Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional

diﬀerential masks for images enhancement

M´ascaras diferenciales fraccionarias de Caputo y Caputo-Fabrizio para la mejora

de im´agenes

Gustavo Asumu MBoro Nchama (asumu@matcom.uh.cu)

Universidad Nacional de Guinea Ecuatorial (UNGE)

Malabo, Guinea Ecuatorial, Calle Hassan II

Leandro Daniel Lau Alfonso (leandro@icimaf.cu)

Instituto de Cibern´etica Matem´atica y F´ısica, ICIMAF

Calle 15, No. 551, entre C y D, Vedado, Habana 4, CP–10400, Cuba

Ana Luisa Morales Galloso (pedrosoaugusto584@gmail.com)

Facultad de Ciencias Medicas, Miguel Enr´ıquez

Ram´on Pinto y Ensenada Luyano, La Habana

Abstract

Image enhancement is one of the most important tasks in the ﬁeld of image processing.

With the help of computer and programming languages many mathematical methods have

been implemented to improve the visual quality of an image. One of the most eﬀective meth-

ods for this purpose is the histogram equalization. The construction of fractional diﬀerential

masks for images enhancement has also been proposed. In this paper, we propose a new

way of construction of fractional diﬀerential mask based on the Caputo and Caputo-Fabrizio

derivatives. The eﬀectiveness of the proposed methods have been compared with the his-

togram equalization method and the multiplication of each pixel of an image by a constant.

The experiments results have shown superiority of the proposed methods, with better visual

quality and higher gray-level co-occurrence matrix values in four directions.

Key words and phrases: Contrast image enhancement, fractional calculus, fractional

diﬀerential mask.

Resumen

La mejora de im´agenes es una de las tareas m´as importantes en el campo del proce-

samiento de im´agenes. Con la ayuda de lenguajes inform´aticos y de programaci´on, se han

implementado muchos m´etodos matem´aticos para mejorar la calidad visual de una imagen.

Uno de los m´etodos m´as eﬁcaces para este prop´osito es la ecualizaci´on del histograma. Tam-

bi´en se ha propuesto la construcci´on de m´ascaras diferenciales fraccionarias para la mejora

de im´agenes. En este art´ıculo, se propone una nueva forma de construcci´on de m´ascara di-

ferencial fraccional basada en las derivadas de Caputo y Caputo-Fabrizio. La eﬁcacia de los

m´etodos propuestos se ha comparado con el m´etodo de ecualizaci´on del histograma y la

Received 9/12/2020. Revised 12/01/2021. Accepted 11/04/2021.

MSC (2010): Primary 34Axx; Secondary 65Lxx.

Corresponding author: Gustavo Asumu MBoro Nchama

2 Gustavo MBoro - Leandro Lau - Ana Morales

multiplicaci´on de cada p´ıxel de una imagen por una constante. Los resultados de los expe-

rimentos han demostrado la superioridad de los m´etodos propuestos, con una mejor calidad

visual y valores de matriz de co-ocurrencia de nivel de gris m´as altos en cuatro direcciones.

Palabras y frases clave: Mejora de la imagen de contraste, c´alculo fraccional, M´ascara

diferencial fraccionaria.

1 Introduction

Digital image processing is a set of techniques applied to digital images with the aim of improving

their quality using a computer. For years, these type of techniques have been investigated and

used in applications for diﬀerent tasks such as image enhancement, image restoration and image

edge detection, among others. For the image enhancement, various methods have been proposed:

Histogram Equalization (HE) is one of the best used methods for image enhancement. It is

provides better quality of images without loss of any information [12]. The multiplication of

each pixel of an image by a constant is also one of the eﬀective methods to make image clearer.

Recently, many authors have proposed the construction of masks, for image enhancement, based

on fractional derivatives [38-47]. A fractional derivative is an integral operator which generalizes

the ordinary derivative, such that if the fractional derivative is represented by D

then, when

α = n, it coincides with the usual diﬀerential operator D

[5]. Such kind of operators are deﬁned

by the help of spaces as:

Deﬁnition 1.1. A function f : [a, b] → R is said to be absolutely continuous on [a, b], denoted

by f ∈ AC[a, b], if given  > 0 there exists some σ > 0 such that

k=1

|f(y

) − f(x

)| < .

whenever {[x

, y

] : k = 1, · · · , n} is a ﬁnite collection of mutually disjoint subintervals of [a, b]

with

− x

) < σ.

