1 Introduction
In the realm of image processing and computer vision, the pursuit of enhancing image quality has been an ongoing endeavour. From the restoration of blurred images to the reduction of noise interference, researchers continually strive to develop sophisticated techniques that can preserve and enhance visual information effectively. One such approach gaining attention is the utilization of fractional derivation in constructing masks for image denoising and deblurring.
Traditionally, linear filtering techniques, such as Gaussian or bilateral filters, have been extensively employed for image denoising and deblurring (Buades et al., 2005). However, these techniques assume that the degradation process and noise follow Gaussian distributions, which might not always reflect real-world scenarios. To address this limitation, researchers have turned their attention towards more advanced methods, such as fractional derivation, to construct masks that facilitate the denoising and deblurring processes.
The concept of fractional calculus, an extension of traditional calculus, has gained traction in various scientific disciplines due to its ability to capture complex, non-local behaviours and characteristics (Podlubny, 1998). By leveraging fractional derivatives and integrals, researchers have been able to tackle challenging problems that evade conventional approaches. In the context of image denoising and deblurring, fractional derivation provides a valuable tool for modelling and capturing intricate image structures and features at different scales.
An image edge analysis method based on Riemann-Liouville fractional derivative was introduced in Amoako-Yirenkyi et al. (2016) that utilizes a fractional derivative mask. Chowdhury et al. (2020) employed fractional-order total variation to address non-blind and blind deconvolution challenges in the presence of Poisson noise. A new image encryption system that combines fractional-order edge detection with generalized chaotic maps has been presented in the referenced paper (Ismail et al., 2020). Ibrahim (2020) introduce a novel image denoising model that utilizes conformable fractional calculus to address multiplicative noise. An adaptive approach to image restoration using fractional-order total variation and the split Bregman iteration has been presented in Li et al. (2018). Wadhwa and Bhardwaj’s research focuses on enhancing MRI images of brain tumours using the Grünwald-Letnikov fractional differential mask (Wadhwa and Bhardwaj, 2020). Liu et al. proposed a blind deblurring method that utilizes fractional-order calculus and a local minimal pixel prior (Liu et al., 2022). Yang et al. (2016) provided a comprehensive review of the application of fractional calculus in image processing. A novel fractional-order mask for image edge detection, utilizing the Caputo-Fabrizio fractional-order derivative without a singular kernel, was introduced in Lavín-Delgado et al. (2020). We refer to (Aboutabit, 2021; Arora et al., 2022; Chandra and Bajpai, 2018; Gholami Bahador et al., 2022; Golami Bahador et al., 2023; Hacini et al., 2020; Irandoust–pakchin et al., 2021; Li and Wang, 2023; Nema et al., 2020) and references in them for more works done in the image processing with fractional derivatives. There are several other methods for image deblurring. For instance, Pooja et al. (2023) present a novel image deblurring algorithm that utilizes region-specific priors and methodologies for enhanced image correction. Zhou et al. (2023) used events to deblur low-light images. The authors introduce a robust unified two-stage framework and a motion-aware neural network designed to reconstruct sharp images by utilizing high-fidelity motion information derived from event data. The authors in Ren et al. (2023) employed multiscale structure-guided diffusion techniques for the purpose of image deblurring. Dong et al. (2023) implemented a Multi-Scale Residual Low-Pass Filter Network to address image deblurring challenges. Li et al. (2023) introduced a novel framework that leverages deep learning techniques to address ongoing challenges in the field, including the limitations of existing methods in handling real-world blur and the problems of over- and under-estimating blur.
In general, constructing masks and applying them for image processing is one of the easiest methods requiring relatively few computations compared to other approaches. However, it is important to note that constructing deblurring masks using Riemann-Liouville and Caputo fractional derivatives can be quite complex, whereas using Grünwald-Letnikov fractional derivatives is more straightforward.
In this paper, we developed an adaptive mask based on Grünwald-Letnikov fractional derivatives and carefully selected the order of the fractional derivatives using the Wakeby distribution. This approach leads to improved results, particularly when compared to other mask construction methods utilizing Grünwald-Letnikov fractional derivatives.
This article is organized as follows: In the next section, we describe the definition of the Grünwald-Letnikov (GL) fractional derivative. In Section 3, we delve into the construction of a mask using the GL fractional derivative and present our proposed method. In Section 4, we present our experimental results and provide a detailed discussion of them.
