Noise detection and image denoising based on fractional calculus

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;20:39]

Chaos, Solitons and Fractals xxx (xxxx) xxx

Contents lists available at ScienceDirect

Chaos, Solitons and Fractals Nonlinear Science, and Nonequilibrium and Complex Phenomena journal homepage: www.elsevier.com/locate/chaos

Noise detection and image denoising based on fractional calculusR Qi Wang, Jing Ma∗, Siyuan Yu, Liying Tan National Key Laboratory of Tunable Laser Technology, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

a r t i c l e

i n f o

Article history: Received 19 July 2019 Revised 15 September 2019 Accepted 25 September 2019 Available online xxx MSC: 46T99 68U10 65Y20 Keywords: Noise detection Image denoising Fractional calculus Algorithm model

a b s t r a c t Fractional-order integral can weaken high-frequency signal and greatly preserve low-frequency signal nonlinearly, that is, the high-frequency noise can be removed while retaining the information of lowfrequency image itself, thus the fractional calculus can achieve good a denoising effect and the application of fractional calculus theory in the digital image processing field has also been favored by more and more scholars. On the basis of summarizing and analyzing previous research works, this paper proposes a new method for detecting noise points in images using fractional differential gradient and an improved image denoising algorithm based on fractional integration. The noise detection method determines the noise position by the fractional differential gradient, and achieves to detect the noise, snowflake and stripe anomaly though utilizing the neighborhood information feature of the image and the contour and direction distribution of various noise anomalies in spatial domain; the image denoising algorithm firstly regards the appearance of noise points in image as a small probability event and divides it, then applies the fractional calculus to subtly transform and adjust signal filter and meanwhile utilizes iterative idea to control image denoising effect for accurately distinguishing between noise and high-frequency signals, so that the original image feature information is more preserved while the image is denoised. The simulation results show that this algorithm model can effectively remove the noise while maintaining the details of image edge and texture, and have the characteristics of simple algorithm and good stability. The model can effectively remove the noise and meanwhile preserve the details of image edges and textures, and has the characteristics of simple algorithm and high stability. The study results of this paper provide a reference for further research on the noise detection and image denoising based on fractional calculus. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Images are an important channel for humans to acquire outside information and play an important role in human activities. However, in the process of digital image acquisition, conversion and transmission, some isolated pixel points with random positions and random values, i.e. image noise, appear in the image due to the factors of the imaging device itself or the influence of the external environment. No matter whether it is improving the imaging device or reducing the environmental interference, the noise can not be avoided. These noises not only affect the visual effect of the image, but also may mask important feature information in the image, which will bring difficulties to the subsequent processing of the image. Therefore, image denoising is one of the important contents of digital image processing research [1]. Fractional

R

Fully documented templates are available in the elsarticle package on CTAN. Corresponding author. E-mail addresses: [email protected] (Q. Wang), [email protected] (J. Ma), [email protected] (S. Yu), [email protected] (L. Tan). ∗

calculus and integer-order calculus theory are born almost at the same time and are generally considered that fractional calculus is a generalization of integer-order calculus; fractional-order calculus theory is one of the mathematical foundations of fractal theory, and its application in the field of digital image processing has been accepted by many scholars. The favor, based on fractional integral denoising is one of the important branches [2]. Mathematically, for the nature of the texture structure, the texture is information with fractional derivative characteristics, and the integer order differential operator is not suitable for processing such information with weak derivatives. At present, most of the image processing methods based on partial differential equation are based on integerorder derivatives. In recent years, image processing methods using fractional derivatives have gradually attracted attention in the field of image processing [3]. The image denoising method can be divided into two categories: spatial domain denoising method and transform domain denoising method. The spatial domain denoising method directly operates the image in the spatial domain, and the early classic methods include noise threshold method, neighborhood averaging

https://doi.org/10.1016/j.chaos.2019.109463 0960-0779/© 2019 Elsevier Ltd. All rights reserved.

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS 2

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

method, weighted average, median filtering and Wiener filtering; the transform domain image denoising method is more classic, and its transform tools include Fourier transform, Laplacian transform, wavelet transform and multi-scale transform; however the main problem of such image denoising is the contradiction between noise removal and image edge preservation [4]. In recent years, many scholars have successfully applied fractional differential theory to solve many problems such as image signal processing, edge detection, image enhancement: fractional differential operators are shown to have higher signal-to-noise ratio when compared with integer differential operators by analyzing the differential characteristics of signals; the fractional spline wavelet is applied to the singularity check and image fusion of image texture respectively by using the memory characteristics of fractional calculus, which achieves better simulation effect than integer calculus [5]; the frequency filter of fractional differential gradient operator is derived by utilizing the definition of fractional calculus; a differential filter is designed by adjusting the differential order and the problem of edge drift caused by traditional operator edge detection can be solved with using the basic theory of fractional calculus; and a dynamic simulation model of fractional differential chaotic system is proposed, which can dynamically observe the evolution law of system variables [6]. On the basis of summarizing and analyzing previous research works, this paper proposes a new method for detecting noise points in images using fractional differential gradient and an improved image denoising algorithm based on fractional integration. The noise detection method determines the noise position by the fractional differential gradient, and achieves to detect the noise, snowflake and stripe anomaly though utilizing the neighborhood information feature of the image and the contour and direction distribution of various noise anomalies in spatial domain; the image denoising algorithm firstly regards the appearance of noise points in image as a small probability event and divides it, then applies the fractional calculus to subtly transform and adjust signal filter and meanwhile utilizes iterative idea to control image denoising effect for accurately distinguishing between noise and high-frequency signals, so that the original image feature information is more preserved while the image is denoised. The study results of this paper provide a reference for further research on the noise detection and image denoising based on fractional calculus. The detailed chapters are organized as follows: Section 2 introduces the research methods and principles; Section 3 proposes the new method for noise detection based on fractional calculus; Section 4 describes the improved image denoising algorithm based on fractional calculus; Section 5 performs a numerical simulation experiment analyzes its results; and Section 6 is conclusion. 2. Methods and principles 2.1. Fractional calculus and gradient operator The three main fractional calculus definitions are: Grmwald– Letnikov (G–L) definition, Riemann–Liouville (R–L) definition and Capotu definition. G–L definition is derived from the differential definition of integer-order differential, while R–L and Capotu definitions are derived from the integral-order Cauchy formula. The Gamma function  (x) is used to generalize the calculus order from integer to fraction, and [·] means the integer part; the variation interval of f(x) is [a, b], and v-order fractional calculus defined by G-L: ν f (x ) = lim h−ν

aDb

h→0

[(b−a )/h]

 i=0

(−1 ) j

(ν + 1 ) f (x − jh ) j!(ν − j + 1 )

