝑦)_Ω(𝐷^𝑣_𝑥𝑄(𝑥,𝑦)−𝐷^𝑣_𝑦𝑃(𝑥,𝑦))=(𝐼^𝑣_𝑥)_𝐶−𝑃(𝑥,𝑦)+(𝐼^𝑣_𝑦)_𝐶−𝑄(𝑥,𝑦), which is the expression of fractional Green’s formula in reference [84]. Second, when 𝑣₁=𝑣₂=1, it follows that (𝐼¹_𝑥𝐼¹_𝑦)_Ω(𝐷¹_𝑥𝑄(𝑥,𝑦)−𝐷¹_𝑦𝑃(𝑥,𝑦))=(𝐼¹_𝑥)_𝐶−𝑃(𝑥,𝑦)+(𝐼¹_𝑦)_𝐶−𝑄(𝑥,𝑦). The classical integer-order Green’s formula is the special case of fractional Green formula.

3.2. The Fractional Euler-Lagrange Formula for Two-Dimensional Image

To implement a fractional partial differential equation-based denoising model, we must obtain the fractional Euler-Lagrange formula first, and thus we furtherly deduce the fractional Euler-Lagrange formula for two-dimensional image based on the above fractional Green’s formula.

Consider the differintegrable numerical function in two-dimensional space to be 𝑢(𝑥,𝑦) and the differintegrable vector function to be 𝜑(𝑥,𝑦)=𝑖𝜑_𝑥+𝑗𝜑_𝑦 [1–6]; the 𝑣-order fractional differential operator is 𝐷^𝑣=𝑖(𝜕^𝑣/𝜕𝑥^𝑣)+𝑗(𝜕^𝑣/𝜕𝑦^𝑣)=𝑖𝐷^𝑣_𝑥+𝑗𝐷^𝑣_𝑦=(𝐷^𝑣_𝑥,𝐷^𝑣_𝑦). Here, 𝐷^𝑣 is a type of linear operator. When 𝑣=0, then 𝐷⁰ represents an equality operator, which is neither differential nor integral, where 𝑖 and 𝑗, respectively, represent the unit vectors in the 𝑥- and 𝑦-directions. In general, the two-dimensional image region Ω is a rectangular simply connected space, and thus the piecewise smooth boundary 𝐶 is also a rectangular closed curve, as shown in Figure 2.

Figure 2 

The two-dimensional simply connected space Ω and its piecewise smooth boundary curve 𝐶.

Referring to (2), it follows that 𝐼^𝑣_𝑥𝑠(𝑥,𝑦)=(1/Γ(𝑣))∫^𝑥_𝑎𝑥(𝑥−𝜂)^𝑣−1𝑠(𝜂,𝑦)𝑑𝜂, 𝐼^𝑣_𝑦𝑠(𝑥,𝑦)  =  (1/Γ(𝑣))∫^𝑦_𝑎𝑦(𝑦−𝜁)^𝑣−1𝑠(𝑥,𝜁)𝑑𝜁 and 𝐼^𝑣_𝑥𝐼^𝑣_𝑦𝑠(𝑥,𝑦)  =  (1/Γ²(𝑣))∫^𝑥_𝑎𝑥∫^𝑦_𝑎𝑦(𝑥−𝜂)^𝑣−1(𝑦−𝜁)^𝑣−1𝑠(𝜂,𝜁)𝑑𝜂𝑑𝜁. From the fractional Green formula (8) and Figure 2, we can derive 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑥𝑄𝑥,𝑦−𝐷^𝑣1_𝑦𝑃𝑥,𝑦=𝐼^𝑣2_𝑥𝐶−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑦𝑃𝑥,𝑦−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃𝑥,𝑦+𝐼^𝑣2_𝑦𝐶−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^{𝑣1−𝑣2}_𝑥𝑄𝑥,𝑦−𝐷^−𝑣1_𝑥𝐷^𝑣1_𝑥𝑄=𝑥,𝑦_𝑥1𝑥0𝐼^𝑣2_𝑥𝐷^{𝑣1−𝑣2}_𝑦𝑃−𝐷^{𝑣1−𝑣2}_𝑦𝑃−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃+^𝑥0_𝑥1𝐼^𝑣2_𝑥𝐷^{𝑣1−𝑣2}_𝑦𝑃−𝐷^{𝑣1−𝑣2}_𝑦𝑃−𝐷^−𝑣1_𝑦𝐷^𝑣1_𝑦𝑃+^𝑦1_𝑦0𝐼^𝑣2_𝑦𝐷^{𝑣1−𝑣2}_𝑥𝑄−𝐷^{𝑣1−𝑣2}_𝑥𝑄−𝐷^−𝑣1_𝑥𝐷^𝑣1_𝑥𝑄+^𝑦0_𝑦1𝐼^𝑣2_𝑦𝐷^{𝑣1−𝑣2}_𝑥𝑄−𝐷^{𝑣1−𝑣2}_𝑥𝑄−𝐷^−𝑣1_𝑥𝐷^𝑣1_𝑥𝑄≡0. (10) Since ∑^∞_𝑚=0∑^𝑚_𝑛=0≡∑^∞_𝑛=0∑^∞_𝑚=𝑛 and (^𝑣_𝑟+𝑛)(_𝑛^𝑟+𝑛)≡(^𝑣_𝑛)(_𝑟^𝑣−𝑛), we have 𝐷^𝑣_𝑥−𝑎𝑓𝑔=^∞_𝑛=0𝑣𝑛𝐷^𝑣−𝑛_𝑥−𝑎𝑓𝐷^𝑛_𝑥−𝑎𝑔. (11) Referring to the homogeneous properties of fractional calculus and (10), we can derive that (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω[𝐷^𝑣1_𝑥(𝑢𝜑_𝑥)+𝐷^𝑣1_𝑦(𝑢𝜑_𝑦)]=(𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω[𝐷^𝑣1_𝑥(𝑢𝜑_𝑥)−𝐷^𝑣1_𝑦(−𝑢𝜑_𝑦)]=0. Then, we have 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1𝜑=𝐼𝑢•_𝑣2𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1_𝑥𝑢𝜑_𝑥+𝐷^𝑣1_𝑦𝑢𝜑_𝑦=−𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω^∞_𝑛=1𝑣₁𝑛𝐷^𝑣1_𝑥^−𝑛𝑢𝐷^𝑛_𝑥𝜑_𝑥+𝐷^𝑣1_𝑦^−𝑛𝑢𝐷^𝑛_𝑦𝜑_𝑦, (12) where the sign • denotes the inner product. Similar to the definition of fractional-order divergence operator div^𝑣𝜑=𝐷^𝑣•𝜑=𝐷^𝑣_𝑥𝜑_𝑥+𝐷^𝑣_𝑦𝜑_𝑦, we consider the 𝑣-order fractional differential operator to be 𝑃^𝑣=∑^∞_𝑛=1(^𝑣_𝑛)[𝑖((𝐷_𝑥^𝑣−𝑛𝑢)/𝑢)𝐷^𝑛_𝑥+𝑗((𝐷_𝑦^𝑣−𝑛𝑢)/𝑢)𝐷^𝑛_𝑦] and the 𝑣-order fractional divergence operator to be div𝑃^𝑣𝜑=𝑃^𝑣•𝜑=𝑃^𝑣_𝑥𝜑_𝑥+𝑃^𝑣_𝑦𝜑_𝑦, where both div𝑃^𝑣 and 𝑃^𝑣are the linear operators. In light of Hilbert adjoint operator theory [85] and (12), we can derive 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1𝑢•𝜑=𝐷^𝑣1𝑢,𝜑^𝑣2=𝑢,𝐷^𝑣1∗𝜑^𝑣2=𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝑢𝐷^𝑣1∗𝜑=−𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝑢𝑃^𝑣1•𝜑=−𝐼^𝑣1_𝑥𝐼^𝑣2_𝑦Ω𝑢div𝑃^𝑣1𝜑, (13) where ⟨,⟩^𝑣2 denotes the integral form of the 𝑣₂-order fractional inner product. (𝐷^𝑣)^∗ is 𝑣-order fractional Hilbert adjoint operator of 𝐷^𝑣. Then, it follows that 𝐷^𝑣∗=−div𝑃^𝑣, (14) where (𝐷^𝑣)^∗ is a fractional Hilbert adjoint operator. When 𝑣₁=𝑣₂=1, (12) becomes 𝐼¹_𝑥𝐼¹_𝑦Ω𝐷¹𝑢•𝜑=𝐷¹𝑢,𝜑¹=𝑢,𝐷^1∗𝜑¹=𝐼¹_𝑥𝐼¹_𝑦Ω𝑢𝐷^1∗𝜑=−𝐼¹_𝑥𝐼¹_𝑦Ω𝑢div¹𝜑, (15) where ⟨,⟩¹ denotes the integral form of the first-order inner product, div¹𝜑=𝐷¹•𝜑=𝐷¹_𝑥𝜑_𝑥+𝐷¹_𝑦𝜑_𝑦 represents the first-order divergence operator, and (𝐷¹)^∗ represents the first-order Hilbert adjoint operator of 𝐷¹. As for digital image, we find that 𝐷^1∗=−div¹. (16) Equations (14) and (16) have shown that the first-order Hilbert adjoint operator is the special case of that of fractional order. When (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω𝐷^𝑣1𝑢•𝜑=0, (14) becomes 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω𝐷^𝑣1𝜑𝐼𝑢•=−^𝑣2_𝑥𝐼^𝑣2_𝑦Ω^∞_𝑛=1𝑣₁𝑛𝐷^𝑣1_𝑥^−𝑛𝑢𝐷^𝑛_𝑥𝜑_𝑥+𝐷^𝑣1_𝑦^−𝑛𝑢𝐷^𝑛_𝑦𝜑_𝑦=0. (17) Since the 𝑥 line and 𝑦 line meet at right angle, it has (𝐷^𝑣1_𝑥^−𝑛𝑢)𝐷^𝑛_𝑥𝜑_𝑥+(𝐷^𝑣1_𝑦^−𝑛𝑢)𝐷^𝑛_𝑦𝜑_𝑦=(𝐷^𝑣1_𝑥^−𝑛𝑢,𝐷^𝑣1_𝑦^−𝑛𝑢)•(𝐷^𝑛_𝑥𝜑_𝑥,𝐷^𝑛_𝑦𝜑_𝑦). As for any two-dimensional function 𝑢, the corresponding 𝐷^𝑣1_𝑥^−𝑛𝑢 and 𝐷^𝑣1_𝑦^−𝑛𝑢 are randomly chosen. According to the fundamental lemma of calculus of variations [83], we know that (𝐷^𝑛_𝑥𝜑_𝑥,𝐷^𝑛_𝑦𝜑_𝑦)=(0,0) is required to make (17) established. Since 𝑛 is the positive integer belonging to 1→∞, we only need (𝐷¹_𝑥𝜑_𝑥,𝐷¹_𝑦𝜑_𝑦)=(0,0) to make (𝐷^𝑛_𝑥𝜑_𝑥,𝐷^𝑛_𝑦𝜑_𝑦)=(0,0) established. Equation (17) can be rewritten as −(𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω^𝑣1₁[(𝐷^𝑣1_𝑥⁻¹𝑢)𝐷¹_𝑥𝜑_𝑥+(𝐷^𝑣1_𝑦⁻¹𝑢)𝐷¹_𝑦𝜑_𝑦]=−(𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω^𝑣1₁(𝐷^𝑣1_𝑥⁻¹𝑢,𝐷^𝑣1_𝑦⁻¹𝑢)•(𝐷¹_𝑥𝜑_𝑥,𝐷¹_𝑦𝜑_𝑦)=0. When and only when the below equation is satisfied, (17) is established: −𝑣₁1𝐷¹_𝑥𝜑_𝑥+𝐷¹_𝑦𝜑_𝑦=Γ1−𝑣₁Γ−𝑣₁𝐷¹_𝑥𝜑_𝑥+𝐷¹_𝑦𝜑_𝑦=0. (18) Equation (18) is the fractional Euler-Lagrange formula, which corresponds to (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω𝐷^𝑣1𝑢•𝜑=0.