2 Fundamental Definition of GL Fractional Derivative
The definition of fractional derivatives can vary depending on the context. Three commonly used definitions are the GL definition, the Riemann-Liouville (R-L) definition, and the Caputo definition. Among these, the GL definition is often considered the most appropriate for image processing applications (Frackiewicz and Palus, 2024; Mortazavi et al., 2023; Zuffi et al., 2024). The Caputo fractional derivative is typically employed in fields such as control theory (Moon, 2023; Frederico, 2008; Kamocki and Majewski, 2015; Sweilam et al., 2021; Bergounioux and Bourdin, 2020), viscoelastic materials (Freed and Diethelm, 2007; Li and Ma, 2023; Mahiuddin et al., 2020; Bhangale et al., 2023), and biological systems (Yusuf et al., 2021; Qureshi, 2020; Rahman et al., 2022; Uçar and Özdemir, 2021). In contrast, the Riemann-Liouville fractional derivative is predominantly utilized in the realms of physics and engineering (Ahmad et al., 2021; Gu et al., 2019; Khan et al., 2024; Liu et al., 2011). Caputo Fractional Derivative usually used in the field of control theory (Moon, 2023; Frederico, 2008; Kamocki and Majewski, 2015; Sweilam et al., 2021; Bergounioux and Bourdin, 2020), Viscoelastic Materials (Freed and Diethelm, 2007; Li and Ma, 2023; Mahiuddin et al., 2020; Bhangale et al., 2023), Biological Systems (Yusuf et al., 2021; Qureshi, 2020; Rahman et al., 2022; Uçar and Özdemir, 2021).
For a function $\tau (t)$, where $t\in R$ and $\tau (t)$ is differentiable n times, the derivatives of different orders can be expressed as follows:
\[\begin{aligned}{}& {\tau ^{\prime }}(t)=\underset{h\to 0}{\lim }\frac{\tau (t)-\tau (t-h)}{h},\\ {} & {\tau ^{\prime\prime }}(t)=\underset{h\to 0}{\lim }\frac{\tau (t)-2\tau (t-h)+\tau (t-2h)}{{h^{2}}},\\ {} & {\tau ^{\prime\prime\prime }}(t)=\underset{h\to 0}{\lim }\frac{\tau (t)-3\tau (t-h)+3\tau (t-2h)-\tau (t-3h)}{{h^{3}}},\end{aligned}\]
and using the concept of mathematical induction, the formula for the nth order derivative is given as:where
The concept has the potential for extension to the widely used GL fractional derivative of order v, enabling the calculation of a non-integer number of derivatives. This is expressed as Atici et al. (2021):
where
For the function $\tau (t)$ where 
, the truncated GL fractional derivative of order α is:
\[ {D^{\alpha }}\tau (t)=\underset{h\to 0}{\lim }\frac{{\textstyle\textstyle\sum _{k=0}^{m}}{(-1)^{k}}\left(\begin{array}{c}\alpha \\ {} k\end{array}\right)\tau (t-kh)}{{h^{\alpha }}},\]
where $m=[(t-{t_{0}})/h]$. Also ${D^{\alpha }}\tau (t)$ can be represented as
\[ {D^{\alpha }}\tau (t)=\underset{h\to 0}{\lim }\frac{{\textstyle\textstyle\sum _{k=0}^{m}}\left(\begin{array}{c}-\alpha \\ {} k\end{array}\right)\tau (t-kh)}{{h^{\alpha }}},\]
where 3 Novel Approach
In this section, we outline our approach for constructing a fractional differential mask using the Grünwald-Letnikov fractional derivative, and we also present a method to choose a suitable value for the order of the fractional derivative at different points in the images.
