scholarly journals Image Denoising of Adaptive Fractional Operator Based on Atangana–Baleanu Derivatives

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Xiaoran Lin ◽  
Yachao Wang ◽  
Guohao Wu ◽  
Jing Hao
Keyword(s):  
Integral Operator ◽  
Image Denoising ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Local Entropy ◽  
Adaptive Function ◽  
Order Function ◽  
Local Variance ◽  
Image Edge ◽  
Fractional Order Integral

A fractional integral operator can preserve an image edge and texture details as a denoising filter. Recently, a newly defined fractional-order integral, Atangana–Baleanu derivatives (ABC), has been used successfully in image denoising. However, determining the appropriate order requires numerous experiments, and different image regions using the same order may cause too much smoothing or insufficient denoising. Thus, we propose an adaptive fractional integral operator based on the Atangana–Baleanu derivatives. Edge intensity, global entropy, local entropy, and local variance weights are used to construct an adaptive order function that can adapt to changes in different regions of an image. Then, we use the adaptive order function to improve the masks based on the Grumwald–Letnikov scheme (GL_ABC) and Toufik–Atangana scheme (TA_ABC), namely, Ada_GL_ABC and Ada_TA_ABC, respectively. Finally, multiple evaluation indicators are used to assess the proposed masks. The experimental results demonstrate that the proposed adaptive operator can better preserve texture details when denoising than other similar operators. Furthermore, the image processed by the Ada_TA_ABC operator has less noise and more detail, which means the proposed adaptive function has universality.

scholarly journals Denoising Algorithm Based on Generalized Fractional Integral Operator with Two Parameters

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Hamid A. Jalab ◽  
Rabha W. Ibrahim
Keyword(s):  
Visual Perception ◽  
Integral Operator ◽  
Image Denoising ◽  
Digital Image ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Smoothing Filter ◽  
Generalized Fractional Integral ◽  
Two Parameters

In this paper, a novel digital image denoising algorithm called generalized fractional integral filter is introduced based on the generalized Srivastava-Owa fractional integral operator. The structures ofn×nfractional masks of this algorithm are constructed. The denoising performance is measured by employing experiments according to visual perception and PSNR values. The results demonstrate that apart from enhancing the quality of filtered image, the proposed algorithm also reserves the textures and edges present in the image. Experiments also prove that the improvements achieved are competent with the Gaussian smoothing filter.

A note on fractional integral operator associated with multiindex Mittag-Leffler functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1931-1939 ◽  
Author(s):  
Junesang Choi ◽  
Praveen Agarwal
Keyword(s):  
Integral Operator ◽  
Fractional Calculus ◽  
Fractional Integral ◽  
Fractional Integration ◽  
Fractional Integral Operator ◽  
Integration Formula ◽  
Special Cases

Recently Kiryakova and several other ones have investigated so-called multiindex Mittag-Leffler functions associated with fractional calculus. Here, in this paper, we aim at establishing a new fractional integration formula (of pathway type) involving the generalized multiindex Mittag-Leffler function E?,k[(?j,?j)m;z]. Some interesting special cases of our main result are also considered and shown to be connected with certain known ones.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals FRACTIONAL INTEGRAL OPERATORS IN NONHOMOGENEOUS SPACES

2009 ◽  
Vol 80 (2) ◽  
pp. 324-334 ◽  
Author(s):  
H. GUNAWAN ◽  
Y. SAWANO ◽  
I. SIHWANINGRUM
Keyword(s):  
Integral Operator ◽  
Fractional Integral ◽  
Type Inequality ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Fractional Integral Operator ◽  
Homogeneous Type ◽  
Fractional Integral Operators

AbstractWe discuss here the boundedness of the fractional integral operatorIαand its generalized version on generalized nonhomogeneous Morrey spaces. To prove the boundedness ofIα, we employ the boundedness of the so-called maximal fractional integral operatorIa,κ*. In addition, we prove an Olsen-type inequality, which is analogous to that in the case of homogeneous type.

scholarly journals From fractional order equations to integer order equations

2017 ◽  
Vol 20 (6) ◽  
Author(s):  
Daniel Cao Labora ◽  
Rosana Rodríguez-López
Keyword(s):  
Integral Equation ◽  
Integral Operator ◽  
Vector Space ◽  
Integral Equations ◽  
Fractional Order ◽  
Fractional Integral ◽  
State Of The Art ◽  
Constant Coefficients ◽  
Fractional Order Integral ◽  
Ordinary Integral

AbstractThe main goal of this article is to show a new method to solve some Fractional Order Integral Equations (FOIE), more precisely the ones which are linear, have constant coefficients and all the integration orders involved are rational. The method essentially turns a FOIE into an Ordinary Integral Equation (OIE) by applying a suitable fractional integral operator.After discussing the state of the art, we present the idea of our construction in a particular case (Abel integral equation). After that, we propose our method in a general case, showing that it does work when dealing with a family of “additive” operators over a vector space. Later, we show that our construction is always possible when dealing with any FOIE under the above-mentioned hypotheses. Furthermore, it is shown that our construction is “optimal” in the sense that the OIE that we obtain has the least possible order.

scholarly journals Certain Hadamard Proportional Fractional Integral Inequalities

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 504 ◽  
Author(s):  
Gauhar Rahman ◽  
Kottakkaran Sooppy Nisar ◽  
Thabet Abdeljawad
Keyword(s):  
Integral Operator ◽  
Difference Equations ◽  
Fractional Integral ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Exponential Functions ◽  
Non Local ◽  
Positive Functions

In this present paper we study the non-local Hadmard proportional integrals recently proposed by Rahman et al. (Advances in Difference Equations, (2019) 2019:454) which containing exponential functions in their kernels. Then we establish certain new weighted fractional integral inequalities involving a family of n ( n ∈ N ) positive functions by utilizing Hadamard proportional fractional integral operator. The inequalities presented in this paper are more general than the inequalities existing in the literature.

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