Optimization of parameters for image denoising algorithm pertaining to generalized Caputo-Fabrizio fractional operator

Abstract

The aim of the present paper is to optimize the values of different parameters related to the image denoising algorithm involving Caputo Fabrizio fractional integral operator of non-singular type with the Mittag-Leffler function in generalized form. The algorithm aims to find the coefficients of a kernel to remove the noise from images. The optimization of kernel coefficients are done on the basis of different numerical parameters like Mean Square Error (MSE), Peak Signal-to-Noise Ratio (PSNR), Structure Similarity Index measure (SSIM) and Image Enhancement Factor (IEF). The performance of the proposed algorithm is investigated through above-mentioned numeric parameters and visual perception with the other prevailed algorithms. Experimental results demonstrate that the proposed optimized kernel based on generalized fractional operator performs favorably compared to state of the art methods. The uniqueness of the paper is to highlight the optimized values of performance parameters for different values of fractional order. The novelty of the presented work lies in the development of a kernel utilizing coefficients from a fractional integral operator, specifically involving the Mittag-Leffler function in a more generalized form.

1 Introduction

Nowadays, fractional calculus has become a significant area of research amongst researchers in the field of image processing. The fractional calculus generalizes the order of derivatives from integers to any arbitrary value, due to which it is compatible in real life situations. The fractional differential equations are being widely used in various technical and scientific fields viz. Applied Sciences, Control Theory, Economics, Signal and image processing, fractal analysis, thermodynamics, image denoising etc. [1,2,3,4,5,6,7,8,9,10]. Since, the mathematical models based on a fractional order are much more realistic than integer order models due to the fractional derivatives which are more effective to express the memory and heredity properties of different materials and processes. Fractional derivative removes noise effectively while preserving other important information of images like texture and edge details, that is why it has been widely used in image denoising processes. Noise can be seen as an unwanted signal that comes during the transmission of images, due to many reasons like temperature, light intensity, Gaussian noise, impulse noise, etc. Many researchers have been studying different efficient image denoising algorithms based on various fractional operators viz. Riemann–Liouville Filter (RLF) [11]. Researchers implemented an image denoising algorithm with Grunwald-Letnikov fractional operator [12]. Lavin-Delgado et al. [13] developed a novel fractional conformable edge detector tailored for medical images for structure extraction focusing on diagnosing cerebral arteriovenous malformations, meningiomas, and medulloblastomas using CT and MRI scan. Jalab et al. [14] designed the Alexander polynomial fractional integral (AFI) to improve the performance of denoising filters by optimizing the order of derivatives. With this concept Li and Xie [15], [16] designed an adaptive algorithm which is an efficient filter to remove the noise. Lavin-Delgado et al. [10] developed an innovative fractional-order technique by merging the Harris–Stephens algorithm with Caputo-Fabrizio and Atangana-Baleanu derivatives for corner object detection and image matching. They integrated a fractional Gaussian filter with generalized image derivatives for crack identification on concrete structural images to get better accuracy. Lavin-Dalgado et al. [16] introduced a novel approach by integrating Caputo-Fabrizio fractional-order derivative into the Sobel operator for improved micro edge detection in medical images. The proposed mask helps in the analysis and monitoring of diseases like breast cancer, benign cysts, and breast calcifications, leading to more effective treatments. In the next study Lavin-Dalgado et al. [17] introduced a novel approach integrating Caputo-Fabrizio fractional-order derivative with the Speeded-Up Robust Feature (SURF) algorithm for enhanced keypoint detection and image matching in regions with low contrast and weak texture. By leveraging the fractional-order derivative, the method significantly improves texture detail detection, leading to more accurate image feature extraction and matching compared to conventional SURF and SIFT algorithms. A fractional Gaussian kernel with the Atangana-Baleanu fractional derivative for blood vessel detection in retinal images was proposed by Solis-Perez et al. [18]. Their method leverages fractional Gaussian matrix eigenvalues analysis to derive principal curvature, optimizing non-integer orders via the Cuckoo Search algorithm for enhanced accuracy. Motivated by all recent research on image denoising using fractional operators [19, 20], we have proposed an algorithm and a corresponding filter kernel based on the generalized Caputo-Fabrizio (CF) fractional operator involving Mittag-Leffler (M-L) function as a kernel. The uniqueness and novelty of the presented work lie in two key aspects. Firstly, it involves demonstrating the optimized parameters of the proposed algorithm, showcasing a refined and efficient approach. Secondly, the work involves the development of a kernel that utilizes coefficients derived from a fractional integral operator, incorporating the Mittag-Leffler function. This integration represents a novel approach, potentially offering advancements in computational techniques and expanding the applicability of the kernel in various domains.