Also, if one considers Φ₁(𝐷^𝑣1𝑢) to be the numerical function of vector function 𝐷^𝑣1𝑢 and Φ₂(𝜑) to be the numerical function of differintegrable vector function 𝜑(𝑥,𝑦)=𝑖𝜑_𝑥+𝑗𝜑_𝑦 [1–6], similarly, when (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_ΩΦ₁(𝐷^𝑣1𝑢)Φ₂(𝜑)𝐷^𝑣1𝑢•𝜑=0, (14) can be written as 𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦ΩΦ₁𝐷^𝑣1𝑢Φ₂𝜑𝐷^𝑣1𝜑=𝐼𝑢•_𝑣2𝑥𝐼^𝑣2_𝑦ΩΦ₁𝐷^𝑣1𝑢𝐷^𝑣1𝑢•Φ₂𝜑𝜑=−𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦Ω^∞_𝑛=1𝑣₁𝑛Φ₁𝐷^𝑣1𝑢𝐷^𝑣1_𝑥^−𝑛𝑢×𝐷^𝑛_𝑥Φ₂𝜑𝜑_𝑥+Φ₁𝐷^𝑣1𝑢𝐷^𝑣1_𝑦^−𝑛𝑢×𝐷^𝑛_𝑦Φ₂𝜑𝜑_𝑦=0. (19) Equation (19) is established, when and only when −𝑣₁1𝐷¹_𝑥Φ₂𝜑𝜑_𝑥+𝐷¹_𝑦Φ₂𝜑𝜑_𝑦=Γ1−𝑣₁Γ−𝑣₁𝐷¹_𝑥Φ₂𝜑𝜑_𝑥+𝐷¹_𝑦Φ₂𝜑𝜑_𝑦=0. (20) Equation (20) is the fractional Euler-Lagrange formula corresponding to (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_ΩΦ₁(𝐷^𝑣1𝑢)Φ₂(𝜑)𝐷^𝑣1𝑢•𝜑=0.

Since it has 𝐷^𝑣_𝑥−𝑎[0]≡0 no matter what 𝑣 is, we know that the fractional Euler-Lagrange formulas (18) and (20) are irrelevant to the integral order 𝑣₂ of fractional surface integral (𝐼^𝑣2_𝑥𝐼^𝑣2_𝑦)_Ω. We therefore only adopt the first-order surface integral (𝐼^𝑣_𝑥𝐼^𝑣_𝑦)_Ω instead of that of fractional order, when we discuss the energy norm of fractional partial differential equation-based model for texture denoising below.

3.3. The Fractional Partial Differential Equation-Based Denoising Approach for Texture Image

Based on the fractional Euler-Lagrange formula for two-dimensional image, we can implement a fractional partial differential equation-based denoising model for texture image.

𝑠(𝑥,𝑦) represents the gray value of the pixel (𝑥,𝑦), where Ω⊂𝑅² is the image region, that is, (𝑥,𝑦)∈Ω. Consider 𝑠(𝑥,𝑦) to be the noised image and 𝑠₀(𝑥,𝑦) to represent the desired clean image. Since the noise can be converted to additive noise by log processing when it is multiplicative noise and to additive noise by frequency transform and log processing when it is convolutive noise, we assume that 𝑛(𝑥,𝑦) is additive noise, that is, 𝑠(𝑥,𝑦)=𝑠₀(𝑥,𝑦)+𝑛(𝑥,𝑦) without loss of generality. Consider 𝑛(𝑥,𝑦) to represent the additive noise, that is, 𝑠(𝑥,𝑦)=𝑠₀(𝑥,𝑦)+𝑛(𝑥,𝑦); we adopted the fractional extreme to form the energy norm. Similar to the fractional 𝛿-cover in the Hausdorff measure [96, 97], we consider the fractional total variation of image 𝑠 to be |𝐷^𝑣1𝑠|^𝑣2=((𝐷^𝑣1_𝑥𝑠)²+(𝐷^𝑣1_𝑦𝑠)²)^𝑣2, where 𝑣₂ is any real number including fractional number and |𝐷^𝑣1𝑠|^𝑣2 is the hypercube measure. We assume the fractional variation-based fractional total variation as 𝐸_FTV(𝑠)=𝐼¹_𝑥𝐼¹_𝑦Ω𝑓𝐷^𝑣1𝑠^𝑣2=_Ω𝑓𝐷^𝑣1𝑠^𝑣2𝑑𝑥𝑑𝑦. (21)