Deﬁnition 1.2. Let n ∈ N and k = 1, 2, · · · , n − 1, the space AC

[a, b] is deﬁned by

[a, b] := {f : [a, b] → R : f

(k)

(t) ∈ C[a, b], f

(n−1)

(t) ∈ AC[a, b]}.

There are many deﬁnitions of fractional derivatives [42-4]. One of the most popular was

deﬁned by Gronwald and Letnikov:

Deﬁnition 1.3. Let a, b ∈ R with a < b, α > 0, f ∈ C

[a, b]. The Gronwald-Letnikov (GL)

fractional derivative of order α, is given by

f(t) = lim

h→0

[

x−a

]

k=0

(−1)





f(t − kh). (1)

where



x−a



denotes the integer part of

x−a

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 3

One of the inconveniences of Gronwald-Letnikov derivative is that the class of functions for

which this derivative is deﬁned is very narrow. To overcome this inconvenience, Riemann and

Liouville proposed the following deﬁnition [17]:

Deﬁnition 1.4. Let a, b ∈ R with a < b, α > 0, f ∈ AC

[a, b]. The Riemann-Liouville (RL)

fractional derivative of order α, is deﬁned by

f(t) =

Γ(n − α)

(t − s)

n−(α+1)

f(s)ds.

Even though the RL approach overcomes the drawbacks related the GL deﬁnition and it has

been applied successfully in many areas of engineering, unfortunately, it leads to initial conditions

containing the limit values of the RL fractional derivative at the lower terminal t = a, for example

lim

t→a

{

α−1

f(t)} = b

, lim

t→a

{

α−2

f(t)} = b

, . . . , lim

t→a

{

α−n

f(t)} = b

In spite of the fact that initial value problems with such initial conditions can be successfully

solved mathematically, their solutions are practically useless, because there is no known physical

interpretation for such types of initial conditions. An alternative solution to this conﬂict was

proposed by M. Caputo [17]:

Deﬁnition 1.5. Let a, b ∈ R with a < b, α > 0, f ∈ AC

[a, b]. The Caputo fractional derivative

of order α, is deﬁned by

f(t) =

Γ(n − α)

(t − s)

n−(α+1)

(n)

(s)ds. (2)

For 0 < α ≤ 1 and a = 0, the numerical approximation of (2) takes the form

u(x) =

Γ(1 − α)

(x − ξ)

−α

(ξ)dξ ≈

Γ(1 − α)

N−1

k=0

(k+1)x

(x − ξ)

−α

(ξ

)dξ. (3)

To describe material heterogeneity and structures with diﬀerent scales which cannot be well

described by classical local theories or by fractional models with singular kernel, Caputo and

Fabrizio introduced a new fractional approach [7]:

Deﬁnition 1.6. Let a, b ∈ R with a < b, 0 < α < 1, f ∈ AC

[a, b]. The Caputo-Fabrizio

fractional derivative of order α, is deﬁned by

u(x) =

M(α)

1 − α

−

1−α

(x−s)

(s)ds,

where M (α) is a function such that M(0) = M(1) = 1.

In [24], Losada and Nieto, suggested the following particular case

u(x) =

1 − α

−

1−α

(x−s)

(s)ds (4)

1 − α



u(x) − e

−

1−α

u(a)



−

(1 − α)

−

1−α

(x−τ )

u(τ)dτ.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

4 Gustavo MBoro - Leandro Lau - Ana Morales

Taking a = 0, formula (4) can be approximated as

u(x) =

1 − α

−

1−α

(x−ξ)

(ξ)dξ ≈

1 − α

N−1

k=0

(k+1)x

−

1−α

(x−ξ)

(ξ

)dξ. (5)

Fractional derivatives provide interesting possibilities for scientiﬁc ﬁelds such as anomalous diﬀu-

sion [40-23], circuit theory [3-43], image processing [33-26] and many others [1-29]. The purpose

of this paper is to improve the visual quality of dark images by using Fractional Diﬀerential

Masks in Caputo (FDMC) and Caputo-Fabrizio (FDMCF) senses. The remainder of this paper

is organized as follows: in section 2, we construct a fractional diﬀerential mask in the Caputo

sense, next, fractional diﬀerential mask in Caputo-Fabrizio sense is given in section 3. Section 4

presents the experimental results of the proposed methods. A conclusion is considered in section