The remaining sections are arranged as follows:

The mathematical prerequisites related to the proposed fractional operator are given in Sect. 2. In Sect. 3, the coefficients are explored for the fractional integral based kernels corresponding to the generalized fractional operator for the proposed algorithm and its pattern. Section 4 gives the details about the development of the algorithm of noise removal for images. Section 5 demonstrates the optimization analysis for different parameters of fractional order, the performance of the parameters are also depicted in the form of various graphs. Finally, conclusions are drawn in Sect. 6.

2 Basics and preliminaries

This section deals with the basic notations and definitions required as prerequisites, which help us to instruct our main results.

Definition 1

The Mittag-Leffler (M-L) function of one parameter is given as [14]:

$$\begin{aligned} E_{\beta }\left( \theta t^{\beta }\right) =\sum ^{\infty }_{j=0}{\frac{{\theta }^jt^{\beta j}}{{\varGamma }\left( \beta j+1\right) },\quad \left( \theta \ne 0,\theta \in R,t\in C,\beta >0\right) }, \end{aligned}$$

(1)

and the M-L function with two parameters $\beta$ and $\gamma$ is given as:

$$\begin{aligned} E_{\beta , \gamma }\left( \theta t^{\beta }\right) =\sum ^{\infty }_{j=0}{\frac{{\theta }^jt^{\beta j}}{{\varGamma }\left( \beta j+\gamma \right) },\quad \left( \theta \ne 0,\theta \in R,t,\gamma \in C,\beta >0\right) }, \end{aligned}$$

(2)

where $E_{\beta ,1}\left( \theta t^{\beta }\right) =E_{\beta }\left( \theta t^{\beta }\right)$.

Definition 2

Let $0<\alpha <1, \beta >0,\; C_{\delta }\left[ {\hat{a}},\; a\right]$ be the weighted space of continuous functions, $f\in C_{\delta }\left[ {\hat{a}},\;a\right]$. The generalized fractional integral of order $\alpha$ and $\beta$ of the function f(t) is defined as [14]:

$$\begin{aligned} \left( K^{\alpha ,\beta }_{{\hat{a}}}f\right) \left( t\right) =\frac{1}{\alpha }\int ^t_{{\hat{a}}}{E_{\beta }\left[ -C_{\alpha }{\left( t-s\right) }^{\beta }\right] f\left( s\right) \text{d}s}, \end{aligned}$$

(3)

where $C_{\alpha }=\frac{1-\alpha }{\alpha }$ and $E_{\beta }\left( z\right) =\sum ^{\infty }_{j=0}{\frac{z^j}{{\varGamma }\left( \beta j+1\right) }}$ is an M-L function.

Let f be a sufficiently smooth function on [a, b],  operator

$$\begin{aligned} \left( K^{\vartheta }_pf\right) \left( x\right) =p\int ^x_{{\hat{a}}}{k_{\vartheta }\left( x,s\right) f\left( s\right) \text{d}s}+q\int ^b_x{k_{\vartheta }\left( s,x\right) f\left( s\right) \text{d}s,\; \alpha >0}. \end{aligned}$$

(4)

The operator $K^{\vartheta }_p$ is called K operator [17], it satisfies the following formula

$$\begin{aligned} \left( K^{\vartheta }_pf\right) \left( x\right) =p\left( K^{\vartheta }_{P1}f\right) \left( x\right) +q\left( K^{\vartheta }_{P2}f\right) \left( x\right) , \end{aligned}$$

(5)

where $P=<a,x,b,p,q>,\;P_1=<a,x,b,1,\;0>$ and ${P}_2=<a,x,b,0,1>.$

For

$$\begin{aligned} k_{\vartheta }\left( x,s\right)&= \frac{E_{\beta }\left[ -C_{\alpha }{\left( x-s\right) }^{\vartheta -1}\right] }{\alpha } \\ \left( k^{\beta +1}_{{\hat{P}}}f\right) \left( t\right)&= \frac{p}{\alpha }\int ^t_0{E_{\beta }\left[ -C_{\alpha }{\left( t-s\right) }^{\beta }\right] f\left( s\right) \text{d}s}. \end{aligned}$$

(6)

Definition 3

The Riemann–Liouville (R–L) fractional integral operator [10] of order $\alpha >0$ of function $f:{R}^+\rightarrow R$ is defined as

$$\begin{aligned} I^{\alpha }f{ (x)}=\frac{1}{{\Gamma }\left( \alpha \right) }\int ^x_0{{\left( x-t\right) }^{\alpha -1}}f\left( t\right) \text{d}t. \end{aligned}$$

(7)