Consider 𝑠 to be the 𝑣₃-order extremal surface of 𝐸_FTV, the test function is the admitting curve surface close to the extremal surface, that is, 𝜉(𝑥,𝑦)∈𝐶^∞₀(Ω), we then correlate and merge 𝑠 and 𝜉 by 𝑠+(𝛽−1)𝜉. When 𝛽=1, it is the 𝑣₃-order extremal surface 𝑠. Assume that Ψ₁(𝛽)=𝐸_FTV[𝑠+(𝛽−1)𝜉] and Ψ₂(𝛽)=∬_Ω𝜆[𝑠+(𝛽−1)𝜉−𝑠₀]𝑠₀𝑑𝑥𝑑𝑦, where Ψ₂(𝛽) is the cross energy of the noise and clean signal s₀, that is, [𝑠+(𝛽−1)𝜉−𝑠₀], and it also the measurement of the similarity between [𝑠+(𝛽−1)𝜉−𝑠₀] and 𝑠₀. We therefore can explain the anisotropic diffusion as energy dissipation process for solving the 𝑣₃-order minimum of fractional energy norm 𝐸_FTV, that is, the process for solving minimum of Ψ₂(𝛽) is to obtain the minimum similarity between the noise and the clean signal. Here, Ψ₂(𝛽) plays the role of nonlinear fidelity during denoising, and 𝜆 is regularized parameter. Fractional total variation-based fractional energy norm in surface family 𝑠+(𝛽−1)𝜉 can be expressed by Ψ𝛽=Ψ₁𝛽+Ψ₂𝛽=𝐼¹_𝑥𝐼¹_𝑦Ω𝑓𝐷^𝑣1𝑠+(𝛽−1)𝐷^𝑣1𝜉^𝑣2+𝜆𝑠+𝛽−1𝜉−𝑠₀𝑠₀=_Ω𝑓𝐷^𝑣1𝑠+𝛽−1𝐷^𝑣1𝜉^𝑣2+𝜆𝑠+𝛽−1𝜉−𝑠₀𝑠₀𝑑𝑥𝑑𝑦. (22)

As for Ψ₁(𝛽), if 𝑣₃-order fractional derivative of 𝑓(|𝐷^𝑣1𝑠+(𝛽−1)𝐷^𝑣1𝜉|^𝑣2) exists, Ψ₁(𝛽) has the 𝑣₃-order fractional minimum or stationary point when 𝛽=1. Referring to the linear properties of fractional differential operator, we have 𝐷^𝑣3_𝛽Ψ₁(𝛽)_𝛽=1=𝜕^𝑣3𝜕𝛽^𝑣3_Ω𝑓𝜑^𝑣2𝑑𝑥𝑑𝑦_𝛽=1=_Ω𝜕^𝑣3𝜕𝛽^𝑣3𝑓𝜑^𝑣2𝑑𝑥𝑑𝑦_𝛽=1=0, (23) where it has 𝜑²=|𝜑|²=[(𝜑)²]²=𝜑•𝜑 as for the vector 𝜑=𝐷^𝑣1𝑠+(𝛽−1)𝐷^𝑣1𝜉, and the signal • denotes the inner product. Unlike the traditional first-order variation, (23) is the 𝑣₃-order fractional extreme of Ψ₁(𝛽), which aims to nonlinearly preserve the complex texture details as much as possible when denoising by using the special properties of fractional calculus that it can nonlinearly maintain the low-frequency contour feature in the smooth area to the furthest degree and nonlinearly enhance the high-frequency edge information in those areas where gray level changes frequently and also nonlinearly enhance the high-frequency texture details in those areas where gray level does not change obviously [33–38].

Provided that 𝑣 is a fractional number, when 𝑛>𝑣, it has (^𝑣_𝑛)=(−1)^𝑛Γ(𝑛−𝑣)/Γ(−𝑣)Γ(𝑛+1)≠0. Referring to (11) and Faà de Bruno formula [95], we can derive the rule of fractional calculus of composite function as 𝐷^𝑣_𝛽−𝑎𝑓𝑔𝛽=𝛽−𝑎^−𝑣Γ()𝑓+1−𝑣_∞𝑛=1𝑣𝑛𝛽−𝑎^𝑛−𝑣Γ()×𝑛−𝑣+1𝑛!_𝑛𝑚=1𝐷^𝑚_𝑔𝑓^𝑛_𝑘=11𝑃_𝑘!𝐷^𝑘_𝛽−𝑎𝑔𝑘!^𝑃𝑘, (24) where 𝑔(𝛽)=|𝜑|^𝑣2 and 𝑎 is the constant. 𝑛=0 is separated from summation item. From (24), we know that the fractional derivative of composite function is the summation of infinite items. Here, 𝑃_𝑘 satisfies ^𝑛_𝑘=1𝑘𝑃_𝑘=𝑛,^𝑛_𝑘=1𝑃_𝑘=𝑚. (25) The third signal ∑ in (25) denotes the summation of {∏^𝑛_𝑘=1(1/𝑃_𝑘!)[𝐷^𝑘_𝛽−𝑎𝑔/𝑘!]^𝑃𝑘}|_{𝑚=1→𝑛} of the combination of 𝑃_𝑘|_{𝑚=1→𝑛} that satisfied (25). Recalling (23), (24) and the property of Gamma function, we can derive _Ω𝜕^𝑣3𝜕𝛽^𝑣3𝑓𝜑^𝑣2𝑑𝑥𝑑𝑦_𝛽=1=_Ω𝛽^−𝑣3𝑓𝜑^𝑣2Γ1−𝑣₃+^∞_𝑛=1𝑣₃𝑛𝛽^𝑛−𝑣3Γ𝑛−𝑣₃×+1𝑛!_𝑛𝑚=1𝐷^𝑚_𝜑^𝑣₂𝑓^𝑛_𝑘=11𝑃_𝑘!𝐷^𝑘_𝛽𝜑^𝑣2𝑘!^𝑃𝑘𝑑𝑥𝑑𝑦_𝛽=1=_Ω𝑓𝐷^𝑣1𝑠^𝑣2Γ1−𝑣₃+^∞_𝑛=1(−1)^𝑛Γ𝑛−𝑣₃Γ−𝑣₃Γ𝑛−𝑣₃×+1_𝑛𝑚=1𝐷^𝑚_𝜑^𝑣₂𝑓_𝛽=1^𝑛_𝑘=11𝑃_𝑘!𝐷^𝑘_𝛽𝜑^𝑣2_𝛽=1𝑘!^𝑃𝑘×𝑑𝑥𝑑𝑦=0. (26) Without loss of the generality, we consider 𝑓(𝜂)=𝜂 for simple calculation; it then has 𝐷¹_𝜂𝑓(𝜂)=1 and 𝐷^𝑚_𝜂𝑓(𝜂)=^𝑚≥20; thus (26) can be reduced as _Ω𝐷^𝑣1𝑠^𝑣2Γ1−𝑣₃+^∞_𝑛=1(−1)^𝑛Γ𝑛−𝑣₃Γ−𝑣₃Γ𝑛−𝑣₃×+1_𝑛𝑘=11𝑃_𝑘!𝐷^𝑘_𝛽𝜑^𝑣2_𝛽=1𝑘!^𝑃𝑘_𝑚=1𝑑𝑥𝑑𝑦=0. (27) We know from (25), when 𝑚=1, 𝑃_𝑘 satisfies 𝑃_𝑛=1 and 𝑃₁=𝑃₂=⋯=𝑃_𝑛−1=0; then it can be furtherly reduced as _Ω𝐷^𝑣1𝑠^𝑣2Γ1−𝑣₃+^∞_𝑛=1(−1)^𝑛Γ𝑛−𝑣₃Γ−𝑣₃Γ𝑛−𝑣₃×𝐷+1_𝑛𝛽𝜑^𝑣2_𝛽=1𝑛!𝑑𝑥𝑑𝑦=0. (28) When 𝑛 takes odd number (𝑛=2𝑘+1, 𝑘=0,1,2,3,…) and even number (𝑛=2𝑘, 𝑘=1,2,3,…), respectively, the expressions of 𝐷^𝑛_𝛽(|𝜑|^𝑣2)|_𝛽=1 are also different: 𝐷^𝑛_𝛽𝜑^{𝑣2𝑛=2𝑘+1}_𝛽=1=^𝑛_𝜏=1𝑣₂−𝜏+1𝐷^𝑣1𝑠^{𝑣2−𝑛−1}×𝐷^𝑣1𝜉^𝑛−1𝐷^𝑣1𝜉•𝐷^𝑣1𝑠,𝐷^𝑛_𝛽𝜑^{𝑣2𝑛=2𝑘}_𝛽=1=^𝑛_𝜏=1𝑣₂−𝜏+1𝐷^𝑣1𝑠^𝑣2−𝑛𝐷^𝑣1𝜉^𝑛. (29) Put (29) into (28), then it becomes _Ω^∞_𝑘=0∏^2𝑘_𝜏=1𝑣₂−𝜏+1𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1𝜉^2𝑘Γ−𝑣₃()!𝐷2𝑘^𝑣1𝑠•𝐷^𝑣1Γ2𝑘−𝑣₃Γ2𝑘−𝑣₃𝑣+1𝑠−²−2𝑘Γ2𝑘−𝑣₃+1(2𝑘+1)Γ2𝑘−𝑣₃𝜉+2×𝑑𝑥𝑑𝑦=0. (30) We consider that ∏^𝑛_𝜏=1(𝑣₂−𝜏+1)=^𝑛=01. Since the corresponding 𝐷^𝑣1𝜉 is randomly chosen for any two-dimensional numerical function 𝜉, 𝐷^𝑣1[(Γ(2𝑘−𝑣₃)/Γ(2𝑘−𝑣₃+1))𝑠  −  ((𝑣₂−2𝑘)Γ(2𝑘−𝑣₃+1)/(2𝑘+1)Γ(2𝑘−𝑣₃+2)𝜉)] is random. Referring to (20), the corresponding fractional Euler-Lagrange formula of (30) is Γ1−𝑣₁Γ−𝑣₁Γ−𝑣₃^∞_𝑘=0∏^2𝑘_𝜏=1𝑣₂−𝜏+1()!×𝐷2𝑘_1𝑥𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑥𝑠+𝐷¹_𝑦𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑦𝑠=0, (31) where it has ∏^𝑛_𝜏=1(𝑣₂−𝜏+1)=^𝑛=01. It is the corresponding fractional Euler-Lagrange formula of (23).