2 Prewitt fractional ﬁlter in the Caputo sense

The goal of this section is to costruct a fractional diﬀerential mask based on the Caputo derivative

deﬁnition. For this purpose, we ﬁrst discretize numerically the Caputo derivative based on the

forward ﬁnite diﬀerence scheme in the interval [0, x] (analogously [0, y]). Let’s take a partition

of N nodes of the interval [0, x], with step ∆x =

. Thus, there are N + 1 nodes. The N + 1

causal pixels can be given by











= u(0)

= u(

)

= u(

)

= u(x),

For α ∈ (0, 1), by approximating, we obtain

(k+1)x

(x − ξ)

−α

(ξ

)dξ ≈



kx+x



− u





∆x

kx+x

(x − ξ)

−α

dξ



kx+x



− u





−(1 − α)(∆x)



(N − k − 1)

1−α

− (N − k)

1−α



. (6)

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 5

Then, taking (6) into (3), we have

u(x) ≈

(∆x)

Γ(2 − α)

N−1

k=0





(k + 1)x



− u







(N − k − 1)

1−α

− (N − k)

1−α





(∆x)

Γ(2 − α)











1−α



1−α

− 2 · 1

1−α



N−1



2 · 2

1−α

− 3

1−α

− 1

1−α



N−2



(N − j − 1)

1−α

+ (N − j + 1)

1−α

−2(N − j)

1−α



+ · · · +



(N − 2)

1−α

−2(N − 1)

1−α

+ N

1−α





(N − 1)

1−α

− N

1−α













. (7)

The anterior approximate diﬀerence of fractional partial diﬀerential on x and y coordinates are

expressed as

u(x, y) ≈

(∆x)

Γ(2 − α)











1−α

u(x, y) +



1−α

− 2 · 1

1−α



u(x − 1, y)



2 · 2

1−α

− 3

1−α

− 1

1−α



u(x − 2, y)

+ · · · +



(N − 1)

1−α

− N

1−α



u(x − n, y)











, (8)

and

u(x, y) ≈

(∆x)

Γ(2 − α)











1−α

u(x, y) +



1−α

− 2 · 1

1−α



u(x, y − 1)



2 · 2

1−α

− 3

1−α

− 1

1−α



u(x, y − 2)

+ · · · +



(N − 1)

1−α

− N

1−α



u(x, y − n)











. (9)

As in a digital 2-D gray image u(x, y), the shortest distance on x and y coordinates is one pixel,

then we put ∆x = ∆y = 1, and from (7), we obtain N + 1 coeﬃcients c

(i = 0, . . . , N), which

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

6 Gustavo MBoro - Leandro Lau - Ana Morales

depend on the fractional order α:











1−α

Γ(2 − α)

1−α

− 2 · 1

1−α

Γ(2 − α)

2 · 2

1−α

− 3

1−α

− 1

1−α

Γ(2 − α)

(N − j − 1)

1−α

+ (N − j + 1)

1−α

− 2(N − j)

1−α

Γ(2 − α)

(N − 1)

1−α

− N

1−α

Γ(2 − α)

3 Prewitt fractional ﬁlter in the Caputo-Fabrizio sense

Following the idea as in the previous section, we obtain

(k+1)x

−

1−α

(x−ξ)

(ξ

)dξ ≈



kx+x



− u





∆x

kx+x

−

1−α

(x−ξ)

dξ,

1 − α



kx+x



− u





∆x



−

α(N−k−1)∆x

1−α

− e

−

α(N−k)∆x

1−α



. (10)

Inserting (10) into (5), we have

u(x) ≈

N−1

k=0













(k+1)x



− u











−

1−α

[N−(k+1)]

− e

−

1−α

[N−k]









α · ∆x













1 − e

−

1−α

∆x





−

α·∆x

1−α

− e

−2

α·∆x

1−α

− 1



N−1



−

2α·∆x

1−α

− e

−3

α·∆x

1−α

− e

−

α·∆x

1−α



N−2

+ · · · +



−

α·(N−j)·∆x

1−α

− e

−

α·(N−j−1)·∆x

1−α

− e

−

α·(N−j+1)·∆x

1−α



+ · · · +



−

α·(N−1)·∆x

1−α

− e

−

α·(N−2)·∆x

1−α

− e

−

α·N·∆x

1−α





−

α·N·∆x

1−α

− e

−

α·(N−1)·∆x

1−α

− e

−

α·(N+1)·∆x

1−α













. (11)