In similar fashion for 0 ${<}$ $\alpha < 1$, the R–L fractional derivative operator of order $\alpha$ is given as

$$\begin{aligned} D^{\alpha }f {(x)}=\frac{1}{{\Gamma }\left( 1-\alpha \right) }\frac{d}{\text{d}x}\int ^x_0{{\left( x-t\right) }^{-\alpha }}f\left( t\right) \text{d}t. \end{aligned}$$

(8)

Definition 4

The Grunwald–Letnikov (G–L) represents a function via a weighted sum around the function s. It is an appropriate tool for application in image processing. The G–L [11] defined $\alpha$ order differential of signal s(t) as

$$\begin{aligned} {}^G_0{D^{\alpha ,s\left( x\right) }_t}= \text{lim}_{h\rightarrow 0} h^{-\alpha }\;\sum ^{n-1}_{m=0}{\frac{{\left( -1\right) }^m{\Gamma }\left( \alpha +1\right) }{{\Gamma }\left( m+1\right) {\Gamma }\left( \alpha -m+1\right) }}\;s(t-mh), \end{aligned}$$

(9)

where the length of s(t) is [0, t],  $\alpha \in R$, $h = (t-a) /n$ is the step size.

Definition 5

For $0 {<}$ $\alpha <1$, Alexander fractional polynomial equation [19] is given as

$${\Delta }^{\alpha }(t) =\prod ^{d-1}_{m=1}{{{\Delta }^{\alpha}_m}}{\left( t\right) }^{l_m},\; m=1, 2 {\dots } d---1,$$

(10)

where $l_m$ is positive integer and

$$\begin{aligned} {\Delta }^{\alpha }_m\left( t\right) =\left( t-E_{\alpha }\left( \frac{2m\pi i}{d}\right) \right) \left( t-E_{\alpha }\left( \frac{-2m\pi i}{d}\right) \right) . \end{aligned}$$

(11)

$E_{\alpha }(t)$ is the M-L function which is given as

$$\begin{aligned} E_{\alpha } {(t)}=\sum ^{\infty }_{m=0}{\frac{t^m}{{\Gamma }\left( \alpha m+1\right) }}. \end{aligned}$$

(12)

The value of d in the above equation is taken as (12) to generate (11) kernel coefficients using the above equation. The coefficient ${\varvec{\Delta }}_{{\varvec{m}}}$ follows the cyclic nature for $d {>}$ 12.

Definition 6

Adaptive fractional differential function [15] is constructed as follows. Let piece wise function of $\vartheta$ order fractional differentiation is given as

$$\vartheta =\left\{ \begin{array}{ll} \frac{M\left( i,j\right) -t}{M(i,j)}, & M\left( i,j\right) \ge t\; \text{and}\; \frac{M\left( i,j\right) -t}{M(i,j)}\ge {\vartheta }_1 \\ {\vartheta }_1, & M\left( i,j\right) \ge t\; \text{and}\; \frac{M\left( i,j\right) -t}{M(i,j)}<{\vartheta }_1 \\ {\vartheta }_{2,} & 2<M\left( i,j\right)<t\; \text{and}\; \frac{M\left( i,j\right) }{t}\ge {\vartheta }_2 \\ \frac{M\left( i,j\right) }{t}, & 2<M\left( i,j\right)<t\; \text{and}\; \frac{M\left( i,j\right) }{t}<{\vartheta }_2 \\ 0, & 0\le M\left( i,j\right) \le 2 \end{array} \right. ,$$

(13)

where t is the optimal threshold gradient of edges in the image, ${\vartheta }_{{1}}$ is the threshold of edge image computed through the adaptive fractional differential order and ${\vartheta }_{{2}}$ is the threshold of weak texture as the result of adaptive fractional order. Further ${\vartheta }_{{1}}$ and ${\vartheta }_{{2}}$ are given as

$${\vartheta }_{{1}} {=}\frac{M_{\text{ed}} -Q}{M_{\text{ed}}}, \quad{\vartheta }_{{2}} {=}\frac{{Q -M}_{\text{tex}}}{Q},$$

where Q is the average gradient of an image and $M_{\text{ed}}\; \text{and} \; M_{\text{tex}}$ are the average gradient of edge pixels and texture pixel segment.

3 Fractional kernel

This section explains the method for finding out the coefficients of the fractional integral based kernels corresponding to the generalized K operator for the proposed algorithm and its pattern.