(1)

According to the derivation, when ν > 0, aDνb represents a fractional differential operator representing a ν -order; when ν ¡0, aDνb

represents a fractional-order integral operator. The ν -order fractional integrals defined by R-L are:

R ν a Dt

f (x ) =

⎧ n d f (x ) ⎪ ⎨ n ν=n∈N n ⎪ ⎩ dn

dx

1 xx (n − ν )



t a

f (y ) dy 0 ≤ n − 1 < ν < n (x − y )ν −n+1 (2)

Caputo definition of fractional calculus:

R ν aDt

f (x ) =

⎧ n d f (x ) ⎪ ⎨ n ν=n∈N n ⎪ ⎩ dn

dx

1 xx (n − ν )



a

t

f n (y ) dy (x − y )ν −n+1

0≤n−1<ν
Under the same conditions, the Caputo definition of fractional calculus is equivalent to the G–L definition of fractional calculus. When the fractional calculus order v is a positive integer and a negative real number, the Caputo definition of the fractional calculus and the R–L definition of the fractional calculus satisfy the following relationship: R ν aDt

f (x ) =

n−1 (k )  f (a )(x − a )k−ν C ν + aDt f (x ) (k − p + 1 )

(4)

k=0

The fractional-order partial differential equation is obtained by substituting the partial derivative of time or space in the traditional integer-order partial differential equation with a fractional partial derivative; and its basic form is:

∂ α f (x, y, t ) ∂ β f (x, y ) ∂ β f (x, y ) = + ∂ αt ∂βx ∂βy

(5)

According to the difference of the partial differential order α and β , the fractional partial differential equation includes three forms of time equation, space equation and space-time equation. 2.2. Effect of fractional calculus on the signal Let u(x, y) be the original clear image, and u0 (x, y) be the image contaminated by noise. That is, u0 (x, y ) = u(x, y ) + n(x, y ), n(x, y ) is random noise with zero mean and variance σ 2 , then the denoising model can be expressed as [7]:

min J (u ) = 



|∇ u(x, y )|dxdy +

λ 2

 

[u(x, y ) − u (x, y )]2 dxdy (6)

The first item on the right side is the regular term of the image, which can suppress the noise during the minimization process; the second term is the fidelity term, which mainly plays the role of maintaining image edge features and reducing image distortion. ∇ = ( ∂∂x , ∂∂y ) is the gradient operator;  is the domain of the image, and the pixel is (x, y) ∈ . For the known arbitrary square integrable energy signal f(t) ∈ L2 (R), the basic theory of signal processing shows that the  Fourier transform is f (ω ) = R f (t )eiωt dt . Assume that the nth derivative of the signal f(t) is fn (t)(n ∈ Z), which can be obtained according to the nature of the Fourier transform:

(Dk fˆ(ω )) = (iω )k fˆ(ω ) = dˆk ω fˆ(ω )

(7)

Thus, the information is followed by the fractional ν derivative of f(x) as fν (t )(ν ∈ R+ ), which is also obtained according to the nature of the Fourier transform:

f ν (t ) ⇔ ( jω )ν F ( jω ) = d ( jω )F ( jω )

(8)

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

3

Fig. 1. Fractional differential (a) and integral (b) frequency response plots for each order.

In the formula,

⎧ ν iθ ν ( ω ) ⎪ ⎨dν (ˆω ) = (iω ) = α ν (ˆω )e n dν (ˆω ) = |ω| u ⎪ ⎩θ ν (ω ) = νπ sgn(ω )

(9)

2

Thus, when ν ∈ R+ , according to the fractional-order operator theory, that is, D − 1 = I, and ν  = −ν, the Fourier transform form of the fractional-order integral operator can be obtained as:

f ν (t ) ⇔ ( jω )ν F ( jω ) = d ( jω )F ( jω ) = θ ν (ω ) =

νπ 2

sgn(ω )F ( jω ) (10)

It can be seen from Fig. 1(a) that the fractional differential operation has an enhancement effect on the high frequency portion of the signal, and as the fractional differential order and frequency increase, the amplitude of the enhancement increases nonlinearly. At the same time, the fractional differential has a certain degree of nonlinear reservation on the low frequency part of the signal. It can be seen from Fig. 1(b) that the fractional integral operation has an attenuation effect on the high frequency portion of the signal, and the attenuation amplitude gradually increases as the fractional integration order and frequency increase; at the same time, the fractional integration enhances the low frequency part of the signal to a large extent [8]. Therefore, the fractional differential has a certain enhancement and retention on the edges and texture of the image, while the fractional integration has a certain denoising effect on the image.