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 7

From (11), we obtain N + 1 nonzero coeﬃcients c

(i = 0, . . . , N), given by











α∆x



1 − e

−

1−α

∆x



α∆x



−

1−α

∆x

− e

−

2α

1−α

∆x

− 1



α∆x



−2

1−α

∆x

− e

−3

1−α

∆x

− e

−

1−α

∆x



α∆x



−

1−α

(N−j)∆x

− e

−

1−α

(N−j−1)∆x

− e

−

1−α

(N−j+1)∆x



N−1

α∆x



−

1−α

(N−1)∆x

− e

−

1−α

(N−2)∆x

− e

−

1−α

N∆x



α∆x



−

1−α

N∆x

− e

−

1−α

(N−1)∆x

− e

−

1−α

(N+1)∆x



Taking ∆x = ∆y = 1, as in the previous section, we obtain the followings two expressions:

u(x, y) ≈













1 − e

−

1−α



u(x, y) +



−

1−α

− e

−2

1−α

− 1



u(x − 1, y)



−2

1−α

− e

−3

1−α

− e

−

1−α



u(x − 2, y) + · · ·



−

1−α

− e

−

1−α

(N−1)

− e

−

1−α

(N+1)



u(x − n, y)











, (12)

and

u(x, y) ≈













1 − e

−

1−α



u(x, y) +



−

1−α

− e

−2

1−α

− 1



u(x, y − 1)



−2

1−α

− e

−3

1−α

− e

−

1−α



u(x, y − 2) + · · ·



−

1−α

− e

−

1−α

(N−1)

− e

−

1−α

(N+1)



u(x, y − n)











. (13)

The next images show the results of applying the proposed FDMCF, with diﬀerent values of

diﬀerential order α, on the following images: goldhill image, drak bedroom and dark room.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

8 Gustavo MBoro - Leandro Lau - Ana Morales

(a) Original image (b) α = 0.0071

(e) α = 0.0074 (f) α = 0.0075

Figure 1: Result of applying the proposed FDMCF on goldhill image with diﬀerent values of

diﬀerential order α.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 9

(a) Original image (b) α = 0.003

(e) α = 0.006 (f) α = 0.007

Figure 2: Result of applying the proposed FDMCF on a dark bedroom image with diﬀerent

values of diﬀerential order α.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

10 Gustavo MBoro - Leandro Lau - Ana Morales

(a) Original image (b) α = 0.003

(e) α = 0.006 (f) α = 0.007

Figure 3: Result of applying the proposed FDMCF on a dark room image with diﬀerent values

of diﬀerential order α.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 11

The next tables show the results of contrast in terms of the Gray-Level Co-occurrence Matrix

(GLCM) in 4 directions, on the following images: goldhill image, drak bedroom and dark room.

Angle

135

Original image 0.2743 0.5685 0.4336 0.5542

HE 0.5018 0.9403 0.6727 0.9352

M MPIT 0.4642 0.8428 0.6060 0.8476

e FDMC for α = 0.0155 7.4922 11.7211 8.8573 11.5499

t FDMC for α = 0.0165 13.0910 18.5857 12.7200 18.6007

h FDMC for α = 0.0170 15.1278 20.5889 13.4156 20.7000

o FDMCF for α = 0.0071 9.0089 12.3465 8.7710 12.4288

d FDMCF for α = 0.0072 9.5492 12.8873 9.0874 13.0488

FDMCF for α = 0.0073 9.9501 13.2338 9.2318 13.4732

Table 1: Gold-hill image.

Angle

135

Original image 0.0007 0.0888 0.0885 0.0888

HE 0.2229 0.3526 0.2241 0.3632

M MPITH 0.1345 0.2625 0.1488 0.2638

e FDMC for α = 0.013 0.7667 1.4983 1.0776 1.5015

t FDMC for α = 0.014 0.7653 1.4477 1.0253 1.4494

h FDMC for α = 0.015 0.7172 1.4236 1.0371 1.4240

o FDMCF for α = 0.005 0.6574 1.0510 0.6768 1.0653

d FDMCF for α = 0.006 0.6459 0.9865 0.6296 0.9923

FDMCF for α = 0.007 0.5014 0.7176 0.4438 0.7180

Table 2: Dark bedroom.