By using the symmetric property in Eq. (6), we get

$$\begin{aligned} \left( k^{\beta +1}_{{\hat{P}}}f\right) \left( t\right) =\frac{p}{\alpha }\int ^t_0{E_{\beta }\left[ -C_{\alpha }{\left( s\right) }^{\beta }\right] f\left( t-s\right) \text{d}s}. \end{aligned}$$

(14)

Let we divide the interval [0, t] in to an n equal subintervals

$$\begin{aligned} \left( k^{\beta +1}_{{\hat{P}}}f\right) \left( t\right) =\frac{p}{\alpha }\sum ^{n-1}_{k=0}{\int ^{\frac{\left( k+1\right) t}{n}}_{\frac{kt}{n}}{E_{\beta }\left[ -C_{\alpha }{\left( s\right) }^{\beta }\right] f\left( t-s\right) \text{d}s.}} \end{aligned}$$

(15)

On approximating f(t) on $\left[ \frac{kt}{n},\frac{\left( k+1\right) t}{n}\right]$ by $\frac{1}{2}\left[ f\left( \frac{kt}{n}\right) +f\left( \frac{\left( k+1\right) t}{n}\right) \right]$ and using Mittag-Leffler definition

$$\begin{aligned} \int ^{\frac{\left( k+1\right) t}{n}}_{\frac{kt}{n}}{E_{\beta }\left[ -C_{\alpha }{\left( s\right) }^{\beta }\right] f\left( t-s\right) \text{d}s}& = \sum ^{\infty }_{j=0}{\frac{{\left( -C_{\alpha }\right) }^j}{{\varGamma }\left( \beta j+1\right) }\int ^{\frac{\left( k+1\right) t}{n}}_{\frac{kt}{n}}{{\left( s\right) }^{\beta j}f\left( t-s\right) \text{d}s}} \\ \left( k^{\beta +1}_{{\hat{P}}}f\right) \left( t\right)&= \frac{p}{\alpha }\sum ^{\infty }_{j=0}{\frac{{\left( -C_{\alpha }\right) }^j}{{\varGamma }\left( \beta j+1\right) }\left\{ \sum ^{n-1}_{k=0}{\int ^{\frac{\left( k+1\right) t}{n}}_{\frac{kt}{n}}{{\left( s\right) }^{\beta j}f\left( t-s\right) \text{d}s}}\right\} } \\&= \frac{p}{\alpha }\sum ^{\infty }_{j=0}{\frac{{\left( -C_{\alpha }\right) }^j}{{\varGamma }\left( \beta j+1\right) }\left\{ \sum ^{n-1}_{k=0}{\frac{1}{2\left( \beta j+1\right) }}\left( f\left( t-\frac{kt}{n}\right) +f\left( t-\frac{\left( k+1\right) t}{n}\right) \right) \right. } \\ & \quad \times \left. \left( {\left( \frac{\left( k+1\right) t}{n}\right) }^{\beta j+1}-{\left( \frac{kt}{n}\right) }^{\beta j+1}\right) \right\} . \end{aligned}$$

(16)

On taking $\frac{t}{n}=1$.

$$\begin{aligned} \left( k^{\beta +1}_{{\hat{P}}}f\right) \left( t\right)& =\frac{p}{2\alpha }\sum ^{\infty }_{j=0}\frac{{\left( -C_{\alpha }\right) }^j}{\left( \beta j+1\right) {\varGamma }\left( \beta j+1\right) }\left\{ \sum ^{n-1}_{k=0}{\left( f\left( t-k\right) +f\left( t-\left( k+1\right) \right) \right) }\right. \\&\quad \times \left. \left( {\left( k+1\right) }^{\beta j+1}-{\left( k\right) }^{\beta j+1}\right) \right\} \\&=\frac{p}{2\alpha }\sum ^{\infty }_{j=0}{\frac{{\left( -C_{\alpha }\right) }^j}{{\varGamma }\left( \beta j+2\right) }\left\{ \sum ^{n-1}_{k=0}{\left( f_k+f_{k+1}\right) }\left( {\left( k+1\right) }^{\beta j+1}-{\left( k\right) }^{\beta j+1}\right) \right\} }, \end{aligned}$$

where, $f_k=f(t-k)$.

$$\begin{aligned} \left( {\kappa }^{\beta +1}_{{\hat{P}}}f\right) \left( t\right)& =\frac{p}{2\alpha }\sum ^{n-1}_{k=0} \left( f_k+f_{k+1}\right) \\& \quad \times \left\{ \left( \left( k+1\right) E_{\beta ,2}\left[ -C_{\alpha }{\left( k+1\right) }^{\beta }\right] -\left( k\right) E_{\beta ,2}\left[ -C_{\alpha }{\left( k\right) }^{\beta }\right] \right) \right\} , \end{aligned}$$

(17)

where $E_{\rho ,\sigma }\left( t^{\rho }\right) =\sum ^{\infty }_{j=0}{\frac{t^{\rho j}}{{\varGamma }\left( \rho j+\sigma \right) }}$ is generalized Mittag-Leffler function of two parameter family.