As for Ψ₂(𝛽), it has 𝐷^𝑣3_𝛽Ψ₂𝛽_𝛽=1=𝐷^𝑣3_𝛽𝐼¹_𝑥𝐼¹_𝑦Ω𝜆𝑠+𝛽−1𝜉−𝑠₀𝑠_0𝛽=1=_Ω𝜆𝑠₀Γ1−𝑣₃Γ2−𝑣₃×Γ2−𝑣₃𝑠−𝑠₀−𝜉+Γ1−𝑣₃𝜉𝑑𝑥𝑑𝑦=0. (32) Since the test function 𝜉 is randomly chosen, Γ(2−𝑣₃)(𝑠−𝑠₀−𝜉)+Γ(1−𝑣₃)𝜉 is also random. According to the fundamental lemma of calculus of variations [83], we know that to make (32) established, it must have 𝜆𝑠₀Γ1−𝑣₃Γ2−𝑣₃=0. (33) Equations (23) and (32), respectively, are the 𝑣₃-order minimal value of Ψ₁(𝛽) and Ψ₁(𝛽). Thus, when we take 𝑣=𝑣₃≠1, 2, and 3, the 𝑣₃-order minimal value of (22) can be expressed by 𝜕^𝑣3𝑠𝜕𝑡^𝑣3=−Γ1−𝑣₁Γ−𝑣₁Γ−𝑣₃×^∞_𝑘=0∏^2𝑘_𝜏=1𝑣₂−𝜏+1()!×𝐷2𝑘_1𝑥𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑥𝑠+𝐷¹_𝑦𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑦𝑠−𝜆𝑠₀Γ1−𝑣₃Γ2−𝑣₃, (34) where it has ∏^𝑛_𝜏=1(𝑣₂−𝜏+1)=^𝑛=01. 𝜕^𝑣3𝑠/𝜕𝑡^𝑣3 is calculated by the approach of fractional difference. We must compute 𝜆(t). If image noise 𝑛(𝑥,𝑦) is the white noise, it has ∬_Ω𝑛(𝑥,𝑦)𝑑𝑥𝑑𝑦=∬_Ω(𝑠−𝑠₀)𝑑𝑥𝑑𝑦=0. When 𝜕^𝑣3𝑠/𝜕𝑡^𝑣3=0, it converges to a stable state. Then, we merely multiply (𝑠−𝑠₀)² at both sides of (34) and integrate by parts over Ω and the left side of (34) vanishes: 𝜆(𝑡)=−Γ1−𝑣₁Γ1−𝑣₃Γ2−𝑣₃𝜎²Γ−𝑣₁Γ−𝑣₃𝑠₀×_Ω^∞_𝑘=0∏^2𝑘_𝜏=1𝑣₂−𝜏+1()!×𝐷2𝑘_1𝑥𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑥𝑠+𝐷¹_𝑦𝐷^𝑣1𝑠^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑦𝑠×𝑠−𝑠₀²𝑑𝑥𝑑𝑦. (35)

Here, we simply note the fractional partial differential equation-based denoising model as FDM (a fractional developmental mathematics-based approach for texture image denoising). When numerical iterating, we need to perform low-pass filtering to completely remove the faint noise in very low-frequency and direct current. We know from (34) and (35) that FDM enhances the nonlinear regulation effects of order 𝑣₂ by continually multiplying function ∏^2𝑘_𝜏=1(𝑣₂−𝜏+1) and power 𝑣₂−2𝑘−2 of |𝐷^𝑣1𝑠| and enhance the nonlinear regulation effects of order 𝑣₃ by increasing Γ(−𝑣₃) in the denominator. Also, we know from (34) that FDM is the traditional potential equation or elliptic equation when 𝑣₃=0, the traditional heat conduction equation or parabolic equation when 𝑣₃=1, and the traditional wave equation or hyperbolic equation when 𝑣₃=2. FDM is the continuous interpolation of traditional potential equation and heat conduction equation when 0<𝑣₃<1 and the continuous interpolation of traditional heat conduction equation and wave equation when 1<𝑣₃<2. FDM has pushed the traditional integer-order partial differential equation-based image processing approach from the anisotropic diffusion of integer-order heat conduction equation to that of fractional partial differential equation in the mathematical and physical sense.

4. Experiments and Theoretical Analyzing

4.1. Numerical Implementation of the Fractional Partial Differential Equation-Based Denoising Model for Texture Image

We know from (34) and (35) that we should obtain the fractional differential operator of two-dimensional digital image before implementing FDM. As for Grümwald-Letnikov definition of fractional calculus in (1), it may remove the limit symbol when 𝑁 is large enough; we then introduce the signal value at nonnode to the definition improving the convergence rate and accuracy, that is, (𝑑^𝑣/𝑑𝑥^𝑣)𝑠(𝑥)|_𝐺-𝐿≅(𝑥^−𝑣𝑁^𝑣/Γ(−𝑣))∑^𝑁−1_𝑘=0(Γ(𝑘−𝑣)/Γ(𝑘+1))𝑠(𝑥+(𝑣𝑥/2𝑁)−(𝑘𝑥/𝑁)). Using Lagrange 3-point interpolation equation to perform fractional interpolation when 𝑣≠1, we can obtain the fractional differential operators of YiFeiPU-2, respectively, on the eight symmetric directions [35, 38], which is shown in Figure 3.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 3 

Fractional differential masks 𝐷𝑣, respectively, on the eight directions. (a) Fractional differential operator on 𝑥-coordinate negative direction, noted as 𝐷𝑣𝑥−. (b) Fractional differential operator on 𝑦-coordinate negative direction, noted as 𝐷𝑣𝑦−. (c) Fractional differential operator on 𝑥-coordinate positive direction, noted as 𝐷𝑣𝑥+. (d) Fractional differential operator on 𝑦-coordinate positive direction, noted as 𝐷𝑣𝑦+. (e) Fractional differential operator on left downward diagonal, noted as 𝐷𝑣ldd. (f) Fractional differential operator on right upward diagonal, noted as 𝐷𝑣rud. (g) Fractional differential operator on left upward diagonal, noted as 𝐷𝑣lud. (h) Fractional differential operator on right downward diagonal, noted as 𝐷𝑣rdd.