3.1 Development of Fractional Differential Masks
Suppose $\tau (t)\in [{t_{0}},t]$, where ${t_{0}}\lt t$ and ${t_{0}},t\in \mathbb{R}$. The fractional differential operator of order α can be expressed mathematically as:
Let’s consider $\tau (t)\in [{t_{0}},t]$, where ${t_{0}}\lt t$ and ${t_{0}},t\in \mathbb{R}$, as a one-dimensional signal. The fractional differential operator of order α for a digital image can be mathematically depicted as:
\[\begin{aligned}{}\frac{{d^{\alpha }}\tau (t)}{d{t^{\alpha }}}& \approx \tau (t)+\frac{(-\alpha )}{1!h}\tau (t-h)+\frac{(-\alpha )(-\alpha +1)}{2!{h^{2}}}\tau (t-2h)\\ {} & \hspace{1em}+\cdots +\frac{(-\alpha )(-\alpha +1)\dots (-\alpha +m-1)}{m!{h^{m}}}\tau (t-mh).\end{aligned}\]
In the realm of digital images, the ${D_{8}}$ (or Chessboard) norm is defined as:
where $({x_{1}},{y_{1}})$ and $({x_{2}},{y_{2}})$ are the coordinates of two pixels. In this metric, the distance between neighbouring pixels—whether they are adjacent vertically, horizontally, or diagonally—is equivalent to 1. This particular distance is commonly denoted as h, where $h=1$. So
\[\begin{aligned}{}\frac{{d^{\alpha }}\tau (t)}{d{t^{\alpha }}}& \approx \tau (t)+\frac{(-\alpha )}{1!}\tau (t-1)+\frac{(-\alpha )(-\alpha +1)}{2!}\tau (t-2)\\ {} & \hspace{1em}+\cdots +\frac{(-\alpha )(-\alpha +1)\dots (-\alpha +m-1)}{m!}\tau (t-m),\end{aligned}\]
or
where Γ denotes the Gamma function, defined as
In a 2D digital image, the backward differences of fractional differentiation for x- and y-directions can be found as:
and
To create a mask of size $m\times m$, the initial m coefficients from the Eqs. (1) and (2) are considered. Table 1 presents the $5\times 5$ templates for the x and y directions, as well as the diagonal direction.
The pixel value in an image is influenced by its neighbouring pixels, with the influence decreasing as the distance from the pixel increases. The template in the x direction is centred at the fifth row and third column in Table 1 (the first mask). There are 5 pixels located at a distance of one pixel from the centre, and the weight $-\alpha $ is equally divided among these neighbours, as the centre pixel’s value depends equally on all of them. Similarly, there are 9 pixels at a distance of two pixels from the centre, and the weight $\frac{\alpha (\alpha -1)}{2}$ is equally distributed among them. Finally, there are 5 pixels at a distance of three and four pixels from the centre, and the weights $-\frac{\alpha (\alpha -1)(\alpha -2)}{3}$ and $\frac{\alpha (\alpha -1)(\alpha -2)(\alpha -3)}{4}$ are equally distributed among them, respectively.
Table 1
$5\times 5$ masks in y, x and diagonal direction.
| 0 | 0 | $\frac{\alpha (\alpha -1)(\alpha -2)(\alpha -3)}{4}$ | 0 | 0 |
| 0 | 0 | $-\frac{\alpha (\alpha -1)(\alpha -2)}{3}$ | 0 | 0 |
| 0 | 0 | $\frac{\alpha (\alpha -1)}{2}$ | 0 | 0 |
| 0 | 0 | $-\alpha $ | 0 | 0 |
| 0 | 0 | 1 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 |
| 1 | $-\alpha $ | $\frac{\alpha (\alpha -1)}{2}$ | $-\frac{\alpha (\alpha -1)(\alpha -2)}{3}$ | $\frac{\alpha (\alpha -1)(\alpha -2)(\alpha -3)}{4}$ |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | 0 |
| 0 | 0 | 0 | 0 | $\frac{\alpha (\alpha -1)(\alpha -2)(\alpha -3)}{4}$ |
| 0 | 0 | 0 | $-\frac{\alpha (\alpha -1)(\alpha -2)}{3}$ | 0 |
| 0 | 0 | $\frac{\alpha (\alpha -1)}{2}$ | 0 | 0 |
| 0 | $-\alpha $ | 0 | 0 | 0 |
| 1 | 0 | 0 | 0 | 0 |
In a similar manner, we apply the aforementioned approach to the second mask in the y direction as presented in Table 1. For the last mask in Table 1, there are $3,5,7$, and 9 pixels at distances of one, two, three, and four pixels from the centre. We extend the same concept to this mask, resulting in the derivation of modified masks in the x, y, and diagonal directions, as detailed in Table 2. In Table 2 we put ${\alpha _{j}}=\alpha (\alpha -1)\dots (\alpha -j),j=1,2,3$.