$$\begin{aligned} \left( K^{\beta +1}_{{\hat{P}}}f\right) \left( t\right) &=\frac{p}{2\alpha }\left[ \left( f_0+f_1\right) E_{\beta ,2}\left( -C_{\alpha }\right) +\left( f_1+f_2\right) \right. \\& \quad \times \left( {2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }{\left( 1\right) }^{\beta }\right) \right) \\ & \quad +\left( f_2+f_3\right) \left( {3E}_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -{2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) \\ & \quad \left. +\left( f_3+f_4\right) \left( {4E}_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right) +\dots \right] \end{aligned}$$

(18)

$$\begin{aligned} \left( {\kappa }^{\beta +1}_{{\hat{P}}}f\right) \left( t\right)& =\frac{p}{2\alpha }\left[ {f_0E}_{\beta ,2}\left( -C_{\alpha }\right) +f_1\left( {2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) \right. \\& \quad +f_2\left( {3E}_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }{\left( 1\right) }^{\beta }\right) \right) \\& \quad +f_3\left( {4E}_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) \\& \quad +f_4\left( 5E_{\beta ,2}\left( -C_{\alpha }{\left( 5\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right) \dots \nonumber \\& \quad \left. +f_n\left( \left( n+1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n+1\right) }^{\beta }\right) -\left( n-1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n-1\right) }^{\beta }\right) \right) \right] \end{aligned}$$

(19)

$$\begin{aligned} \left( {\kappa }^{\beta +1}_{{\hat{P}}}f\right) \left( t\right) & =\frac{p}{2\alpha }\left[ E_{\beta ,2}\left( -C_{\alpha }\right) f\left( t\right) +\left( {2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( t-1\right) \right. \\ & \quad + \left( {3E}_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }{\left( 1\right) }^{\beta }\right) \right) f\left( t-2\right) \\ & \quad+\left( 4E_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( t-3\right) \\ & \quad +\left( 5E_{\beta ,2}\left( -C_{\alpha }{\left( 5\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right) f\left( t-4\right) \dots \\ & \quad \left. +\left( \left( n+1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n+1\right) }^{\beta }\right) -\left( n-1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n-1\right) }^{\beta }\right) \right) f\left( t-n\right) \right] .\\ C_n & = \frac{p}{2\alpha }\left[ \left( n+1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n+1\right) }^{\beta }\right) -\left( n-1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n-1\right) }^{\beta }\right) \right] . \end{aligned}$$

(20)

Thus, the coefficients of the filter kernel are given as:

$$\begin{aligned}&C_0=\frac{p}{2\alpha }\left[ E_{\beta ,2}\left( -C_{\alpha }\right) \right] \\ & C_1=\frac{p}{2\alpha }\left[ 2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right] \nonumber \\&C_2=\frac{p}{2\alpha }\left[ 3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }\right) \right] \\ & C_3=\frac{p}{2\alpha }\left[ 4E_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right] \\ & C_4=\frac{p}{2\alpha }\left[ 5E_{\beta ,2}\left( -C_{\alpha }{\left( 5\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right] . \end{aligned}$$

(21)

The above mentioned coefficients of fractional integral kernel ${C}_{0}$, ${C}_{1}$, ${C}_{2}$, ${C}_{3}$,...${C}_{n}$ are now used to remove the noise by the proposed algorithm. The pattern of such kernels of size (7 × 7) is shown in Fig. 1.

Fig. 1

Proposed pattern for kernel

As the image is a 2-D signal, so it requires a kernel of two dimensions which can remove the noise in both directions. Fractional order integral of 2-D signal along with X- and Y direction is given as-

$$\begin{aligned} {\kappa }^{\beta +1}_{{\hat{P}}}f_x\left( x,y\right) & =\frac{p}{2\alpha }\left[ E_{\beta ,2}\left( -C_{\alpha }\right) f\left( x,y\right) +\left( {2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( x-1,y\right) \right. \\ & \quad +\left( {3E}_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }{\left( 1\right) }^{\beta }\right) \right) f\left( x-2,y\right) \\ & \quad +\left( 4E_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( x-3,y\right) \\ & \quad +\left( 5E_{\beta ,2}\left( -C_{\alpha }{\left( 5\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right) f\left( x-4,y\right) \dots \\ & \quad +\left( \left( n+1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n+1\right) }^{\beta }\right) -\left( n-1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n-1\right) }^{\beta }\right) \right) \\& \quad \left. \times f\left( x-n, y\right) \right] . \end{aligned}$$

(22)