Define G(i, j) as the gradient value of the current voxel point, Nε G(i, j) is the gradient value of the adjacent 18 voxels centered on the current voxel, ε =0,1,...,18. First define the gradient discriminant factor J(i, j) as:

(11)

It can be seen from the above equation that the gradient discrimination factor J(i, j) is the minimum value of the gradient modulus of the current voxel and the gradient of the 18 adjacent points in the three-dimensional discrete template. Then define an adaptive fractional order expression as:



ν=

G(i, j ) max G(i, j ) + α 0

J (i, j ) ≥ T J (i, j ) < T

I (i, j, k ) =

18 1  V − Nε ((i, j, k )) 18

(13)

ε=1

where I(i, j, k) is the new voxel after denoising by adaptive fractional integration. According to this rule, the three-dimensional fractional discrete template is used to traverse the entire threedimensional noisy image, that is, an adaptive fractional integral denoising is completed. The pure anisotropic diffusion model improves the recovery effect of the basic model to some extent, and can better maintain the boundary in image denoising, but it still spreads excessively in the flat region, resulting in a staircase effect. To solve this problem, fourth-order partial differential equation is proposed and its energy functional is:

E (u ) =

 K

(|∇ 2 u| + λ(u − u0 )2 /2 )dxdy

(14)

Reintroducing the time variable t and using the gradient descent method:

∂u = −∇ [ f (|∇ 2 u| )∇ 2 u] + λ(u0 − u ) = −[|u|u] + λ(u0 − u ) ∂t

2.3. Template construction of fractional calculus operator

J (i, j ) = min ||G(i, j )| − |Nε G(i, j )||

the gradient value of the surrounding voxel change greatly, that is, when the gradient discrimination factor is greater than a given threshold, the voxel is determined to be high frequency noise and adaptive denoising is performed. The new I(I, j, k) filtered by the current voxel point is:

(12)

where α is a positive adjustment factor; max G(i, j) is the gradient maximum of 18 voxel points; T is the threshold chosen for different images. When the gradient value of the processed voxel and

(15) 2 2 where  is a Laplacian, u = ∂∂ x2u + ∂∂ y2u ; f is a non-negative function, and strictly monotonically decreasing in the form of f (s ) = 1 , and k is the gradient threshold. 1+(s/k )2

As shown in Fig. 2(a)–(h) respectively show the mask +W in the direction of the positive X-axis, the mask -W in the direction of the negative X-axis, and the direction along the positive direction of the Y-axis. Mask +W f , mask −W f in the direction opposite ◦ to the negative direction of the Y axis, mask W f45 in the direction ◦ of 45 counterclockwise with respect to the positive axis of the X axis, and 225◦ in the counterclockwise direction from the positive ◦ ◦ axis of the X axis W f225 , a direction mask W f135 that is 135◦ counterclockwise with respect to the X-axis positive axis, and a direc◦ tion mask W f315 that is 315◦ counterclockwise with respect to the X-axis positive axis [9].

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS 4

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

Fig. 2. Fractional differential (a) and integral (b) frequency response plots for each order.

In practice, usually let h=1, then the fractional derivative of the one-dimensional signal J (t )(t = 0, 1, . . . , n ) from the filter point of view is:

3. Noise detection based on fractional calculus 3.1. Regularization model of fractional calculus According to the R–L definition of the fractional integral, the ν order fractional integral of the signal s(x) over the duration [0, x] is:

J ν s(x )|R−L =

1 (−ν )

1 = (−ν )



x

0



x

0

s (ξ ) dξ (x − ξ )ν +1 s (x − ξ ) dξ ν < 0 ν +1

(16)

ξ

In order to numerically calculate the above equation, the continuous integral is converted into a discrete sum, that is, the integral interval [0, x] is N-divided. When N is sufficiently large, the ν -order fractional integral of s(x) can be approximated as:

J ν s(x )|R−L ∼ =

N−1   (kx+x )/N s(x − ξ ) 1 dξ (−ν ) ξ ν +1 kx/N

ν<0

(17)

k=0

In the image denoising model based on partial differential equations, a noise image I(x, y), x, y ∈  is known, assuming that at time t, the noise image is I(x, y, t), if So that the energy functional E(· , t) rapidly reaches the minimum value with time t, then the numerical calculation method of the image denoising model can be obtained by using the gradient descent flow:

⎧ ∂I ⎪ ⎨ =0 ∂t ⎪ ⎩ ∂ F − d ( f rac∂ F ∂ Ix ) − d ( f rac∂ F ∂ Iy ) ∂ I dx dy ∂I d d ∂F = 0 ( f rac∂ F ∂ Ix ) + ( f rac∂ F ∂ Iy ) − ∂t dx dy ∂I

(18)

(19)

In order to be able to understand the basic concept of the fractional step-down flow, it is uncommon, assuming energy function:

E = a(χ − χνop=1 )2 + Eνmin =1

(20)

where a represents the rate of change of energy; represents the optimal stagnation of order ν =1; represents the minimum of the energy functional of the optimal stagnation pair.

f α (t ) = 0Dtα f (t ) =

τ 

ωαj f (t − j ) t = 0, 1, . . . , n

(21)

j=0

In the two-dimensional gray image, the edge and the noise are locally discontinuous points, and the gray value of the neighboring pixels corresponding to the noise and the edge changes drastically. The so-called image edge refers to the set of pixels whose gray value or brightness value of the neighborhood pixel has a step change or a roof change and it exists between the target and the background, between the target and the target, between the region and the region, and between the pixel and the image. The edge has order and directivity, and has a high correlation with the pixels of the neighborhood, while the noise signal has randomness and has no correlation with the pixels of the neighborhood. In the signal processing process, the correlation of neighboring pixels can be used to cancel the influence of noise and strengthen the edge signal and the image that needs to be processed inevitably has noise. The noise tends to blur the edge of the image, which leads to some details that cannot be detected; the edge information is not continuous enough. However, it can be seen from the experimental results that the edge information is basically increased with the increase of the order. It does not change, but the noise increases, indicating that noise is well suppressed when taking lower orders. The differential operator constructed by the description can effectively extract the edge information, and can obtain the continuous edge information by changing the order of the fractional differential to satisfy the different requirements of image processing. The method of final estimation is similar to the method of preliminary estimation; the difference is that the preliminary estimation result map obtained by the first step estimation is used as the basis of block matching since the preliminary estimate is smoothed at this time, the difference in block matching will be smaller. At this time, a second three-dimensional matrix formed on the basis of preliminary estimation results is obtained. The two obtained matrices are all three-dimensionally transformed, and the threedimensional matrix formed by the noise image is subjected to coefficient scaling by Wiener filtering, and the coefficient is obtained by the value of the matrix estimated by the foundation and the