Table 2
Modified mask of size $5\times 5$ in x, y and diagonal direction.
| $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{3}}}{20}$ |
| $-\frac{\alpha }{5}$ | $-\frac{\alpha }{5}$ | $\frac{{\alpha _{1}}}{18}$ | $-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{3}}}{20}$ |
| 1 | $-\frac{\alpha }{5}$ | $\frac{{\alpha _{1}}}{18}$ | $-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{3}}}{20}$ |
| $-\frac{\alpha }{5}$ | $-\frac{\alpha }{5}$ | $\frac{{\alpha _{1}}}{18}$ | $-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{3}}}{20}$ |
| $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{3}}}{20}$ |
| $\frac{{\alpha _{3}}}{20}$ | $\frac{{\alpha _{3}}}{20}$ | $\frac{{\alpha _{3}}}{20}$ | $\frac{{\alpha _{3}}}{20}$ | $\frac{{\alpha _{3}}}{20}$ |
| $-\frac{{\alpha _{2}}}{15}$ | $-\frac{{\alpha _{2}}}{15}$ | $-\frac{{\alpha _{2}}}{15}$ | $-\frac{{\alpha _{2}}}{15}$ | $-\frac{{\alpha _{2}}}{15}$ |
| $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ | $\frac{{\alpha _{1}}}{18}$ |
| $\frac{{\alpha _{1}}}{18}$ | $-\frac{\alpha }{5}$ | $-\frac{\alpha }{5}$ | $-\frac{\alpha }{5}$ | $\frac{{\alpha _{1}}}{18}$ |
| $\frac{{\alpha _{1}}}{18}$ | $-\frac{\alpha }{5}$ | 1 | $-\frac{\alpha }{5}$ | $\frac{{\alpha _{1}}}{18}$ |
| $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ |
| $-\frac{{\alpha _{2}}}{21}$ | $-\frac{{\alpha _{2}}}{21}$ | $-\frac{{\alpha _{2}}}{21}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
| $\frac{{\alpha _{1}}}{15}$ | $\frac{{\alpha _{1}}}{15}$ | $\frac{{\alpha _{1}}}{15}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
| $-\frac{\alpha }{3}$ | $-\frac{\alpha }{3}$ | $\frac{{\alpha _{1}}}{15}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
| 1 | $-\frac{\alpha }{3}$ | $\frac{{\alpha _{1}}}{15}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
Once the masks in the eight directions $\frac{k\pi }{4}$, $k=0,1,2,\dots ,7$, have been acquired, we combine them to create a $9\times 9$ mask as shown in Table 3. Next, we divide each element by the total sum of the coefficients $(8-12\alpha +4{\alpha ^{2}})$ to create a $9\times 9$ differential mask, ensuring that the resulting image’s intensity values fall within the range of $[0,255]$.
Table 3
Resultant mask of size $9\times 9$.
| $\frac{{\alpha _{3}}}{20}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{10}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{19{\alpha _{3}}}{180}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ |
| $\frac{7{\alpha _{3}}}{90}$ | $-\frac{{\alpha _{2}}}{21}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{17{\alpha _{2}}}{105}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
| $\frac{7{\alpha _{3}}}{90}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{{\alpha _{1}}}{5}-\frac{{\alpha _{2}}}{15}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $\frac{3{\alpha _{1}}}{10}-\frac{2\alpha }{9}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $\frac{3{\alpha _{1}}}{10}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{7{\alpha _{3}}}{90}$ |
| $\frac{7{\alpha _{3}}}{90}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $-\frac{5\alpha }{9}$ | $-\alpha $ | $-\frac{5\alpha }{9}$ | $\frac{2{b_{2}}}{5}-\frac{\alpha }{9}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{7{\alpha _{3}}}{90}$ |
| $\frac{19{\alpha _{3}}}{180}$ | $-\frac{17{\alpha _{2}}}{105}$ | $-\frac{2\alpha }{9}+\frac{3{\alpha _{1}}}{10}$ | $-\alpha $ | 8 | $-\alpha $ | $\frac{3{\alpha _{1}}}{10}-\frac{2\alpha }{9}$ | $-\frac{2{\alpha _{2}}}{21}-\frac{{\alpha _{2}}}{15}$ | $\frac{19{\alpha _{3}}}{180}$ |
| $\frac{7{\alpha _{3}}}{90}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $-\frac{5\alpha }{9}$ | $-\alpha $ | $-\frac{5\alpha }{9}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{7{\alpha _{3}}}{90}$ |