Similarly,

$$\begin{aligned} {\kappa }^{\beta +1}_{{\hat{P}}}f_y\left( x,y\right) & =\frac{p}{2\alpha }\left[ E_{\beta ,2}\left( -C_{\alpha }\right) f\left( x,y\right) +\left( {2E}_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( x,y-1\right) \right. \\ & \quad +\left( {3E}_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) -E_{\beta ,2}\left( -C_{\alpha }{\left( 1\right) }^{\beta }\right) \right) f\left( x,y-2\right) \\ & \quad +\left( 4E_{\beta ,2}\left( -C_{\alpha }{\left( 4\right) }^{\beta }\right) -2E_{\beta ,2}\left( -C_{\alpha }{\left( 2\right) }^{\beta }\right) \right) f\left( x,y-3\right) \\ & \quad +\left( 5E_{\beta ,2}\left( -C_{\alpha }{\left( 5\right) }^{\beta }\right) -3E_{\beta ,2}\left( -C_{\alpha }{\left( 3\right) }^{\beta }\right) \right) f\left( x,y-4\right) \dots \\ & \quad +\left( \left( n+1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n+1\right) }^{\beta }\right) -\left( n-1\right) E_{\beta ,2}\left( -C_{\alpha }{\left( n-1\right) }^{\beta }\right) \right) \\ & \quad \left. f\left( x,y-n\right) \right] \end{aligned}$$

(23)

4 Noise removal algorithm

This section deals with the development of the algorithm of noise removal for images. Let a function be defined on the following two sets of angle for each pixel $f(x,y)\; \notin$ corners, as follows:

$$\begin{aligned}{} & {} {\phi }_1=\left\{ \phi \,\phi =\frac{n\pi }{2},\; n=0,1,2,3\right\} \end{aligned}$$

(24)

$$\begin{aligned}{} & {} {\phi }_2=\left\{ \phi \,\phi =\frac{\pi }{4}+\frac{\left( n-1\right) \pi }{2},\; n=1,\;2,\;3,\;4 \right\} \end{aligned}$$

(25)

$$\begin{aligned}{} & {} \varkappa :{\phi }_1\cup {\phi }_2\rightarrow R, \end{aligned}$$

(26)

where

$$\begin{aligned} \varkappa \left( \phi \right) =\text{min}\left\{ \left| h-h_1\right| ,\left| h-h_2\right| \right\} \; \text{for}\; \phi ={\phi }_1 \end{aligned}$$

(27)

and

$$\begin{aligned} \varkappa \left( \phi \right)= \text{min}\left\{ \left| h-h_3\right| ,\left| h-h_4\right| \right\} \; \text{for}\; \phi ={\phi }_2 \end{aligned}$$

(28)

$$\begin{aligned} h & = f\left( x,y\right) ,\\ {h}_1 & = f\left( x- \sin\left( \phi \right) ,y+\cos\left( \phi \right) \right) , \\ h_2 & = f\left( x-2\sin\left( \phi \right) ,y+2\cos\left( \phi \right) \right) , \\ h_3 & = f\left( x-\sqrt{2}\sin\left( \phi \right) ,y+\sqrt{2}\cos\left( \phi \right) \right) , \\ h_4 & = f\left( x-2\sqrt{2}\sin\left( \phi \right) ,y+2\sqrt{2}\cos\left( \phi \right) \right) . \end{aligned}$$

(29)

The algorithm consists of the following steps:

• Step 1: We let ${{f}}_{{n}}(x,y)$ denote the image with noise and the size is $m{\times }n$.

• Step 2: For all 1${\le } x {\le }m$, and $1{\le } y$ ${\le }n$, the pixel $f{}_{n}(x,y)$ of intensity with value either 0 or 255. Let $f{}_{n}(x,y)$ be a noisy pixel only if

  $$\begin{aligned} 10 {<}{\text{max}}_{{\phi }{\in }{{\phi }}_{{1}}{\cup }{{\phi }}_{{2}}}{\varkappa }\left( {\phi }\right) {<} 240. \end{aligned}$$

  (30)

  The intensity value f(x, y) that belongs to noise is replaced by the convoluted value with a kernel of 5 × 5 with $f{}_{n}(x, y)$ as the central pixel. All pixels with intensity value will remain unchanged except the pixel with intensity values 0 or 255.

• Step 3: Stop the algorithm, if the stopping conditions satisfy.