Angle

135

Original image 0.4711 0.6691 0.2139 0.6694

HE 0.6178 0.8568 0.5153 0.8977

M MPITH 0.4221 0.7419 0.4225 0.7560

e FDMC for α = 0.013 3.5218 5.4161 3.4127 5.4255

t FDMC for α = 0.014 3.6872 5.8437 3.7464 5.8422

h FDMC for α = 0.015 3.8128 6.0092 3.8800 6.0260

o FDMCF for α = 0.005 2.4504 3.9630 2.6457 3.9593

d FDMCF for α = 0.006 2.5697 4.2178 2.8633 4.1469

FDMCF for α = 0.007 2.7114 4.0842 2.6762 4.0415

Table 3: Dark room.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

12 Gustavo MBoro - Leandro Lau - Ana Morales

The following images show the comparison of contrast enhancement capability between the

methods: MPIT; MPITH; HE; FDMC; and FDMCF applied to the Goldhill image.

(a) Original image (b) MPIT

(e) FDMC with α = 0.0155 (f) FDMCF with α = 0.0072

Figure 4: Comparison of contrast enhancement capability between methods.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 13

The next images show the comparison of contrast enhancement capability between the meth-

ods: MPIT; MPITH; HE; FDMC; and FDMCF applied to the Dark Bedroom image.

(a) Original image (b) MPIT

(e) FDMC with α = 0.014 (f) FDMCF with α = 0.006

Figure 5: Comparison of contrast enhancement capability between methods.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

14 Gustavo MBoro - Leandro Lau - Ana Morales

The next images show the comparison of contrast enhancement capability between the meth-

ods: MPIT; MPITH; HE; FDMC; and FDMCF applied to the Dark room image.

(a) Original image (b) MPIT

(e) FDMC with α = 0.013 (f) FDMCF with α = 0.005

Figure 6: Comparison of contrast enhancement capability between methods.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 15

(a) Dark bedroom image

(b) Dark room image (c) Gold-hill image

Figure 7: Original images used in the experimental result.

4 Experimental results

The aim of this section is to demonstrate that fractional diﬀerential masks based on Caputo

(FDMC) and Caputo-Fabrizio (FDMCF) deﬁnitions have better capability in texture-enhancing

than the traditional approaches for texture-rich image. To this purpose, we analyze the texture-

enhancing capability of the proposed masks and discuss the relationship between fractional power

parameter α and texture-enhancing details by using Gray-Level Co-occurrence Matrix (GLCM).

Finally, we discuss the capability of texture enhancement of the proposed masks by making

comparison with Histogram Equalization (HE), Multiplication of each Pixel of an Image by Two

(MPIT) and Multiplication of each Pixel of an Image by Three (MPITH) methods. Three images

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

16 Gustavo MBoro - Leandro Lau - Ana Morales

used in the experimental results are shown in Figures 7. First is dark bedroom image, second

is a dark room image, the third one is the gold-hill image. To obtain the fractional diﬀerential

on the eight symmetric directions and make the fractional diﬀerential masks have anti-rotation

capability, eight fractional diﬀerential masks which are respectively on the directions of 0

, 45

, 135

, 180

, 225

, 270

and 315

are implemented in Fig. 8. Considering

sum =

+ c

+ · · · + c

and taking into account the eight neighboring pixels of a given one, we propose the fractional

diﬀerential mask, given by Table 4.

0 0 C

. 0

0 C

· · · C

sum C

· · · C

. 0 C

. 0

0 0 C

Table 4: Fractional diﬀerential mask.

For the implementation of the FDMCF method, we have taken only the following three

coeﬃcients:



1 − e

−

1−α





−

1−α

− e

−

2α

1−α

− 1





−2

1−α

− e

−3

1−α

− e

−

1−α



while for the FDMC method, we considered the coeﬃcients:

1−α

Γ(2 − α)

, c

1−α

− 2 · 1

1−α

Γ(2 − α)

, c

2 · 2

1−α

− 3

1−α

− 1

1−α

Γ(2 − α)