| $\frac{7{\alpha _{3}}}{90}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{3{\alpha _{1}}}{10}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $-\frac{2\alpha }{9}+\frac{3{\alpha _{1}}}{10}$ | $\frac{{\alpha _{1}}}{5}-\frac{\alpha }{9}$ | $\frac{3{\alpha _{1}}}{10}$ | $-\frac{4{\alpha _{2}}}{35}$ | $\frac{7{\alpha _{3}}}{90}$ |
| $\frac{{\alpha _{3}}}{36}$ | $-\frac{{\alpha _{2}}}{21}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{2{\alpha _{2}}}{21}-\frac{{\alpha _{2}}}{15}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{4{\alpha _{2}}}{35}$ | $-\frac{{\alpha _{2}}}{21}$ | $\frac{{\alpha _{3}}}{36}$ |
| $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{19{\alpha _{3}}}{180}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{7{\alpha _{3}}}{90}$ | $\frac{{\alpha _{3}}}{36}$ | $\frac{{\alpha _{3}}}{36}$ |
3.2 Choosing the Appropriate Fractional Order
The fractional order of derivatives in different parts of an image should be chosen in such a way that enhances the edges, preserves smooth areas and highlights the textural features. To achieve these objectives, the desired image is initially segmented into three distinct regions: edges, smooth areas, and textures. Subsequently, specific fractional derivative orders are selected for each region in order to tailor their effects accordingly. For edge enhancement, a higher value of the fractional derivative parameter (represented as α) close to 1 is chosen. This helps in enhancing the edges and capturing fine details in the image gradient. In the case of textured areas, a smaller value of the fractional derivative parameter α is considered. This allows for a more nuanced differentiation and highlight of textural features present in the image. Finally, for smooth areas, a value of α close to zero is utilized. This choice ensures that the smooth regions are preserved without introducing unnecessary high-frequency components. By carefully selecting suitable fractional derivative orders for different image regions, it becomes possible to effectively construct masks that optimize edge enhancement, maintain smooth areas, and highlight textural features, contributing to the overall quality and interpretability of the processed image.
Using statistical analysis, we can examine the data from the image matrix and identify that it conforms to the Wakeby distribution. The Wakeby distribution is characterized by its quantile function:
\[ \texttt{W}\text{(}\texttt{p}\text{)}=\xi +\frac{\theta }{\beta }\big(1-{(1-\texttt{p})^{\beta }}\big)-\frac{\eta }{\nu }\big(1-{(1-\texttt{p})^{-\nu }}\big)\]
and its density function:
\[ \frac{d}{dp}\texttt{W}(p)=\texttt{w}(\texttt{p})=\theta {(1-\texttt{p})^{\beta -1}}+\eta {(1-\texttt{p})^{-\nu -1}}.\]
In these equations, $0\leqslant \texttt{p}\leqslant 1$ represents the probability, ξ is the location parameter, θ and η are scale parameters, and β and ν are shape parameters.To utilize this distribution, we can employ the fifth and ninety-fifth percentiles in the following manner: let X be the values of the pixels in our image, arranged in a vector and sorted from least to greatest. Let n be the size of vector X and let q represent the percentile quantile. To find the qth quantile, we use the formula:
Let’s assume
and
Then the qth quantile can be found as:
So the fifth and ninety-fifth percentiles can be obtained by setting $q=0.05$ and $q=0.95$ respectively.
Table 4
$3\times 3$ filter for gradient image.
| $-\frac{1}{8}$ | $-\frac{1}{8}$ | $-\frac{1}{8}$ |
| $-\frac{1}{8}$ | 1 | $-\frac{1}{8}$ |
| $-\frac{1}{8}$ | $-\frac{1}{8}$ | $-\frac{1}{8}$ |
Table 5
$5\times 5$ filter for gradient image.
| $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ |
| $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ |
| $-\frac{1}{24}$ | $-\frac{1}{24}$ | 1 | $-\frac{1}{24}$ | $-\frac{1}{24}$ |
| $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ |
| $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ | $-\frac{1}{24}$ |
Table 6
$7\times 7$ filter for gradient image.