In Figure 3, 𝐶_𝑠−1 is the coefficient of operator 𝑠₋₁=𝑠(𝑥+𝑥/𝑁) at noncausal pixel, and 𝐶_𝑠0 is the coefficient of operator at pixel of interest 𝑠₀=𝑠(𝑥). 𝑛 is often taken odd number. When 𝑘→𝑛=2𝑚−1, we can implement fractional differential operator of (2𝑚+1)×(2𝑚+1), whose coefficients of the operator, respectively, are 𝐶_𝑠−1  =  (𝑣/4)+(𝑣²/8), 𝐶_𝑠0  =  1−(𝑣²/2)−(𝑣³/8), 𝐶_𝑠1  =  −(5𝑣/4)+(5𝑣³/16)+(𝑣⁴/16),…, 𝐶_𝑠𝑘  =  (1/Γ(−𝑣))[(Γ(𝑘−𝑣+1)/(𝑘+1)!)⋅((𝑣/4)+(𝑣²/8))+(Γ(𝑘−𝑣)/𝑘!)⋅(1−(𝑣²/4))+(Γ(𝑘−𝑣−1)/(𝑘−1)!)⋅(−(𝑣/4)+(𝑣²/8))],…, 𝐶_𝑠𝑛−2  =  (1/Γ(−𝑣))[(Γ(𝑛−𝑣−1)/(𝑛−1)!)⋅((𝑣/4)+(𝑣²/8))+(Γ(𝑛−𝑣−2)/(𝑛−2)!)⋅(1−(𝑣²/4))+(Γ(𝑛−𝑣−3)/(𝑛−3)!)⋅(−(𝑣/4)+(𝑣²/8))], 𝐶_𝑠𝑛−1  =  (Γ(𝑛−𝑣−1)/(𝑛−1)!Γ(−𝑣))⋅(1−(𝑣²/4))+(Γ(𝑛−𝑣−2)/(𝑛−2)!Γ(−𝑣))⋅(−(𝑣/4)+(𝑣²/8)), and 𝐶_𝑠𝑛  =  (Γ(𝑛−𝑣−1)/(𝑛−1)!Γ(−𝑣))⋅(−(𝑣/4)+(𝑣²/8)) [35, 38]. Spatial filtering of convolution operator is adopted as the numerical algorithm of fractional differential operator. We take the maximum value of fractional partial differential on the above 8 directions as the pixel’s fractional calculus. As for fractional differential operator 𝐷^𝑣=𝑖₁𝐷^𝑣_𝑥−+𝑖₂𝐷^𝑣_𝑦−+𝑖₃𝐷^𝑣_𝑥++𝑖₄𝐷^𝑣_𝑦++𝑖₅𝐷^𝑣_ldd+𝑖₆𝐷^𝑣_rud+𝑖₇𝐷^𝑣_lud+𝑖₈𝐷^𝑣_rdd in (34) and (35), when 𝑣≠1  𝐷^𝑣_𝑥−, 𝐷^𝑣_𝑦−, 𝐷^𝑣_𝑥+, 𝐷^𝑣_𝑦+, 𝐷^𝑣_ldd, 𝐷^𝑣_rud, 𝐷^𝑣_lud, 𝐷^𝑣_rdd can perform numerical computing referring to Figure 3 and their concerning coefficients [35, 38], and when 𝑣=1, we take a special difference as the approximate of first-order differential to maintain calculation stability, that is, 𝐷¹_𝑥𝑠(𝑥,𝑦)  =  (2[𝑠(𝑥+1,𝑦)−𝑠(𝑥−1,𝑦)]  +  𝑠(𝑥+1,𝑦+1)−𝑠(𝑥−1,𝑦+1)  +  𝑠(𝑥+1,𝑦−1)−𝑠(𝑥−1,𝑦−1))/4, 𝐷¹_𝑦𝑠(𝑥,𝑦)  =  (2[𝑠(𝑥,𝑦+1)−𝑠(𝑥,𝑦−1)]  +  𝑠(𝑥+1,𝑦+1)−𝑠(𝑥+1,𝑦−1)  +  𝑠(𝑥−1,𝑦  +  1)−𝑠(𝑥−1,𝑦−1))/4.

If time equal interval is Δ𝑡, time 𝑛 is noted as 𝑡_𝑛=𝑛Δ𝑡 when 𝑛=0,1,…. 𝑡₀=0 represents the initial time. The digital image at time 𝑛 is 𝑠^𝑛_𝑥,𝑦=𝑠(𝑥,𝑦,𝑡_𝑛). Thus, we have 𝑠⁰_𝑥,𝑦=𝑠₀(𝑥,𝑦,𝑡₀)+𝑛(𝑥,𝑦,𝑡₀), where 𝑠⁰_𝑥,𝑦 is the original image and 𝑠₀ is the desired clean image that is a constant, that is, 𝑠₀(𝑥,𝑦,𝑡₀)=𝑠₀(𝑥,𝑦,𝑡_𝑛). We can derive the numerical implementation of fractional calculus about time from (1) as 𝜕^𝑣𝑠𝜕𝑡^𝑣=Δ𝑡^−𝑣𝑠^𝑛+1_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦+2𝜇𝜂Γ()𝑠3−𝑣^{𝑛𝑥,𝑦}−𝑠^𝑣∗_𝑥,𝑦²𝑠^𝑛_𝑥,𝑦^−𝑣,when𝑣≠1, (36) where 𝜇 and 𝜂 are the factors for ensuring iterative convergence. Also, the desired clean image 𝑠₀(𝑥,𝑦,𝑡₀) is unknown, but the intermediate denoising results is an approximate to the desired clean image, that is, 𝑠^𝑛_𝑥,𝑦→𝑠₀(𝑥,𝑦,𝑡₀)=𝑠₀(𝑥,𝑦,𝑡_𝑛). We consider that (𝑠−𝑠₀)_𝑥,𝑦≅𝑠⁰_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦 for making 𝑠 close to 𝑠₀ as much as possible. We should compute the fractional calculus on the 8 directions simultaneously in practice in order to improve the calculation accuracy and antirotation capability.

The best image 𝑠^𝑣∗_𝑥,𝑦 in (36) is unknown in numerical iteration, but the intermediate result 𝑠^𝑛_𝑥,𝑦 is an approximate to 𝑠^𝑣∗_𝑥,𝑦, that is, 𝑠^𝑛_𝑥,𝑦→𝑠^𝑣∗_𝑥,𝑦. We consider that (𝑠^𝑛_𝑥,𝑦−𝑠^𝑣∗_𝑥,𝑦)²≅(𝑠^𝑛−1_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦)² to make the iterative result approximate (𝑠^𝑛_𝑥,𝑦−𝑠^𝑣∗_𝑥,𝑦)² as much as possible. Take 𝜂=1 in (36) and 𝑘=0,1 in (34) and (35), then we can derive the numerical implementation of (34) and (35) as 𝑠^𝑛+1_𝑥,𝑦=𝑃𝑠^𝑛_𝑥,𝑦Δ𝑡^𝑣3−𝜆^𝑛Δ𝑡^𝑣3Γ1−𝑣₃Γ2−𝑣₃𝑠^𝑛_𝑥,𝑦+𝑠^𝑛_𝑥,𝑦−2𝜇Γ3−𝑣₃𝑠^𝑛−1_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦²𝑠^𝑛_𝑥,𝑦^−𝑣3,when𝑣₃𝜆≠1,2and3,_𝑛=Γ1−𝑣₃Γ2−𝑣₃𝜎^𝑛2𝑠^𝑛_{𝑥,𝑦𝑥,𝑦}𝑃𝑠^𝑛_𝑥,𝑦𝑠⁰_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦², (37)(38) where ∏^𝑛_𝜏=1(𝑣₂−𝜏+1)=^𝑛=01, 𝜎^𝑛2=∑_𝑥,𝑦(𝑠⁰_𝑥,𝑦−𝑠^𝑛_𝑥,𝑦)², 𝑃(𝑠^𝑛_𝑥,𝑦)=(−Γ(1−𝑣₁)/Γ(−𝑣₁)Γ(−𝑣₃))∑¹_𝑘=0(∏^2𝑘_𝜏=1(𝑣₂−𝜏+1)/(2𝑘)!)[𝐷¹_𝑥(|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑥𝑠^𝑛_𝑥,𝑦)+𝐷¹_𝑦(|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|^{𝑣2−2𝑘−2}𝐷^𝑣1_𝑦𝑠^𝑛_𝑥,𝑦)].