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

5

noise intensity. After filtering, the matrix of the noise image is inversely transformed back to the original image position by inverse transformation to form a final estimate [10]. The image texture detail feature generally appears as the relative change of the gray value between the pixels in the domain. This relative change is reflected as the relative change of the instantaneous equilibrium state of each point on the envelope curve of the vertical and horizontal projection of the image gray value. The gray value of each point on the envelope curve is the instantaneous amplitude, which is the zero-order metric of its instantaneous equilibrium state; the fractional derivative of the corresponding points on the envelope curve is the fractional measure of its instantaneous equilibrium state. In addition to the instantaneous amplitude, it is necessary to investigate the relative change speed and direction fractional metric of the gray value. The numerical operation rule of the fractional differential mask is to use the fractional differential mask convolution scheme to realize the spatial differential filtering of order differential digital image score. The mechanism of this spatial filtering is to move the mask point by point in the digital image to be processed. 3.2. Noise detection algorithm based on fractional calculus The total variation regularization model models the image denoising problem as an optimization problem, and is expressed as:

min u∈X

λ u 1 + u − g 22

 (22)

2

where X represents a finite dimensional vector space, · ν denotes a ν -norm, ∇ denotes a gradient operator, u denotes a denoised image, g denotes an observed image, and λ denotes a regularization parameter. The first item in the model is called a regular term and acts to suppress noise during the optimization process and the second term is called the fidelity term. The main function is to maintain the similarity between the image after denoising and the observed image, so as to maintain the edge features of the image. The regularization parameter λ is used to balance the role of the regular term with the fidelity term. Let the original variable x be fixed and the dual variable y be derived. The pre-solution of the variable y is:

y = (I + ∇ F ∗ )−1 (y + Ax )

(23)

F ∗

where and G correspond to the gradients of the functions F∗ and G, respectively. In order to effectively remove Gaussian noise and salt and pepper noise, and to better maintain the details of the edge texture features of the image, inspired by the model, the following denoising model is proposed [11]:



ut − |∇ε

u|α ∇

ε

∇ε u |∇ε u|1+ω¯



= β ( u0 − u )

α , β , ω¯ ≥ 0, 1 ≤ ε ≤ 2 (24)

where u is the real image; u0 is the image after the noise is added; ut is the image after the tth iteration; |∇ u| and ∇ u are mathematically the same, both are gradient modes. ∇ ε u and |∇ ε u| is a fractional step operator and a fractional step mode, respectively. Given a grid point P(i, j), let (k, l) be one of the points in the 8-neighborhood of P(i, j): ) D((k,l u= i, j )



|ui, j − uk,l | ( i − k )2 + ( j − l )2

(25)

The difference format for calculating ∇ u at P(i, j) is:

|∇ u|i, j =

2



D((ki,1j,l) 1 ) u + D((ki,2j,l) 2 ) u

2

(26)

Fig. 3. Schematic diagram of noise, snowflake and stripe detection.

The difference format combines the two smallest difference values to approximate the gradient value. Therefore, it is not the exact value of the gradient mode in the mathematical sense. That is, when the spatial step size of the mesh tends to zero, it does not necessarily converge toward this point. Gradient mode, however, from the numerical experimental results, it can effectively protect the image edge texture details and other information. Fig. 3 shows schematic diagram of noise, snowflake and stripe detection. The regions, texture regions, textures, and edge regions in which the pixels are invariant in the image correspond to the low frequency, intermediate frequency, and high frequency of the signal, respectively. Through the graphical analysis generated by the experiment, it can be seen that when the differential order is reduced to 0.1, the differential operation does not increase the high frequency too much, and the low frequency does not excessively suppress. It can be seen that at the 0.1th order, the differential has no significant effect on the enhancement of the edge contour of the image and the enhancement of the image stripe and the texture information of the smoothed area has a large influence on the detection of the edge of the image [12]. Therefore, it can be said that the image edge enhancement algorithm based on integer differential is not an effective method for detecting texture details in a smooth region. In the image processed by fractional differential, the texture details of the smooth region are not weakened, and some nonlinearity is retained. Based on the order continuity of the fractional differential, by adjusting the value of the fractional order, the best image edge information can be obtained, so the fractional differential is more favorable to the extraction of the edge information of the image than the integer derivative. Edges and noise have something in common, and they all have a lot of high-frequency components. The reason is that the edge and noise are discontinuous grayscale image features and local abrupt changes, and corresponding pixel grayscales appear. According to the amplitude-frequency characteristics of the fractional differential, when the fractional differential operator with order of 1–2 is selected, the high-frequency features of the image can be greatly enhanced, but many internal texture details

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS 6

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

of the image that are not related to the target contour may be obtained. When the fractional differential operator is between -1-0 is selected, it is possible to suppress the image noise and the local detail interference in the image, and obtain the approximate target contour information of the image. More precisely, the fractional order (1 < n < 2) the differential operator has the function of increasing the high-frequency component of the signal and increases nonlinearly with the increase of the differential order, and weakens the low-frequency signal. For the fractional differential (0 < n < 1), the high-frequency signal is improved by less than the 1- and 2-order differentials, but it has also been improved enough; at the same time, the signal is also greatly enhanced, but the low-frequency signal is not greatly attenuated and is nonlinearly preserved. It can be seen that the fractional differential is nonlinearly enhanced and preserved for the intermediate frequency component and the low frequency signal while improving the high frequency component of the image [13]. 4. Image denoising based on fractional calculus 4.1. Theoretical analysis of fractional calculus denoising For a noisy image, there are noises of various intensities; if only the same fractional order is used to process the noise, it is difficult to achieve better denoising effect. Therefore, this paper proposes a global adaptive fractional-order integral image denoising algorithm, which performs fractional-order integral operations of different orders for each pixel. Let the average value of the gradient magnitudes in the eight directions of each pixel in the image f(i, j) be M(i, j), normalize it to find the integral order corresponding to the pixel, and take the maximum value of M(i, j) as Y and the minimum value as X. After normalizing the gradient magnitude of the pixel, the dynamic fractional order is obtained:

ν = (−1 ) ×

M (i, j ) − X Y − M (i, j )

(27)

Therefore, it can be realized that when the gradient mean value is large, there is a small negative order, and the fractional integral of the order has a large attenuation effect on the noise; for the gradient amplitudes of medium and small with the corresponding size of the integration order, has a certain enhancement and retention of the image texture. The pixel with the larger radius of the neighborhood has less correlation with the pixel at the center point. Pixels farther from the center point will suppress the change of the center point, while pixels near the center point will increase the trend of the center point. Therefore, the improvement method is obviously better than other methods. To quantitatively evaluate the image enhancement effect, define the image sharpness as [14]: De f inition m2 ,n2 =



i, j=m1 ,n1

[u(i, j ) − u(i − 1, j )]2 + [u(i, j ) − u(i, j − 1 )]2

(m2 − m1 )(n2 − n1 )

(28)

In the formula, u is the gray value of the image enhanced by various algorithms; m1 , n1 , m2 , and n2 are window coordinates. The larger the image sharpness value, the better the image enhancement effect. In order to further verify the validity and advancement of the algorithm, two common objective indicators, Relative Error (ReErr) and Peak Signal-to-Noise Ratio (PSNR), were used to evaluate the final effect of the denoising method:





0≤i≤M



ReErr =

RSNR = 10lg

0≤ j≤N [

(u(i, j ) − f (i, j ))]2 2 0≤ j≤N u (i, j )



0≤i≤M

2252 MSE



1/2

(29) (30)

MSE =

M−1 N−1 1  [ f (i, j ) − u(i, j )]2 MN

(31)

i=0 j=0

where u is the original clear image, f is the image to be evaluated; it can be seen that the smaller the ReErr, the more similar the two images are, the better the denoising effect is; on the contrary, the larger the PSNR, the better the denoising effect. The edge contour of an object in an image contains most of the information of the image; therefore, image edge detection is one of the important contents of digital image processing technology. The relationship between fractional order integral order ν and RSNR, original dual interval, contract ratio and entropy are shown in Fig. 4. The principle of the image edge detection algorithm is to determine whether a pixel point is on the edge contour of an object by detecting the gray value of the pixel point and its neighboring pixel points. If a pixel is on the edge contour of the object, then the gray value of the pixel of the neighborhood of the pixel is relatively large. Therefore, it can be defined that the region where the gray value of the pixel changes sharply is the edge contour portion of the object in the image, and usually exhibits a high frequency characteristic. The noise portion in the image is also a region where the gray value of the pixel point changes abruptly and thus is also a high frequency component of the image. The noise in the image and the edge contour of the object are both regions where the gray value of the neighboring pixel points changes greatly, but the difference between the two is that the appearance of image noise is random, the gray value of the noise and the neighboring pixel points. The gray value is irrelevant, and the gray value of the pixel on the edge contour of the object is related to the gray value of the neighboring pixel. In addition, the edge contour of the object is also characterized by order, orientation and structure and inside the relatively smooth object in the image, the gray value of the pixel does not change much, and generally exhibits a low frequency characteristic. In the texture detail part of the image, the gray value of the pixel has a certain change, which is the medium frequency component of the image [15]. The fractional-order model can effectively suppress the staircase effect, that is, the segmentation smoothing phenomenon, compared with the first-order model. Fig. 5 shows the Comparison of image Lena denoising effects under different fractional integral orders. Compared with the second-order model, noise can be removed more effectively; as the fractional order increases, the detail protection of the image can be effectively enhanced, but more noise components remain. Considering that the shortest distance at which the gray value change of the digital image occurs is between two adjacent pixels, the conventional fractional differential template for edge detection uses a fixed step size, which is, measured in units of pixels. Although this has obtained a good edge extraction effect, it also has the following disadvantages: the characteristics of the high autocorrelation of the gray value of the digital image are not fully utilized, and the calculation result shows a certain deviation; if the image is interfered by noise, the differential operation is directly performed and the error is large, which will inevitably greatly reduce the edge extraction effect. The use of fractional differentials with non-integer steps can effectively overcome the above two shortcomings [16]. The data processing method in the theory of calculus has clearly pointed out that in the practical work of image processing, the staff can calculate the signal energy index of the arbitrary image processing interface, and can convert the calculus according to the basic operation theory of signal processing. At the same time of transforming properties, realtime control of the corresponding signal transmission content is realized. According to the characteristics of the image and the degree of attention in the image processing, the high-frequency components

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

7

Fig. 4. Relationship between fractional order integral order and RSNR(a), original dual interval(b), contract ratio(c) and entropy(d).

of the signal, the nonlinear enhanced signal component, and the low-frequency component of the reserved signal can be effectively improved by selecting the fractional differential of different orders. The frequency component corresponds to the edge and noise of the image, the intermediate frequency component corresponds to the texture detail of the image, and the low frequency component corresponds to the smooth region of the image. Therefore, when the image edge detection is performed by the fractional differential, the high frequency characteristic of the image can be better extracted and the edge information can also extract the complex texture edge details with low frequency characteristics in the image more effectively [17]. By applying the calculation mode of calculus real-time conversion, the staff can better grasp the amplitude and frequency characteristics of image processing object in signal transmission. The calculus denominator operators of different existence forms can affect the transmission of image processing information commands and play a signal strengthening role. Under normal circumstances, the increase and decrease of the order quantity in the calculus accounting process can affect the frequency of the image processing signal transmission. When the fractional order parameter is controlled above the reference value, the corresponding algorithm equation does not have any form of influence on the image processing signal. 4.2. Image denoising model based on fractional calculus The predecessors derived a non-convex multiplicative noise removal model according to the assumption of Gamma distribution and the modeling method of maximum posterior probability:

 min

u∈S 

|∇ u|du + λ



 

logu +



f du u

(32)

where f is the original noisy image; u is the approximate image to be solved; S() = uBV (), U > 0, BV () is the bounded variation function defined on the region  space. In order to remove the noise and better maintain the edge and texture details, a fractional derivative operator and a first derivative operator are combined to obtain a new fractional variation model:

min α1 u∈S

 

|∇ u|du + α2

 

|∇ ν u|duλ

 

logu +



f du u

(33)

where |∇ u| = u2x + u2y , |∇ ν u| = (uνx )2 + (uνy )2 ; uνx and uνy are the v-order partial derivatives of u with respect to x and y, respectively; α 1 and α 2 are parameters between the equilibrium model regular term and the fractional model regular term. In particular, when α1 = 1 and α2 = 0, the above equation becomes a classical model; when α1 = 0 and α2 = 1, it becomes a multiplicative noise pattern denoising model. The edge of the object is continuous and there must be more than one edge point around the pixel on one edge, and the noise is random. The probability of a noise point with little difference in value is small, and the discrimination function of the noise edge is constructed accordingly:



E=

0 1

min ||∇ Ix,y | − |∇neighbour (Ix,y )|| ≤ T min ||∇ Ix,y | − |∇neighbour (Ix,y )|| > T

(34)

In the formula, the gradient modulus of the central pixel point Q is represented; the gradient modulus of the pixel of the eight fields is represented, and if the minimum value of the absolute values of the two is less than a certain threshold T, it means that the Q point is an edge point, and vice versa, it is noise point. It can be seen from Fig. 6 that the differential operation has the effect of increasing the high frequency of the signal, and it

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

JID: CHAOS 8

ARTICLE IN PRESS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

Fig. 5. Comparison of image Lena denoising effects under different fractional integral orders.

Fig. 6. Comparison of convergence under different fractional orders.

increases nonlinearly with the increase of frequency and differential order, and weakens the very low frequency of the signal. It is obvious from the figure. The 2-order differential is much higher than the 1st order differential of the high frequency component of the signal, and the weakening of the very low frequency signal is also stronger than the 1st order differential. Corresponding to ω >  , the amplitude of the high-frequency component of the signal is greatly improved. Although it is much smaller than the 1- and 2-order differential, it is also sufficiently improved; at the same time, in the  > ω > 1 segment, the signal intermediate frequency is also strengthened. The differential increase of 1st and 2nd order is small. In the 0 < ω < 1 segment, the amplitude of the very low frequency of the signal decreases nonlinearly with the decrease of frequency, and the attenuation amplitude is obviously smaller than the differential of 1st and 2-order. It indicates that the fractional differential is in the high signal frequency of the enhanced signal and the composition also has a nonlinear retention of the very low frequency components of the signal [18]. The noise information is generally mixed in the image information in the form of high frequencies. Therefore, when image denoising, the spatial domain image information is often converted

to the frequency domain by Fourier transform, and the high frequency part is filtered by the low pass filter, and then inversely transformed into the air domain to achieve the purpose of noise removal. Since the contour or edge portion of the image signal is also high frequency information, it is easily confused with the high frequency noise signal. If only the low frequency filter is used to filter out the high frequency portion, this not only filters out the noise signal but also filters out the image and the edges and contours of the signal make the image after denoising chaotic and unclear. After the fractional-order calculus change is applied to the low-pass filter, the filtering algorithm is slightly improved, and the originally denoized image is clearer. Therefore, the filtering algorithm is more detailed between the image signal and the noise, and some of the high-frequency signal portions are retained as much as possible. With the fractional-order calculus transformation and the fine adjustment of the fractional calculus derivative for the low-pass filter, the outline of the image contaminated by the salt and pepper noise becomes more and more obvious, and becomes clear step by step. Diffusion is performed in the airspace, and in order to maintain the edge and contour information of the image, an edge stop func-

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

ARTICLE IN PRESS

JID: CHAOS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

tion must be added. In order to solve the problem that the integer step-stop operator is sensitive to noise, the gradient modulus of the image Gaussian smoothing is used as the independent variable of the edge stop function. Although the Gaussian smoothed image can eliminate certain noise, however, it is inevitable that some details such as edges and textures will be lost, which will result in the gradient modulus of the smoothed image not well describing the intrinsic properties of the image. In addition, in the process of iterative denoising, it is necessary to calculate the gradient modulus of the Gaussian smoothed image each time, which will greatly increase the computation time complexity and increase the system overhead. When the fractional differential order 0 < ν < 1, the information of the smooth region can be nonlinearly preserved while improving the detail information such as image edges and textures. It can be considered that the fractional differential operator can effectively extract the edge at the same time. The noise of the image is suppressed, and since the enhancement of the high-frequency component of the signal by the fractional-order differential operator is weaker than that of the integer-order differential operator, it is not sensitive to the superimposed noise like the integerorder differential operator. Therefore, using the modulus value of the fractional step operator as a measure of the strength of the fractional diffusion can more accurately describe the essential characteristics of the image [19].