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | 1 | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}\hspace{2.5pt}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
| $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ | $-\frac{1}{48}$ |
The categorization of pixels into edge, texture, or smooth classes relies on the computation of a gradient image. For this purpose, the image I is convolved with an averaging mask, such as the masks in Tables 4, 5 and 6 to obtain the gradient matrix G. Then, based on specific threshold values, pixels are divided into these three categories. Here we put
as thresholding values to categorize the pixels. Pixels with a gradient value less than or equal to ${t_{1}}$ are categorized as smooth pixels, and pixels with a gradient value exceeding or equal to ${t_{2}}$ are classified as edge pixels. The pixels falling within the range of ${t_{1}}$ to ${t_{2}}$ are classified as texture pixels.
Smooth pixels receive a low fractional order to ensure preservation of smooth regions. The fractional order κ is applied to process the texture region in order to preserve weaker textures and enhance strong texture pixels. The value of κ is determined by the gradient value, with larger values assigned to pixels with high gradients in the range $({t_{1}},{t_{2}})$ and smaller values assigned to weak texture pixels (Wadhwa and Bhardwaj, 2020; Saadia and Rashdi, 2016). The equation used to determine the fractional differential mask order is as follows:
where
Figure 1 shows the values of PSNR, SD, Entropy, ENL, BRISQE, AG, PIQE and CV criteria for 8 test images with different values of λ. Based on the information provided, we see that different values of λ yield the best results for different criteria. Finding the optimal value of λ may be a challenging problem. Nevertheless, we choose $\lambda =0.5$ in relation (5).
(4)
\[ \alpha =\left\{\begin{array}{l@{\hskip4.0pt}l}0.1,\hspace{1em}& G(i,j)\leqslant {t_{1}},\\ {} \kappa ,\hspace{1em}& {t_{1}}\lt G(i,j)\lt {t_{2}},\\ {} 0.9,\hspace{1em}& G(i,j)\geqslant {t_{2}},\end{array}\right.\]Fig. 1
The values of PSNR, SD, Entropy, ENL, BRISQE, AG, PIQE and CV for $\lambda =0.1,0.2,\dots ,0.9$ for 8 test images.
By applying an appropriate fractional order differential mask, every pixel in the image undergoes convolution, resulting in an improved image that enhances edges and highlights texture while maintaining the integrity of smooth areas.
The following algorithm illustrates the necessary steps for image deblurring.
-
1. Start with a grayscale image I.
-
4. Assign a fractional order using the relation (4) for each pixel.
4 Empirical Findings and Discussions
The efficiency of the proposed algorithm was evaluated using eight brain images, as shown in Fig. 2. These images were blurred, as depicted in Fig. 3, and subsequently deblurred using the presented method, illustrated in Fig. 4. To demonstrate the efficiency of the presented method, we use different criteria and compare our results to other methods. We also provide brief definitions of these criteria below.
4.1 PSNR
Peak Signal-to-Noise Ratio (PSNR) is a metric to measure the quality of a reconstructed or processed image compared to the original image. It is calculated by comparing the peak signal strength to the noise level in the image, providing a quantitative measure of the fidelity of the reconstructed image. The higher values of PSNR indicate a higher level of similarity between the original and modified images. The formula for calculating PSNR is:
where $\textsf{MAX}$ is the maximum pixel value (for example, 255 for an 8-bit grayscale image) and MSE is the mean squared error between the original and modified images defined as:
where N is the number of pixels in the image and $I(j,k)$ and $\bar{I}(j,k)$ represent the gray levels of the original and enhanced images, respectively, at position $(j,k)$.
Table 7 shows the PSNR values of deblurred test images and compares the results to other methods in the literature.