We should pay attention to the following when performing numerical iterative implementation. First, 𝜇 is a small number in (37) to ensure convergence, and here we take 𝜇=0.005. Second, we do not need to know or estimate the variance of noise, but we need to assume 𝜎¹² to be a small positive number in the first iteration. We therefore assume that 𝜎¹²=0.01 in the experiment below. We take 𝜎¹² to (38) and perform numerical iteration. Each iterative result 𝜎^𝑛2 may be different, but it is the approximate to the variance of noise. Third, it is possible that |𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|=0 in numerical iterative calculation, so we take |𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|=0.0689 when |𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|≤0.0689 to ensure that (37) and (38) are meaningful. Fourth, we take 𝑠^𝑛_𝑥,𝑦=0.00001 when 𝑠^𝑛_𝑥,𝑦=0 to make (𝑠^𝑛_𝑥,𝑦)^−𝑣3 have meaning. Fifth, to completely denoise faint noise in very low-frequency and direct current, FDM takes the simple way by reducing the convex in the area where gradient is not changed obviously. We therefore need to perform low-pass filtering for very low-frequency and direct current in numerical iterating. The practices of (37) and (38) are as follows. For one-dimensional signal, we consider that 𝑠_𝑥^𝑛+1=(𝑠^𝑛+1_𝑥−1+2𝑠_𝑥^𝑛+1+𝑠^𝑛+1_𝑥+1)/4 when |𝐷^𝑣1𝑠^𝑛_𝑥|≺𝛼_𝐴 and 𝛼_𝑁𝐴≺𝛼_𝐴 and if the noise is not severe in order to ensuring denoising effect, and in the rest conditions we consider that 𝑠_𝑥^𝑛+1=𝑠_𝑥^𝑛+1, where 𝛼_𝐴=(1/𝑁_𝑥)∑^𝑁𝑥_𝑥|𝐷^𝑣1𝑠^𝑛_𝑥|, 𝛼_𝑁𝐴=(|𝐷^𝑣1𝑠^𝑛_𝑥−1|+2|𝐷^𝑣1𝑠^𝑛_𝑥|+|𝐷^𝑣1𝑠^𝑛_𝑥+1|)/4, and if the noise is very strong, we consider that 𝑠_𝑥^𝑛+1=(𝑠^𝑛+1_𝑥−1+2𝑠_𝑥^𝑛+1+𝑠^𝑛+1_𝑥+1)/4 to accelerate the denoising speed. For two-dimensional signal, we consider 𝑠^𝑛+1_𝑥,𝑦  =  (^𝑥𝑠^𝑛+1_𝑥,𝑦+^𝑦𝑠^𝑛+1_𝑥,𝑦+^𝑟𝑠^𝑛+1_𝑥,𝑦+^𝑙𝑠^𝑛+1_𝑥,𝑦)/4. When 𝛼^𝑥_𝑁𝐴  =  min(𝛼^𝑥_𝑁𝐴,𝛼^𝑦_𝑁𝐴,𝛼^𝑟_𝑁𝐴,𝛼^𝑙_𝑁𝐴), they are ^𝑥𝑠^𝑛+1_𝑥,𝑦  =  (𝑠^𝑛+1_{𝑥−1,𝑦}+2𝑠^𝑛+1_𝑥,𝑦+𝑠^𝑛+1_𝑥+1,𝑦)/4 and ^𝑦𝑠^𝑛+1_𝑥,𝑦  =  ^𝑟𝑠^𝑛+1_𝑥,𝑦  =  ^𝑙𝑠^𝑛+1_𝑥,𝑦  =  𝑠^𝑛+1_𝑥,𝑦, when 𝛼^𝑦_𝑁𝐴  =  min(𝛼^𝑥_𝑁𝐴,𝛼^𝑦_𝑁𝐴,𝛼^𝑟_𝑁𝐴,𝛼^𝑙_𝑁𝐴) they are ^𝑦𝑠^𝑛+1_𝑥,𝑦  =  (𝑠^𝑛+1_{𝑥,𝑦−1}+2𝑠^𝑛+1_𝑥,𝑦+𝑠^𝑛+1_𝑥,𝑦+1)/4 and ^𝑥𝑠^𝑛+1_𝑥,𝑦  =  ^𝑟𝑠^𝑛+1_𝑥,𝑦  =  ^𝑙𝑠^𝑛+1_𝑥,𝑦  =  𝑠^𝑛+1_𝑥,𝑦, when 𝛼^𝑟_𝑁𝐴  =  min(𝛼^𝑥_𝑁𝐴,𝛼^𝑦_𝑁𝐴,𝛼^𝑟_𝑁𝐴,𝛼^𝑙_𝑁𝐴) they are ^𝑟𝑠^𝑛+1_𝑥,𝑦=(𝑠^𝑛+1_{𝑥−1,𝑦+1}+2𝑠^𝑛+1_𝑥,𝑦+𝑠^𝑛+1_{𝑥+1,𝑦−1})/4 and ^𝑥𝑠^𝑛+1_𝑥,𝑦  =  ^𝑦𝑠^𝑛+1_𝑥,𝑦  =  ^𝑙𝑠^𝑛+1_𝑥,𝑦  =  𝑠^𝑛+1_𝑥,𝑦, and in the rest occasions, it has ^𝑥𝑠^𝑛+1_𝑥,𝑦  =  ^𝑦𝑠^𝑛+1_𝑥,𝑦  =  ^𝑟𝑠^𝑛+1_𝑥,𝑦  =  ^𝑙𝑠^𝑛+1_𝑥,𝑦  =  𝑠^𝑛+1_𝑥,𝑦, where 𝛼^𝑥_𝑁𝐴  =  (|𝐷^𝑣1𝑠^𝑛_{𝑥−1,𝑦}|  +  2|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|+|𝐷^𝑣1𝑠^𝑛_𝑥+1,𝑦|)/4, 𝛼^𝑦_𝑁𝐴  =  (|𝐷^𝑣1𝑠^𝑛_{𝑥,𝑦−1}|+2|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|+|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦+1|)/4, and 𝛼^𝑟_𝑁𝐴=(|𝐷^𝑣1𝑠^𝑛_{𝑥−1,𝑦+1}|+2|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|+|𝐷^𝑣1𝑠^𝑛_{𝑥+1,𝑦−1}|)/4, 𝛼^𝑙_𝑁𝐴  =  (|𝐷^𝑣1𝑠^𝑛_{𝑥−1,𝑦−1}|+2|𝐷^𝑣1𝑠^𝑛_𝑥,𝑦|+|𝐷^𝑣1𝑠^𝑛_{𝑥+1,𝑦+1}|)/4. Here, 𝑥 denotes 𝑥-coordinate direction, 𝑦 denotes 𝑦-coordinate direction, 𝑟 denotes right diagonal direction, and 𝑙 denotes left diagonal direction. Sixth, we expand the edge of the image for better denoising edge pixels. Seventh, since we perform low-pass filtering in (37) and (38), in order to remove the possible divergence point in numerical iteration, we consider that 𝑠_𝑥^𝑛+1=𝑠^𝑛_𝑥 when |𝑠_𝑥^𝑛+1|≻6|𝑠^𝑛_𝑥| for one-dimensional signal, and we consider that 𝑠^𝑛+1_𝑥,𝑦=𝑠^𝑛_𝑥,𝑦 when |𝑠^𝑛+1_𝑥,𝑦|≻6|𝑠^𝑛_𝑥,𝑦| for two-dimensional signal.

4.2. Denoising Capabilities Analysis of the Fractional Partial Differential Equation-Based Denoising Model for Texture Image

To analyze and explain the denoising capabilities of fractional partial differential equation-based denoising model for texture image, we perform the comparative experiments using the composite one-dimensional signal combined by the rectangle wave, the sine wave and the sawtooth wave. The numerical iteration will stop at the point where the peak signal-to-noise ratio (PSNR) is the highest, as shown in Figure 4.

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Figure 4 

Denoising for integrated one-dimensional signal comprising rectangle wave, sine wave, and sawtooth wave. (a) Original clean signal and noisy signal (adds white Gaussian noise to the original clean signal, peak signal-to-noise ratio (PSNR) = 25.2486), (b) Gaussian denoising, (c) fourth-order TV denoising [71–73], (d) bilateral filtering denoising [86–88], (e) contourlet denoising [89, 90], (f) wavelet denoising [88, 91], (g) nonlocal means noise filtering (NLMF) denoising [92, 93], (h) fractional-order anisotropic diffusion denoising [82], and (i) FDM denoising (𝑣1=1.025, 𝑣2=2.25, 𝑣3=1.05, Δ𝑡=0.0296).

From subjective visual effect we know the following from Figure 4. First, the denoising effect of Gaussian denoising and fourth-order TV denoising is comparatively worse than others, and the high-frequency singularity component has been greatly diffused and smoothed. We can see from Figures 4(b) and 4(c) that the convexes of high-frequency edge of rectangle wave and sawtooth wave are remarkably smoothed and their energy of high-frequency singularity is obviously diffused in neighboring. Second, the denoising capability of fractional-order anisotropic diffusion denoising is in the middle; that is, the capability of maintaining high-frequency singularity is better than that of Gaussian denoising, fourth-order TV denoising, and contourlet denoising but less than bilateral filtering denoising, wavelet and NLMF denoising. Also, the denoising is not completed. We can see from Figure 4(h) that the convexes of high-frequency edge of the rectangle wave and the sawtooth wave are weakly smoothed, and their energy is weakly diffused in neighboring. Also, the denoised signal has tiny burr. Third, the denoising capabilities of bilateral filtering denoising, contourlet denoising, wavelet denoising, and NLMF denoising are better, which can well maintain the high-frequency singularity but the denoising is still uncomplete. We can see from Figures 4(d)–4(g) that the convexes of high-frequency edge of rectangle wave and sawtooth wave are well retained, but the denoised signal still has many small burrs. Finally, the denoising capability of FDM is the best, which is not only well maintains the high-frequency edge of rectangle wave and sawtooth wave but also denoises completely. We can see from Figure 4(i) that the high-frequency edge singularity of rectangle and sawtooth wave is well maintained and little burr is left.