5. Numerical simulation and result analysis 5.1. Simulation experiment design For the denoising algorithm and image super-resolution reconstruction algorithm introduced above, the experimental results of the fractional-order-based algorithm and the integer-order-based algorithm are compared through several experiments. In the simulation of the denoising algorithm, the objective evaluation criterion used is the Peak Signal-to-Noise Ratio (PSNR). In the superresolution reconstruction of the simulated image, the evaluation standard used is Structural Similarity Index (SSIM), also the larger its value, the higher the similarity of the image, the better the effect. (1) Using fractional-order p-Laplace and fractional tensor denoising algorithms to compare with algorithms in various literatures. In the images Barbara, Linfa, Baboon and Elaine, the Gaussian white noise with noise standard deviation of 25, 35, and 45 was added to experiment, and the PSNR between the denoised image and the original image was calculated. The last three algorithms in Table 1 are based on fractional-order denoising algorithms; it can be clearly seen that the calculated PSNR values are larger than the PSNR values calculated by other integer-order-based denoising algorithms, indicating fractional-based denoising. Compared with several other integer-order based denoising algorithms, the algorithm has advantages in denoising efficiency. (2) In the simulation experiment of image super-resolution reconstruction, the image is first reduced to 1/3 of the original image size, and then the algorithm based on fractional-order image super-resolution reconstruction and the most Nearest Neighbor Interpolation (NNI) algorithm are used. The allvariation (TV) algorithm, the Fractional Bidirectional (FB) algorithm, and the Edge Prior (EP) algorithm amplify the reduced low-resolution image and calculate the SSIM value between the magnified image and the original image. The value of the structural similarity calculated when the 6 Lena images are magnified 3 times.

9

5.2. Simulation result analysis Through experimental simulation, it is known that as the fractional differential order increases, the extracted edge information increases. This is consistent with the amplitude-frequency analysis results of the previous fractional differential. When 0 < ν < 1, the differential effect on the amplitude of the high-frequency part of the image signal is gradually increased, and the edge information extracted is also more. In the experiment, when ν > 0.6, the extracted edge information sharply increases, and most of them exceed the image display range. It can be seen that the detection effect is best when the differential order is between 0.4 and 0.7. Through observation, it is easy to find that the first-order differential Roberts, Sobel and Prewitt operators extract relatively few edge information and the edge continuity is poor; Laplacian operator, Canny operator and traditional fractional differential method extract the edge information is richer; the edge information of the gradient operator detection results presented in this paper is the most abundant and accurate, and the continuity is the best; compared with the previous methods, it has better edge extraction effect. It can be seen from Fig. 7 that the evaluation value is approximately linear with the order when the order is less than 0.7, and is no longer linear when the order is greater than 0.7. In the experiment, where the evaluation index is abrupt, the edge of the image will appear excessive sharpening; it can be considered that the image does not sharpen excessively when the information entropy and the sharpness change linearly with the order, and the processing effect is better. In the actual algorithm, the sharpness of the linear effect is taken as the index, and the order is made as large as possible while ensuring that the sharpness index is in the linear interval. From the data in Fig. 7, the fractional order is generally 0.7. Since the images used in the process of enhancement are all operators of the same fractional order, each region in the image is enhanced by the same intensity, which causes local distortion of the enhanced image, and the noise in the image is not plus enhanced. Two parameters of local mean and local standard deviation of the image are introduced to define the region of the image and the region to be enhanced is processed by a fractional differential mask. The differential order can also be constructed according to the image gradient modulus and local entropy, thereby achieving the adaptation of the differential order. When −1 < ν < 0, the fractional integral has an enhancement effect on the very low frequency signal in the very low frequency range of 0 < ν < 1, and the enhancement of the very low frequency signal increases with the increase of the absolute value of the fractional integral fraction. The degree becomes larger, indicating that the fractional integral operation can preserve the low frequency information of the image. In the frequency range of ω > 1, the fractional integral has a certain attenuation effect on the highfrequency signal. When the absolute value of the fractional integral increases, the attenuation of the high-frequency signal increases and indicates the fractional integral operation. The high-frequency signal can be attenuated to achieve image denoising; however, as the signal frequency increases, the frequency characteristic curve is stable, indicating that the high-frequency component can be nonlinearly preserved, so that the image denoising can be applied while applying fractional integration. It is not difficult to see that as the order increases, the intermediate frequency and high frequency components of the signal can be effectively enhanced. For the image, the intermediate frequency component corresponds to the texture portion of the image, and the high frequency component corresponds to the edge and noise portion of the image. It can be seen from Fig. 8 that when the fractional order is fixed, the relative error gradually decreases with the increase of the sampling times. When the sampling times are constant, the

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

JID: CHAOS 10

ARTICLE IN PRESS

[m5G;October 18, 2019;20:39]

Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

Fig. 7. Comparison of abnormality experiment results in image denoising.

Fig. 8. Comparison of image Lena denoising results with new algorithm model and other models.

relative error gradually increases with the increase of the fractional integration steps, and when the number of uses is large enough, the transformation of the fractional order has little effect on the relative error. It can be seen that the accuracy of the fractionalorder integral denoising mask is sufficiently high. When the number of samples is large enough, the relative error can be taken on the order of 10−6 , which is almost zero error in engineering. With the variation of the integral order v, the fractional-order integral denoising operator can nonlinearly preserve the high-frequency edge features of the gray-scale transition amplitude in the digital image and the gray-scale jump amplitude and frequency variation in the digital image. Relatively small high-frequency texture detail features. The gray level co-occurrence matrix contrast parameter of the image reflects the sharpness of the image and the degree of texture groove depth [20]. As the fractional order of the fraction changes from −1.0 to 0, the contrast parameter gradually increases,

that is, the texture groove gradually increases, and the image gradually becomes clear. The gray level co-occurrence matrix correlation parameter of the image measures the degree of similarity of the gray level co-occurrence matrix elements in all directions. 6. Conclusions On the basis of summarizing and analyzing previous research works, this paper proposes a new method for detecting noise points in images using fractional differential gradient and an improved image denoising algorithm based on fractional integration. The noise detection method determines the noise position by the fractional differential gradient, and achieves to detect the noise, snowflake and stripe anomaly though utilizing the neighborhood information feature of the image and the contour and direction distribution of various noise anomalies in spatial domain; the im-