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 17

Mask in the direction of 315

Mask in the direction of 45

Mask in the direction of 90

Mask in the direction of 270

Mask in the direction of 180

Mask in the direction of 0

Mask in the direction of 135

Mask in the direction of 225

Figure 8: Diﬀerent mask

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

18 Gustavo MBoro - Leandro Lau - Ana Morales

Images a) of the Figures 1, 2 and 3 are the original images while images b), c), d) and f) of

Figures 1, 2 and 3 are the results of applying the FDMCF method on the original images with

diﬀerent values of diﬀerential order. Images a), b), c), d) and f) of the Figures 4, 5 and 6 are the

original image, enhancing result of a) by MPIT, enhancing result of a) by MPITH, enhancing

result of a) by HE, enhancing result of a) by FDMC and enhancing result of a) by FDMCF,

respectively. On theses ﬁgures, we can see that images obtained by the proposed methods look

better than those obtained by other methods. For the comparison purpose, we use the contrast of

Gray-Level Co-occurrence Matrix (GLCM) in four directions. Tables 1, 2 and 3 are the contrasts

of GLCM in four directions. From these tables we can conclude that the proposed methods

outperform HE, MPIT and MPITH methods. Based on the results shown in Figures 4, 5 and

6 we can see that the proposed methods are more eﬀectives than the HE, MPIT and MPITH

methods, since they enhance better the visual appearance of an image and make it clearer.

5 Conclusion

In this paper, we proposed construction of fractional diﬀerential masks using Caputo and Caputo-

Fabrizio fractional derivatives. Experiments results showed that ﬁltered images by the proposed

methods have better visual appearance. Moreover, the proposed techniques have demonstrated

a good performance with higher GLCM values.

References

[1] Ajou, A.; Oqielat, M. N.; Al-Zhour, Z.; Kumar, S. and Momani, S.; Solitary solutions for

time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative,

Chaos (2019), 29 (2019), 093102.

[2] Al-Zhour, Z.; Barfeie, M.; Soleymani, F. and Tohidi, E.; A computational method to price

with transaction costs under the nonlinear Black-Scholes model, Chaos, Solitons & Fractals,

127 (2019), 291–301.

[3] Andrade, A. M. F.; Lima, E. G. and Dartora, C. A.; An introduction to fractional calculus

and its Applications in Electric Circuits, Revista Brasileira de Ensino de F´ısica, 40 (2018),

e3314.

[4] Atangana, A. and Baleanu, D.; New fractional derivatives with nonlocal and non-singular

kernel: Theory and application to heat transfer model, arXiv preprint, arXiv:1602.03408

(2016).

[5] Baleanu, D.; Diethelm, K.; Escalas, E.; and Trujillo, J. J.; Fractional Calculus: Models and

numerical methods, Series on Complexity, Nonlinearity and Chaos, 3 (2012).

[6] Baleanu, D.; Jajarmi, A.; Sajjadi, S. S. and Mozyrska, D.; A new fractional model and op-

timal control of a tumor-immune surveillance with non-singular derivative operator, Chaos,

29 (2019), 083127.

[7] Caputo, M. and Fabrizio, M.; A new Deﬁnition of Fractional Derivative without Singular

Kernel, Progr. Fract. Diﬀer. Appl., 1 (2005), 73–85.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 19

[8] Concezzi, M. and Spigler, R.; Identifying the Fractional Orders in Anomalous Diﬀusion

Models from Real Data, Fractal Fract., 2(1) (2014), 14.

[9] El-Ajou, A.; Al-Zhour, Z.; Oqielat, M.; Momani, S. and Hayat, T.; Series Solutions of

Nonlinear Conformable Fractional KdV-Burgers Equation with Some Applications, The Eu-

ropean Physical Journal Plus, 134 (2019), 402.

[10] El-Ajou, A.; Oqielat, M.; Al-Zhour, Z. and Momani, S.; Analytical Numerical Solutions of

the Fractional Multi-Pantograph System: Two Attractive Methods and Comparisons, Results

in Physics, 14 (2019), 102500.

[11] El-Ajou, A.; Oqielat, M.; Al-Zhour, Z.; Kumar, S. and Momani, S.; Solitary solutions for

time-fractional nonlinear dispersive PDEs in the sense of conformable fractional derivative,

Chaos, 29 (2019), 093102.

[12] Garg, P. and Jain, T.; A comparative study on histogram equalization and cumulative his-

togram equalization. International Journal of technology and research, 3(9) (2017), 41–43.