5 Experimental results

In this section, we demonstrate the performance of the fractional kernel due to Caputo-Fabrizio and the proposed algorithm on different images. In the experiment, nine standard sampled images are taken to compare the results of the proposed algorithm. The proposed algorithm is compared to seven relevant algorithms viz. Median Filter (MF) [21], Riemann Liouville Filter (RLF) [11], Improved Grunwald Letnikov filter (IGLF) [22], Alexander fractional differential filter (AFDF) [14], Alexander fractional integral filter (AFIF) [14], Algorithm based on small probability strategy (AFCSPS) [15] and ENAFC [23], Algorithm based on small probability strategy (AFC-SPS) and Algorithm based on entropy and gradient feature (ENA-FC) for validation. The four most popular performance parameters MSE, PSNR, SSIM, and IEF are taken to analyze the performance related to denoising. From the experimental study, it has been found that the 5 × 5 size kernel performs better for 256 × 256 pixels size images, whereas for higher dimension images 9 × 9 performs better.

The MSE is defined as follows:

$$\begin{aligned} \text{MSE}=\frac{1}{mn}\sum ^m_{x=1}{\sum ^n_{y=1}{{\left( f\left( x,y\right) -f_0\left( x,y\right) \right) }^2}}, \end{aligned}$$

(31)

where $f\left( x,y\right)$ is the denoised image and $f_0\left( x,y\right)$ is the sample image without noise.

The PSNR for an 8-bit image is given as:

$$\begin{aligned} \text{PSNR}=10 \log\left( \frac{{255}^2}{\text{MSE}}\right) . \end{aligned}$$

(32)

The SSIM is given by

$$\begin{aligned} \text{SSIM}=\frac{\left( 2{\mu }_x{\mu }_y+A_1\right) +\left( 2{\sigma }_{xy}+A_2\right) }{\left( {{\mu }_x}^2+{{\mu }_y}^2+A_1\right) +\left( {{\sigma }_x}^2+{{\sigma }_y}^2+A_2\right) }, \end{aligned}$$

(33)

where ${\sigma }_x,\; {\text{and}\; \sigma }_y$ are the standard deviations, ${\sigma }_{xy},$ is the cross covariance and ${\mu }_{x}$ and ${\mu }_y$ are the local means of the sampled image without noise and denoise image.

The IEF is defined as:

$$\begin{aligned} \text{IEF}= \frac{\sum ^m_{x=1}{\sum ^n_{y=1}{{\left( f_n\left( x,y\right) -f_0\left( x,y\right) \right) }^2}}}{\sum ^m_{x=1}{\sum ^n_{y=1}{{\left( f\left( x,y\right) -f_0\left( x,y\right) \right) }^2}}}, \end{aligned}$$

(34)

where $f_n\left( x,y\right)$ is the noisy image, ${f}_0\left( x,y\right)$ is the sampled image and $f\left( x,y\right)$ is the denoise image.

For experimental purposes, we added 10% noise density in the cameraman image to check the performance of the proposed kernel. In this paper, we tried to optimize the value of p, $\alpha$, and $\beta$ to get better results. In the first experiment, we changed the value of p by keeping a random value of $\alpha= 0.75$ and $\beta= 0.5$ constant. The standard image ‘cameraman’ is taken for optimizing the performance on the basis of the following parameters PSNR, SSIM, and IEF. The values of MSE, PSNR, SSIM and IEF are shown in Table 1. Results from the table show that the results are better for p = 0.3.

Table 1 Analysis of parameters for $p = [0,1]$ at a fixed value of $\alpha= 0.75$ and $\beta = 0.5$

After optimizing the value of p, in the second experiment, we kept $\alpha= 0.75$ and p = 0.3, and changed the value of $\beta$ from 0.1 to 1 with a difference of 0.1. The same ‘cameraman’ has been taken to optimize the performance. The values of MSE, PSNR, SSIM and IEF are shown in Table 2. Results from the table show that the results are better for $\beta= 0.2$ .

Table 2 Analysis of parameters for $\beta = [0,1]$ at a fixed value of $\alpha = 0.75$ and p = 0.3

In the third experiment, we optimized the final parameter $\alpha$ by keeping $\beta= 0.2$ and p = 0.3, and changed the value of $\alpha$ from 0.1 to 1 at a difference of 0.05 for the above-mentioned ‘cameramen’ image. The values of MSE, PSNR, SSIM and IEF are shown in Table 3. Results from the table show that results are better for $\alpha=0.75$.

Table 3 Analysis of parameters for $\alpha = [0,1]$at a fixed value of $\beta =0.2$ and p = 0.3

The performance of MSE for different values of p, $\alpha$ and $\beta$ are plotted in Fig. 2. The figure clearly indicates that the mean square error is minimum at values of p= 0.3, $\alpha$=0.75 and $\beta$=0.2.