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463

JID: CHAOS

ARTICLE IN PRESS Q. Wang, J. Ma and S. Yu et al. / Chaos, Solitons and Fractals xxx (xxxx) xxx

age denoising algorithm firstly regards the appearance of noise points in image as a small probability event and divides it, then applies the fractional calculus to subtly transform and adjust signal filter and meanwhile utilizes iterative idea to control image denoising effect for accurately distinguishing between noise and high-frequency signals, so that the original image feature information is more preserved while the image is denoised. The simulation results show that this algorithm model can effectively remove the noise while maintaining the details of image edge and texture, and have the characteristics of simple algorithm and good stability. The model can effectively remove the noise and meanwhile preserve the details of image edges and textures, and has the characteristics of simple algorithm and high stability. The image edge enhancement algorithm based on integer differential is not an effective method to detect the texture details in the smooth region; and the image processed by the fractional differential, the texture details of the smooth region will not be weakened, but also somehow there is nonlinearity. Through experimental simulation, it is known that as the fractional differential order increases, the extracted edge information increases. This is consistent with the amplitude-frequency analysis results of the previous fractional differential. When 0 < ν < 1, the differential effect on the amplitude of the high-frequency part of the image signal is gradually increased, and the edge information extracted is also more. In the simulation, when ν > 0.6, the extracted edge information sharply increases, and most of them exceed the image display range. It can be seen that the detection effect is best when the differential order is between 0.4 and 0.7. When −1 < ν < 0, the fractional integral has an enhancement effect on the very low frequency signal in the very low frequency range of 0 < ω < 1, and the enhancement of the very low frequency signal increases with the increase of the absolute value of the fractional integral fraction. The degree becomes larger, indicating that the fractional integral operation can preserve the low frequency information of the image. When the order is less than 0.7, the evaluation value is approximately linear with the order, and when the order is greater than 0.7, it is no longer linear. Since the images used in the process of enhancement are all operators of the same fractional order, each region in the image is enhanced by the same intensity, which causes local distortion of the enhanced image, and the noise in the image is not plus enhanced. The research results of this paper provide a reference for further research on noise detection and image denoising based on fractional calculus. Declaration of Competing Interest None.

[m5G;October 18, 2019;20:39] 11

Acknowledgments This work was supported by the program of excellent team in Harbin Institute of Technology. References [1] Lei Y. Technique for image de-noising based on non-subsampled shearlet transform and improved intuitionistic fuzzy entropy. Optik-Int J Light Electron Opt 2015;126(4):446–53. [2] Xiao Q, Li J, Bai Z, Sun J, Zhou N, Zeng Z. A small leak detection method based on VMD adaptive de-noising and ambiguity correlation classification intended for natural gas pipelines. Sensors 2016;16(12):1–16. [3] Chen YT, Lai WN, Sun EW. Jump detection and noise separation by a singular wavelet method for predictive analytics of high-frequency data. Comput Econ 2019;54(2):809–44. [4] Khan A, Waqas M, Ali MR, Altalhi A, Alshomrani S, Shi S. Image de-noising using noise ratio estimation, k-means clustering and non-local means-based estimator. Comput Electr Eng 2016;54:370–81. [5] Li J. An image de-noising algorithm based on k-SVD and BM3d. Appl Mech Mater 2014;596:333–6. [6] Youssef IK, El Dewaik MH. Solving poissons equations with fractional order using haarwavelet. Appl Math Nonlinear Sci 2017;2(1):271–84. [7] Liu S, Shi M, Hu S, Xiao Y. Synthetic aperture radar image de-noising based on shearlet transform using the context-based model. Phys Commun 2014;13:221–9. [8] Chen M, Liu Q. Blow-up criteria of smooth solutions to a 3d model of electro-kinetic fluids in a bounded domain. Electron J Differ Equ 2016;2016(128):1–8. [9] Chen MC, Lu SQ, Liu QL. Global regularity for a 2d model of electro-kinetic fluid in a bounded domain. Acta Mathematicae Applicatae Sinica, English Series 2018;34(2):398–403. [10] Hellassa S, Souag-Gamane D. Improving a stochastic multi-site generation model of daily rainfall using discrete wavelet de-noising: a case study to a semi-arid region. Arabian J Geosci 2019;12(2):53. [11] Brzeziski DW. Comparison of fractional order derivatives computational accuracy right hand vs left hand definition. Appl Math Nonlinear Sci 2017;2(1):237–48. [12] Shi GC, Wang FX. Mixed noise image de-noising based on EM algorithm. Appl Mech Mater 2014;556–562:4734–41. [13] Burgos C, Corts JC, Villafuerte L, Villanueva RJ. Mean square calculus and random linear fractional differential equations. Theory Appl 2017;2(2):317–28. [14] Jiao Y, Huang BW. A new adaptive threshold image-denoising method based on edge detection. Appl Mech Mater 2014;678(5):137–42. [15] Qiang ZJ, Zhang HX. Laser active image de-noising algorithm based on lifting-wavelet. Adv Mater Res 2014;1044–1045:1194–200. [16] Zhang J, Mo FJ, Yao F, Wang XJ. De-noising method for partial discharge signal based on wavelet transform and partial differential equation. Appl Mech Mater 2014;529:444–7. [17] Xia H, Montresor S, Picart P, Guo R. Comparative analysis for combination of unwrapping and de-noising of phase data with high speckle decorrelation noise. Opt Lasers Eng 2018;107:71–7. [18] Liu H, Wang W, Xiang C, Han L, Nie H. A de-noising method using the improved wavelet threshold function based on noise variance estimation. Mech Syst Signal Process 2018;99:30–46. [19] Pritamdas K, Singh KM, Singh LL. Removal of impulse noise from color images based on the localized image characteristics and noise level. Signal Image Video Process 2018;12(3):1–9. [20] Khaw HY, Soon FC, Chuah JH, Chow C. High-density impulse noise detection and removal using deep convolutional neural network with particle swarm optimisation. IET Image Process 2019;13(2):365–74.

Please cite this article as: Q. Wang, J. Ma and S. Yu et al., Noise detection and image denoising based on fractional calculus, Chaos, Solitons and Fractals, https://doi.org/10.1016/j.chaos.2019.109463