From the viewpoint of quantitative analysis, we take PSNR and correlation coefficients between noisy signal or denoised signal and original clean signal to measure the denoising effect of the above algorithms, as shown in Table 1.

Table 1 

Denoising effect for composite one-dimensional signal that combined rectangle wave, sine wave, and sawtooth wave.

We know from Table 1 as follows. First, PSNR of Gaussian denoising, fourth-order TV denoising, and contourlet denoising are relatively small among the above approaches with 26.9611≤PSNR≤28.2945 and their correlation coefficients are also small with 0.9953≤correlationcoefficients≤0.9959, which indicates that their denoising capability is worse and the similarity between denoised signal and original clean signal is also low. Second, PSNR of fractional-order anisotropic diffusion denoising is in the middle with PSNR=29.8692, and its correlation coefficients are in the middle with correlationcoefficients=0.9975, which shows that its denoising capability and the similarity between denoised signal and original clean signal is also in the middle. Third, PSNR of bilateral filtering denoising, wavelet denoising, NLMF denoising, and FDM denoising is relatively big with 33.3088≤PSNR≤39.0434, and its correlation coefficients are also big with 0.9975≤correlationcoefficients≤0.9996, which shows that their denoising capabilities are better and the similarity between denoised signal and original clean signal is also high. PSNR and correlation coefficients of FDM denoising is the highest, which shows that its denoising capability is the best and the similarity between denoised signal and original clean signal is also the highest.

From the subjective visual of Figure 4 and quanlitative analysis of Table 1, we know the following. First, FDM has the best denoising capability no matter in high-frequency, middle-frequency, and low-frequency components, the denoised signal fits for the edge of high-frequency and the outline of middle-frequency component, and also the noise in low-frequency component is removed clearly and little blur is left. Second, the high-frequency edge singularity of rectangle signal and sawtooth wave signal in Figure 4 has strong high-frequency component which corresponds to high-frequency edge and texture details of two-dimensional signal, while the high-frequency singularity is small and the middle-frequency components are big in the slope of sine wave and sawtooth signal which corresponds to middle-frequency of two-dimensional signal. The constant direct current of regulation signal corresponds to low-frequency and direct current components and backgrounds.

To analyze and explain the good denoising capability of FDM, we choose the better models, including bilateral filtering denoising, wavelet denoising, NLMF denoising, and FDM denoising to perform the contrast experiments for texture-rich metallographic images of an iron ball. The numerical iterative process will stop at the point where the PSNR is the highest, as shown in Figure 5.

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Figure 5 

Denoising effects of texture-rich metallographic image of iron ball. (a) Original clean image, (b) noisy image (adds white Gaussian noise to the original clean image, PSNR = 13.6872), (c) partial enlarged details of 1/4 party of (a) red box, (d) denoised image of bilateral filtering denoising [88, 89, 93], (e) residual plot of bilateral filtering denoising, (f) partial enlarged details of 1/4 party of (d) red box, (g) denoised image of wavelet denoising [92, 93], (h) residual plot of wavelet denoising, (i) partial enlarged details of 1/4 party of (g) red box, (j) denoised image of NLMF denoising [94, 95], (k) residual plot of NLMF denoising, (l) partial enlarged details of 1/4 party of (j) red box, (m) denoised image of FDM denoising (𝑣1=1.75, 𝑣2=2.25, 𝑣3=1.05, Δ𝑡=10−10), (n) residual plot of FDM denoising, (o) partial enlarged details of 1/4 party of (m) red box, (p) denoised Image of FDM denoising (𝑣1=2.25, 𝑣2=2.5, 𝑣3=1.25, Δ𝑡=10−10), (q) residual plot of FDM denoising, (r) partial enlarged details of 1/4 party of (p) red box.

From a subjective view of the visual effect, we know the following from Figure 5. First, the denoising capabilities of bilateral filtering and wavelet denoising are worse than the other methods, because they obviously diffuse and smooth the high-frequency edge and texture details. We can see that the edge and texture details are clear from Figures 5(e) and 5(h); that is, the denoised noise by bilateral filtering and wavelet denoising may not be the same as the added noise. And we can see from Figures 5(f) and 5(i) that the denoised image is blurry. The denoising is not completed in Figure 5(i), which shows that their capabilities for preserving edge and texture details are worse. Second, the capability for preserving edge and texture details of NLMF denoising is better, but its denoising capability for edge and texture neighboring is worse. From Figure 5(k), we know that though the edge and texture details of denoised image are weaker than those in Figures 5(e) and 5(h), but they still can be seen, the denoised noise by NLMF denoising is close to the added noise. In Figure 5(l), the edge and texture details are small blurred, that is, NLMF denoising can well preserve the edge and texture details. Also, the neighboring of edge and texture details is smooth in Figure 5(k), while the residual noise in the edge and texture neighboring is stronger than other parts in Figure 5(l), that is, the denoising capability of NLMF denoising is worse at the edge and texture neighboring. Third, the denoising capabilities of FDM denoising are the best, which preserve the high-frequency edge and texture details well and also denoise comparatively completed. We can see indistinctly the edge and texture detail from Figures 5(n) and 5(q), which shows that the denoised noise by FDM denoising is the most close to the added noise; the edge and texture details are the smallest blurred and the noise is the clearest removed in Figures 5(o) and 5(r), that is, the capabilities for preserving the edge and texture details of FDM are the best.

From the viewpoint of quantitative analysis, we take the PSNR, the correlation coefficients between the noisy image or the denoised image and the original clean image [94], and the average gray level concurrence matrix to comprehensively estimate the denoised effect. We calculate the gray level concurrence matrix coefficient in 5 pixel distance in Figure 5 and export the typical coefficients: contrast, correlation, energy, and homogeneity taking four directions of 0^∘, 45^∘, 90^∘, and 135^∘. Here, 0^∘ represents the projection in the positive 𝑦-coordinate direction, and 90^∘ represents the projection in the 𝑥-coordinate direction. We then average the above values, as seen in Table 2.

Table 2 

Comprehensive denoising effects results for texture-rich metallographic images of an iron ball.

We know the denoising capabilities of the above algorithms are as follows from Table 2. First, the denoising capabilities of bilateral filtering and wavelet denoising are worse than the other methods and their PSNR values and correlation coefficients are relatively small. This shows that the high-frequency edge and texture details are greatly diffused and smoothed and the noise is not completely removed. The similarity between denoised image and original clean image is small. Also, the contrast of average gray level concurrence matrix for bilateral filtering denoising is small, which shows that fewer pixels have great contrast and that the texture depth is light and seems fuzzy. The contrast of the average gray level concurrence matrix of wavelet denoising is the largest. This shows that there are more pixels with great contrast, but we cannot say that the texture depth is deeper and the visual effects are clearer because the denoising is incomplete. Second, the denoising capabilities of NLMF and FDM denoising are better. Their PSNR and correlation coefficients are comparatively higher, which indicates that the high-frequency edge and texture details of the denoised image are well preserved, that the denoising is completed, and that the similarity between the denoised image and original clean image is also great. PSNR and correlation coefficients of FDM denoising are the highest, that is, the denoising is most completed and the similarity is also the greatest. The contrast of the average gray level concurrence matrix of FDM denoising is the highest. This shows that there are more pixels with great contrast, that the texture depth is the deepest, and that the image looks clearer. The correlation is small, which shows that the partial gray correlation is weak and that the texture details are obvious. The energy is rather smaller, which shows that the texture changing is not uniform and regular, and thus the texture details are obvious. The homogeneity is also small, which indicates the regional changing is dramatic and that the texture details are obvious. Therefore, we can conclude that FDM denoising is the best denoising algorithm.

To consider a scenario where the Gaussian noise is very strong, and especially when the original clean signal is completely drowned in noises, we perform comparison experiments using the well-performed algorithms discussed above, including bilateral filtering denoising, wavelet denoising, NLMF denoising, and FDM denoising for further analysis of the denoising capability of FMD for robust noise. The numerical iterative process will also stop at the point where peak signal-to-noise ratio is the highest, as seen in Figure 6.

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Figure 6 

Denoising effects of abdomen MRI of texture-rich internal organ, when Gaussian noise is very strong. (a) Original clean MRI, (b) noisy image (adds white Gaussian noise to the original clean MRI, PSNR=5.4389), (c) denoised image of bilateral filtering denoising [88, 89, 93], (d) partial enlarged details of 1/4 party of (c) red box, (e) denoised image of wavelet denoising [88, 91], (f) partial enlarged details of 1/4 party of (e) red box, (g) denoised image of NLMF denoising [94, 95], (h) partial enlarged details, 1/4 party of (g) red box, (i) denoised image of FDM denoising (𝑣1=1.75, 𝑣2=2.75, 𝑣3=1.05, Δ𝑡=5×10−6), (j) partial enlarged details of 1/4 party of (i) red box, (k) denoised image of FDM denoising (𝑣1=2.25, 𝑣2=2.5, 𝑣3=1.25, Δ𝑡=5×10−6), and (l) partial enlarged details of 1/4 party of (k) red box.