[13] Goufo, E. F. D. and Mugisha, S. B.; Similarities in a ﬁfth-order evolution equation with and

with no singular kernel, Chaos, Solitons & Fractals, 130 (2020), 10946.

[14] Jajarmi, A.; Arshad, S. and Baleanu, D.; A new fractional modelling and control strategy

for the outbreak of dengue fever, Physica A., 535 (2019), 122524.

[15] Jajarmi, A.; Baleanu, D.; Sajjadi, S. S. and Asad, J. H.; A new feature of the fractional

Euler-Lagrange equations for a coupled oscillator using a nonsingular operator approach,

Frontiers in Physics, 7 (2019), 00196.

[16] Jajarmi, A.; Ghanbari, B. and Baleanu, Dumitru. A new and eﬃcient numerical method for

the fractional modelling and optimal control of diabetes and tuberculosis co-existence, Chaos,

29 (2019), 093111.

[17] Kilbas, A. A.; Srivastava, H. M. and Trujillo, J. J.; Theory and Applications of Fractional

Diﬀerential Equations, North Holland Mathematics Studies, 204 (2006).

[18] Kumar, S.; A new fractional modeling arising in engineering sciences and its analytical

approximate solution, Alexandria Engineering Journal, 52 (2013), 813–819.

[19] Kumar, S.; Kumar, A.; Abbas, S.; Al Qurashi, M. and Baleanu, D.; A modiﬁed analytical

approach with existence and uniqueness for fractional Cauchy reaction-diﬀusion equations,

Advances in Diﬀerence Equations, 2020(1) (2020), 1–18.

[20] Kumar, S.; Kumar, A. and Nisar, K. S.; Numerical solutions of nonlinear fractional model

arising in the appearance of the strip patterns in two dimensional systems, Advances in

Diﬀerence Equations, 413 (2019).

[21] Kumar, R.; Kumar, S.; Singh, J. and Al-Zhour, Z. A comparative study for fractional chem-

ical kinetics and carbon dioxide CO

absorbed into phenyl glycidyl ether problems, Mathe-

matics, 5(4) (2020), 3201–3222.

[22] Kumar, S.; Nisar, K. S.; Kumar, R.; Cattani, C. and Samet, B.; A new Rabotnov fractional-

exponential function based fractional derivative for diﬀusion equation under external force,

Mathematical Methods in Applied Science, 43 (2020), 4460–4471.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

20 Gustavo MBoro - Leandro Lau - Ana Morales

[23] Liang, X.; Gao, F.; Zhou, C. B.; Wang, Z. and Yang, X. J.; An anomalous diﬀusion model

based on a new general fractional operator with the Mittag-Leﬄer function of Wiman type,

Advances in Diﬀerence Equations, 18(1) (2018), 25.

[24] Losada, J. and Nieto, J.; Properties of a New Fractional Derivative without Singular Kernel,

Progr. Fract. Diﬀer. Appl., 1 (2015), 87–92.

[25] Mboro Nchama, G. A.; Alfonso, L. L.; Le´on Mec´ıas, A. M. and Rodr´ıguez Richard, M.; Con-

struction of Caputo-Fabrizio fractional diﬀerential mask for image enhancement, Progress

in Fractional Diﬀerentiation and Application (2020).

[26] Mboro Nchama, G. A.; Le´on Mec´ıas, A. M. and Rodr´ıguez Ricard, M.; Perona-Malik model

with diﬀusion coeﬃcient depending on fractional gradient via Caputo-Fabrizio derivative,

Abstract and Applied Analysis (2020), 2020 (2020), 15 pages.

[27] Mboro-Nchama, G. A.; Mec´ıas, A. L. and Ricard, M. R.; The Caputo-Fabrizio fractional

integral to generate some new inequalities, Information Sciences Letters, 8 (2019), 73–80.

[28] Mboro-Nchama, G. A.; Mec´ıas, A. L. and Ricard, M. R.; Properties of the Caputo-Fabrizio

fractional derivative, Applied Mathematics & Information Sciences, 14 (2020), 1–10.

[29] Mboro Nchama, G. A.; New fractional integral inequalities via Caputo-Fabrizio operator and

an open problem concerning an integral inequality, New trends in Mathematical Sciences

(2020), 8(2) (2020), 9–21.