Table 7
Comparing PSNR rate for test images.
| Proposed method | MGL (Hemalatha and Anouncia, 2018) | |||||
| Image | Blurred image | Deblurred image | Adaptive (Wadhwa and Bhardwaj, 2020) | ADFA (Li and Xie, 2015) | $\alpha =0.5$ | $\alpha =0.9$ |
| a | 22.5824 | $\mathbf{39.1837}$ | 31.0507 | 17.7232 | 28.5643 | 15.3211 |
| b | 19.5427 | $\mathbf{37.7322}$ | 30.8592 | 16.916 | 24.5577 | 12.6556 |
| c | 22.5329 | 36.3296 | $\mathbf{38.3346}$ | 20.6807 | 33.2279 | 17.1651 |
| d | 22.6419 | $\mathbf{40.2910}$ | 35.9398 | 19.7146 | 28.2865 | 13.9391 |
| e | 21.5470 | $\mathbf{37.6767}$ | 33.9197 | 19.5176 | 29.3097 | 16.7414 |
| f | 21.6181 | $\mathbf{38.6181}$ | 35.3671 | 19.847 | 30.7506 | 17.3028 |
| g | 20.8060 | $\mathbf{39.6382}$ | 32.5681 | 18.4106 | 27.6775 | 13.783 |
| h | 20.1791 | $\mathbf{38.7736}$ | 31.9848 | 16.9364 | 24.5164 | 12.6009 |
4.2 AMBE
The Average Mean Brightness Error (AMBE) is a metric used to quantify the average difference in brightness between a reference image and a processed image. It is calculated by taking the mean of the absolute differences in brightness values for corresponding pixels in the two images. AMBE provides a measure of the overall brightness distortion introduced during image processing or restoration. The best result is a zero AMBE value.
Table 8 presents the AMBE values for the deblurred test images and contrasts these results with those from other methods found in the literature. Meanwhile, Table 9 offers an in-depth comparison of the PSNR and AMBE metrics, detailing their average values and standard deviations. The findings indicate that the average scores for both PSNR and AMBE notably surpass those of existing state-of-the-art methods in nearly all cases. This highlights the effectiveness of the proposed approach in enhancing the quality of medical images.
Table 8
Comparing AMBE rate for test images.
| Proposed method | MGL (Hemalatha and Anouncia, 2018) | ||||
| Image | Deblurred image | Adaptive (Wadhwa and Bhardwaj, 2020) | ADFA (Li and Xie, 2015) | $\alpha =0.5$ | $\alpha =0.9$ |
| a | 0.0184 | 0.3521 | 5.0795 | 0.6157 | 8.0261 |
| b | 0.0380 | 0.5516 | 6.9137 | 1.7944 | 15.881 |
| c | 0.1048 | 0.1595 | 2.6175 | 0.4138 | 4.0873 |
| d | 0.0268 | 0.205 | 3.8883 | 0.7204 | 11.3506 |
| e | 0.0357 | 0.2615 | 2.9863 | 0.5015 | 4.3518 |
| f | 0.0338 | 0.1754 | 2.6159 | 0.3969 | 4.155 |
| g | 0.0342 | 0.4024 | 5.5945 | 1.031 | 12.9937 |
| h | 0.0366 | 0.3983 | 6.5385 | 1.6116 | 14.7137 |
Table 9
Assessment of average values and standard deviations of performance metrics for advanced methods.
| $\text{METHODS}$ | $\text{PSNR}$ | $\text{AMBE}$ |
| Proposed | $38.31\pm 1.98$ | $0.062\pm 0.04$ |
| Fuzzy (Wadhwa and Bhardwaj, 2024) | $37.03\pm 2.25$ | $1.91\pm 0.57$ |
| Morphology (Wadhwa and Bhardwaj, 2021) | $20.78\pm 2.72$ | $8.03\pm 3.95$ |
| Chaira (Ensafi and Tizhoosh, 2005) | $30.04\pm 1.82$ | $4.7\pm 1.24$ |
| CLAHE (Li and Xie, 2015) | $17.07\pm 1.73$ | $31.29\pm 5.97$ |
| HE (Li and Xie, 2015) | $4.80\pm 0.59$ | $146.15\pm 9.42$ |
| GC (Li and Xie, 2015) | $20.17\pm 1.46$ | $11.40\pm 1.26$ |
| ADFA (Li and Xie, 2015) | $18.99\pm 1.49$ | $4.21\pm 1.67$ |
| MGL (Hemalatha and Anouncia, 2018) | $22.75\pm 2.50$ | $7.31\pm 5.26$ |
4.3 Entropy
Entropy refers to a measure of the amount of information or uncertainty present in an image. It quantifies the randomness or disorder in the distribution of pixel values within the image. Images with higher entropy contain more diverse and unpredictable pixel values, while images with lower entropy have more uniform pixel distributions. Entropy is commonly used in image analysis to assess the complexity or texture of an image, as well as to evaluate the quality of compression and encoding algorithms. The entropy of an image can be calculated as:
where $H(I)$ represents the entropy of the image I, $P({x_{k}})$ denotes the probability of occurrence of each pixel value ${x_{k}}$ in the image and n is the total number of unique pixel values in the image.