From the viewpoint of visual effects, we know the following from Figure 6, when noise is very strong, especially when MRI is drawn completely. First, the denoising capabilities of bilateral filtering and wavelet denoising are worse than the other methods. We can see indistinctly that the contour and the texture details of inner organ can hardly be recognized from Figures 6(c)–6(f). Second, the denoising capability of NLMF is better because we see that the contour is clearer, but the edge and texture details are still blurry as in Figures 6(g) and 6(h). Finally, the denoising capabilities of FDM denoising are the best because we can see from Figures 6(i)–6(l) that the contour is clearest and the edge and texture details can be recognized also.

For quantitative analysis, we measure the denoising effects in terms of the PSNR, the correlation coefficients between the noisy image or the denoised image and the original clean MRI [94], and the average gray level concurrence matrix, as seen in Table 3.

Table 3 

Denoising effects of abdomen MRI of texture-rich internal organ, when Gaussian noise is very strong.

From Table 3, we know that the denoising capabilities of the above algorithms are as follows, when noise is very strong, especially when MRI is completely drowned by the noise. First, the denoising capabilities of bilateral filtering denoising and wavelet denoising are rather poor, and their PSNR and correlation coefficients are relatively small. This shows that the noise cannot be clearly denoised and the similarity between the denoised image and the original clean MRI is small. Also, the contrast of average gray level concurrence matrix of wavelet algorithm is the greatest, which indicates greater pixels with greater contrast but we cannot say that the texture depth is deeper and that the visual effects are clearer because the denoising is incomplete. Second, the denoising capabilities of NLMF denoising and FDM denoising are better and their PSNR and correlation coefficients are comparatively higher, which shows that the noise is denoised completely and the similarity between denoised image and original clean MRI is great. The PSNR and the correlation coefficients of FDM denoising is the highest and its contrast of average gray level concurrence matrix is the greatest. Also, its correlation, energy, and homogeneity are smaller. We therefore can conclude that FDM denoising is the best model of the above models.

When the noise is very strong and especially when original clean signal is completely drowned in noises, we take a texture-rich meteorite crater remote sensing image of moon satellite to perform further comparison experiments using the above well-performed algorithms, including bilateral filtering denoising, wavelet denoising, NLMF denoising, and FDM denosing to test the denoising capability of FDM for robust noise. The added noise is the composite noise combined by white Gaussian noise, salt and pepper noise, and speckle noise. Also, the numerical iterative process will stop at the point where peak signal-to-noise ratio is the highest, as shown in Figure 7.

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Figure 7 

Denoising effects of meteorite crater texture-rich moon satellite remote sensing image, when white Gaussian noise, salt, and pepper noise and speckle noise are added together. (a) Original clean image, (b) noisy image (adds white Gaussian noise (its standard variance is 0.02), salt and pepper noise (its noise density is 0.2), and speckle noise (its standard variance is 0.1) to the original clean image, PSNR=8.8564), (c) denoised image of bilateral filtering denoising [88, 89, 93], (d) partial enlarged details of 1/4 party of (c) red box, (e) denoised image of wavelet denoising [92, 93], (f) partial enlarged details of 1/4 party of (e) red box, (g) denoised image of NLMF denoising [94, 95], (h) partial enlarged details of 1/4 party of (g) red box, (i) denoised image of FDM denoising (𝑣1=1.75, 𝑣2=2.75, 𝑣3=1.05, and Δ𝑡=10−10), (j) partial enlarged details of 1/4 party of (i) red box, (k) denoised image of FDM denoising (𝑣1=2.25, 𝑣2=2.5, 𝑣3=1.25, and Δ𝑡=10−10), and (l) Partial enlarged details of 1/4 party of (k) red box.

From view of visual effects, we know from Figure 7 the following; when the composite noise is added by white Gaussian noise, salt and pepper noise, and speckle together, especially when texture-rich meteorite crater remote sensing image of moon satellite is completely drown in noises. First, the denoising capabilities of bilateral filtering and wavelet and NLMF denoising are comparatively worse. We can see from Figures 7(c)–7(h) that the contour may be seen indistinctly and the edge and texture details can hardly be recognized. Second, the denoising capability of FDM denoising is the best. We can see from Figures 7(i)–7(l) that the contour is not only the clearest and also the edge and texture details can be clearly recognized.

For quantitative analysis, we measure the denoising effects in terms of the PSNR, correlation coefficients between noisy image or denoised image and original clean remote sensing image [94], and average gray level concurrence matrix, as seen in Table 4.

Table 4 

Denoising effects on texture-rich meteorite crater remote sensing image of moon satellite, when white Gaussian noise, salt and pepper noise and speckle noise are added together.

We know denoising capabilities of the above algorithms from Table 4 are as follows, when composite noise added by white Gaussian noise, salt and pepper noise, and speckle noise, especially when the original image is completely drowned in noises. First, the denoising capabilities of bilateral filtering, wavelet, and NLMF denoising are poor, and their PSNR and correlation coefficients are relatively small. This shows that the added noise cannot be denoised completely and the similarities between denoised image and original clean MRI are small. Also, the contrasts of average gray level concurrence matrices of NLMF algorithm are the greatest, which indicates that there are more pixels with great contrast, but we cannot say that the texture depth is deeper and the visual effect is clearer because the denoising is uncompleted. Second, the denoising capability of FDM denoising is the best. Its PSNR and correlation coefficients are comparatively high, which shows that the noise is completely denoised and the similarity between denoised image and original clean MRI is the greatest. Also, the PSNR and the correlation coefficients of FDM denoising are the highest. The contrast of average gray level concurrence matrix of FDM denoising is high, while its correlation, energy, and homogeneity are small. We therefore can say that FDM denoising is the best denoising model of the above models.

By comparing the visual effects of Figures 5, 6, and 7 and quantitative analysis in Figures 1, 2, and 3, we find the following. First, the denoising capability of FDM is the best, irrespective of the strength and type of added noise. Its PSNR and correlation coefficients are the highest and the denoising is relatively completed. The similarity between the denoised image and the original image is the highest. Second, FDM denoising can nonlinearly preserve the low-frequency contour feature in the smooth area to the furthest degree, nonlinearly enhance high-frequency edge information in those areas where gray level changes obviously, nonlinearly enhance the high-frequency edge and texture details, and also nonlinearly maintain the fractional multiscale denoising capability in those areas where gray level does not change obviously.

5. Conclusions

We propose the introduction of a new mathematical method—fractional calculus to the field of image processing and the implementation of fractional partial differential equation. First, it presents three common-used definitions of Grümwald-Letnikov, Riemann-Liouville, and Caputo, which is the premise of fractional partial differential equation-based denoising model. Second, we derive fractional Green’s formula for two-dimensional image processing by extending classical integer order to fractional order and then derive fractional Euler-Lagrange formula. Based on the above fractional formulas, a fractional partial differential equation is proposed. Finally, we show the denoising capability of the proposed model by comparing Gaussian denoising, fourth-order TV denoising, bilateral filtering denoising, contourlet denoising, wavelet Denoising, nonlocal means noise filtering (NLMF) denoising, and fractional-order anisotropic diffusion denoising. The experimental results prove that FDM can preserve the low-frequency contour feature in the smooth area, nonlinearly maintain the high-frequency edge and texture details in those areas where gray level change greatly and also nonlinearly retain the texture details in those areas where gray level has little changed. As for texture-rich images, the denoising capability of the proposed FDM denoising model is obviously superior to traditional integral based algorithm when denoising.

Acknowledgment

The work is supported by Foundation Franco-Chinoise Pour La Science Et Ses Applications (FFCSA), China National Nature Science Foundation (Grants no. 60972131 and no. 61201438), Returned Overseas Chinese Scholars, State Education Ministry of China (Grant no. 20111139), Sichuan Science and Technology Support Project of China (Grant nos. 2011GZ0201 and 2013SZ0071), Open fund of State key Laboratory of Networking and Switching Technology of Beijing University of Posts and Telecommunications (SKLNST-2010-1-03), and Project of Sichuan Province of China under Grant 2011GZ0201and Grant 2013SZ0071, the Soft Science Project of Sichuan Province of China under Grant 2013ZR0010, and the initial funding for young teachers of Sichuan University under Grant 2011SCU11.