Fig. 2

Performance analysis of Mean Square error for different values of parameters p, $\alpha$ and $\beta$

The performance of PSNR for different values of p, $\alpha$ and $\beta$ are plotted in Fig. 3. The figure clearly indicates that Peak Signal-to-Noise Ratio is maximum at values of p = 0.4, $\alpha =0.75$ and $\beta =0.1$.

Fig. 3

Performance analysis of Peak Signal-to-Noise Ratio for different values of parameters p, $\alpha$ & $\beta$

The performance of SSIM for different values of p, $\alpha$ and $\beta$ are plotted in Fig. 4. The figure clearly indicates that Structural Similarity Index Measure is maximum at values of p = 0.3, $\alpha =0.75$ and $\beta =0.2$.

Fig. 4

Performance analysis of Structural Similarity Index Measure for different values of parameters p, $\alpha$ and $\beta$

The performance of IEF for different values of p, $\alpha$ and $\beta$ are plotted in Fig. 5. The figure clearly indicates that the Image Enhancement Factor is maximum at values of p = 0.3, $\alpha= 0.75$ and $\beta = 0.2$.

Fig. 5

Performance analysis of Image Enhancement Factor for different values of parameters p, $\alpha$ and $\beta$

In Table 4, the nine standard images are taken and compared with their performance on the basis of performance parameters as mentioned above. The value of PSNR shown in Table 4, of all the images, denoise by the proposed kernel is improved than all discussed methods. The value of IEF is quite high in comparison to other methods. SSIM is near to the one which is also higher than the other methods, this validates our claim that our proposed method is more efficient to remove the noise with different noise density, and at the same time it preserves the shape, texture and other useful information.

Table 4 The values of performance parameter of test images denoised by prevailing filters at 10% noise density

Fig. 6, explicitly depicted that the images denoises by the presented algorithm are much improved than the other prevailed filters, hence on visual perception, we can claim that the proposed method is much better than the others mentioned in the table. For experimental purposes, we added 10% noise density in the above mentioned images as shown in Fig. 2. (aL2), (bC2), (cB2), (dH2), (eE2), (fT2), (gL2), (hP2) and (iB2) to get the corresponding output through the proposed filter.

Fig. 6

Results using an optimized kernel with salt and pepper noise (ND = 0.1), where Grey-scale image of size (256x256) pixels (aL1). ‘Lena’, (bC1). ‘Cameraman’, (cB1). ‘Blonde’, (dH1). ‘House’, (eE1). ‘Elaine’, (fT1). ‘Tank’, (gL1). ‘Living room’, (hP1). ‘Pirate’, (iB1). ‘Boats’ are the Original images. (aL2), (bC2), (cB2), (dH2), (eE2), (fT2), (gL2), (hP2), and (iB2) are Images with the Salt and Pepper noise. (aL3), (bC3), (cB3), (dH3), (eE3), (fT3), (gL3), (hP3) and (iB3) are the corresponding outputs of the proposed filter

In the next experiment as tabulated in Table 5, the proposed kernel is tested with the optimized values of p, $\alpha$ and $\beta$ on all mentioned images for MSE, PSNR, SSIM and IEF at different level of noise density viz. 2%, 10%, 20%, 50% and 90%. The tabulated results explicitly depict that the image denoise by the present algorithm is much better, even at 90% noise density. The highest image recovery is 25%, that is, in the case of tank image.

Table 5 Performance of the proposed kernel at different noise density

6 Conclusion

The present research work proposed an optimization analysis for different fractional orders for an image noise removal algorithm involving the generalized Caputo-Fabrizio fractional operator. The fractional kernel related to Caputo-Fabrizio fractional operator has been developed by equation (21). First, we have optimized the values of different parameters (p, $\alpha$ and $\beta$ ) related to the image denoising algorithm involving Caputo-Fabrizio fractional integral operator of a non-singular type with the Mittag-Leffler function and then the results of the final kernel, are compared with the existing filters viz. Median Filter (MF), RLF, IGLF, AFDF, AFIF, AFC-SPS and EN-AFC. The performance of the standard parameters PSNR, SSIM, and IEF along with the visual perception for nine standard images are taken and compared, as shown in Table 4. The value of PSNR is shown in Table 4, of each single image, denoised by the proposed method is improved than all other prevailing methods. The performance parameters, i.e., IEF and SSIM, are quite high compared to other existing methods. SSIM is near to one because the kernel preserves the shape, texture, and other useful information. The results in the Table 5, are explicitly depicted that the image denoised by the present algorithm is much better, even at 90% noise density. The highest image recovery is 25%, that is, in the case of Tank. The SSIM yielded values ranging from 0.72 to 0.88 when evaluating the proposed method across various images. These results indicate superior stability and performance in comparison to alternative approaches. Thus, our results validate our claim that our proposed method is more efficient to remove the noise.