[30] Mboro Nchama, G. A.; On open problems concerning Riemann-Liouville fractional integral

inequality, Mediterranean Journal of Modeling and Simulation, 11 (2019), 001–008.

[31] Mboro-Nchama, G. A.; Properties of Caputo-Fabrizio fractional operators, New Trends in

Mathematical Sciences, 8 (2020), 1–25.

[32] Morales-Delgado, V. F.; G´omez-Aguilar, J. F. and Taneco-Hern´andez, M. A.; Analytical

solutions of electrical circuits described by fractional conformable derivatives in Liouville-

Caputo sense, International Journal of Electronics and Communications, 85 (2018), 108–

117.

[33] Nandal, A.; Gamboa-Rosales, H.; Dhaka, A.; Celaya-Padilla, J. M.; Galvan-Tejada, J. I.;

Galvan-Tejada, C. E.; Martinez-Ruiz, F. J. and Guzman-Valdivia, C.; Image edge detec-

tion using fractional calculus with feature and contrast enhancement. Circuits, Systems, and

Signal Processing, 37(9) (2018), 3946–3972.

[34] Odham, K. B. and Spanier, J.; The Fractional Calculus, Academic Press (1984), New York.

[35] Oqielat, M.; El-Ajou, A.; Al-Zhour, Z.; Alkhasawneh, R. and Alrabaiah, H.; Series solutions

for nonlinear time-fractional Schrdinger equations: Comparisons between conformable and

Caputo derivatives, Alexandria Engineering Journal (2020).

[36] Polubny, I.; Fractional Diﬀerential Equations, Academic Press (1999), New York.

[37] Pu, Y.; Application of fractional diﬀerential approach to digital image processing. Journal of

Sichuan University, 39(3) (2007), 124–132.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21

Caputo and Caputo-Fabrizio fractional diﬀerential masks for image enhancement 21

[38] Pu, Y.; Wang, W.; Zhou, J. et al. Fractional diﬀerential approach to detecting texture features

of digital image and its fractional diﬀerential ﬁlter implementation, Sci. China Ser. F, Inf.

Sci.; 38(12) (2008), 2252–2272.

[39] Qing-li, C.; H. Guo, H. and Xiu-qiong, Z.; A Fractional Diﬀerential Approach to Low Con-

trast Image Enhancement, International Journal of Knowledge and Language Processing

(2012), 3(2) (2012), 20–27.

[40] Ram´ırez, C.; Astorga, V.; Nu˜nez, H.; Jaques, A. and Simpson, R.; Anomalous diﬀusion

based on fractional calculus approach applied to drying analysis of apple slices: The eﬀects of

relative humidity and temperature, Food Process Engineering (2017), 40 (5) (2017), e12549.

[41] Rchid, M.; Ammi, S. and Jamiai, I.; Finite diﬀerence and Legendre spectral method for a

time-fractional diﬀusion-convection equation for image restoration, Discrete and Continuous

Dynamical Systems Series, 11 (2018), 103–117.

[42] Sabatier, J.; Agrawal, O. P. and Machado, J. A. T.; Advanced in Fractional Calculus: The-

oretical Developments and Applications in Physics and Engineering, Springer (2007).

[43] Sikora, R.; Fractional derivatives in electrical circuit theory critical remarks, Archives of

Electrical Engineering, 66 (2017), 155–163.

[44] Surya Prasath, V. B.; Image denoising by anisotropic diﬀusion with inter-scale information

fusion, Pattern Recognition and Image Analysis, 27 (2017), 748–753.

[45] Yirenkyi, P. A.; Appati, J. K. and Dontwi, I. K.; A new construction of a fractional derivative

mask for image edge analysis based on Riemann-Liouville fractional derivative. Advances in

Diﬀerence Equations (2016), DOI 10.1186/s13662-016-0946-8, pp. 1-21.

[46] Yu, J.; Zhai, R.; Zhou, S. and Tan, L.; Image Denoising Based on Adaptive Fractional

Order with Improved PM Model, Mathematical Problems in Engineering (2018), Article ID

9620754.

[47] Zhang, Y.; Pu, Y. and Zhou, J.; Construction of Fractional diﬀerential Masks Based on

Riemann-Liouville Deﬁnition. Journal of Computational Information Systems, 6 (2010),

3191–3198.

Divulgaciones Matem´aticas Vol. 22, No. 1(2021), pp. 1–21