Table 10 displays the Entropy values for deblurred test images and contrasts the findings with those from other methods documented in the literature.
Table 10
Comparing Entropy rate for test images.
| Proposed method | MGL (Hemalatha and Anouncia, 2018) | |||||
| Image | Blurred | Deblurred image | Adaptive (Wadhwa and Bhardwaj, 2020) | ADFA (Li and Xie, 2015) | $\alpha =0.5$ | $\alpha =0.9$ |
| a | 3.9288 | 4.3115 | 3.2035 | 2.8469 | 3.0903 | 2.4986 |
| b | 4.7521 | 5.0517 | 3.8652 | 3.295 | 3.5492 | 2.4328 |
| c | 4.1223 | 4.3227 | 3.3237 | 3.1764 | 3.2533 | 3.0656 |
| d | 5.1789 | 5.5790 | 4.5531 | 4.299 | 4.4412 | 3.7025 |
| e | 4.4722 | 4.7190 | 3.6527 | 3.2962 | 3.5781 | 3.2188 |
| f | 4.0355 | 4.3960 | 3.2814 | 2.9551 | 3.2641 | 2.9198 |
| g | 4.7184 | 5.1486 | 4.1601 | 3.8553 | 4.005 | 3.2697 |
| h | 4.7142 | 5.0675 | 4.0502 | 3.5105 | 3.7824 | 2.6725 |
4.4 Standard Deviation (SD)
The standard deviation (SD) is the square root of the noise variance and is commonly used to analyse the contrast-level of an image. It finds applications in various fields of image processing, such as image denoising and image fusion. A higher SD value typically corresponds to better perceptual image quality.
Table 11 displays the SD values for deblurred test images.
4.5 Equivalent Number of Look (ENL)
This metric is used to evaluate the level of smoothing in an image, especially in its homogeneous areas. It is computed by taking the ratio of the mean squared value to the variance. This measure is commonly applied in the context of reducing speckle noise in images.
Table 12 displays the ENL values for deblurred test images.
4.6 Blind/Referenceless Image Spatial Quality Evaluator (BRISQUE)
BRISQUE utilizes a probabilistic analysis of local normalized luminance signals to evaluate the naturalness of an image. A lower BRISQUE value indicates better perceptual image quality.
Table 13 displays the BRISQUE values for deblurred test images.
4.7 Perceptual Image Quality Evaluator (PIQE)
PIQE criteria in image processing refers to a set of objective measures used to assess the quality of an image based on human perception. These criteria take into account various factors such as contrast, sharpness, colour accuracy, and noise levels to determine the overall visual quality of an image. PIQE criteria are used to evaluate and compare the performance of different image processing algorithms and to ensure that the processed images meet certain quality standards.
Table 14 displays the PIQE values for deblurred test images.
4.8 Coefficient of Variation (CV)
This metric assesses the preservation of texture in non-uniform image areas, often used in the context of reducing speckle noise. It is computed as the ratio of the standard deviation to the mean value, expressed as a percentage.
Table 15 displays the CV values for deblurred test images.
4.9 Average Gradient (AG)
The Average Gradient (AG) is used to evaluate image sharpness, especially in image fusion applications. It assesses changes in texture and contrast features resulting from the fusion process. A higher AG value suggests enhanced perceptual image quality.
Table 16 displays the AG values for deblurred test images.
5 Conclusion
In conclusion, our proposed method for image deblurring, based on Grünwald-Letnikov fractional derivation and image segmentation using the Wakeby distribution, has shown promising results. By selecting optimal fractional derivative values for different image categories, we have effectively improved image quality and reduced blurring. The evaluation of our method using criteria such as PSNR, AMBE, Entropy, SD, ENL, BRISQUE etc. has demonstrated its efficiency in image restoration. This approach has the potential to contribute to advancements in image processing and restoration techniques.
