Suppose τ ( t ) ∈ [ t 0 , t ], where t 0 < t and t 0 , t ∈ R. The fractional differential operator of order α can be expressed mathematically as:

Let’s consider τ ( t ) ∈ [ t 0 , t ], where t 0 < t and t 0 , t ∈ R, as a one-dimensional signal. The fractional differential operator of order α for a digital image can be mathematically depicted as: d α τ ( t ) d t α ≈ τ ( t ) + ( − α ) 1 ! h τ ( t − h ) + ( − α ) ( − α + 1 ) 2 ! h 2 τ ( t − 2 h ) + ⋯ + ( − α ) ( − α + 1 ) … ( − α + m − 1 ) m ! h m τ ( t − m h ) . In the realm of digital images, the D 8 (or Chessboard) norm is defined as: D 8 = max { | x 2 − x 1 | , | y 2 − y 1 | } , 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 d α τ ( t ) d t α ≈ τ ( t ) + ( − α ) 1 ! τ ( t − 1 ) + ( − α ) ( − α + 1 ) 2 ! τ ( t − 2 ) + ⋯ + ( − α ) ( − α + 1 ) … ( − α + m − 1 ) m ! τ ( t − m ) , or d α τ ( t ) d t α ≈ τ ( t ) + ( − α ) 1 ! τ ( t − 1 ) + ( − α ) ( − α + 1 ) 2 ! τ ( t − 2 ) + ⋯ + Γ ( − α + m ) Γ ( − α ) Γ ( m + 1 ) τ ( t − m ) , where Γ denotes the Gamma function, defined as Γ ( z ) = ∫ 0 ∞ x z − 1 e − x d x , R e ( z ) > 0 .

In a 2D digital image, the backward differences of fractional differentiation for x- and y-directions can be found as: (1) ∂ α τ ( x , y ) ∂ x α ≈ τ ( x , y ) + ( − α ) 1 ! τ ( x − 1 , y ) + ( − α ) ( − α + 1 ) 2 ! τ ( x − 2 , y ) + ⋯ + Γ ( − α + m ) Γ ( − α ) Γ ( m + 1 ) τ ( x − m , y ) , and (2) ∂ α τ ( x , y ) ∂ y α ≈ τ ( x , y ) + ( − α ) 1 ! τ ( x , y − 1 ) + ( − α ) ( − α + 1 ) 2 ! τ ( x , y − 2 ) + ⋯ + Γ ( − α + m ) Γ ( − α ) Γ ( m + 1 ) τ ( x , y − m ) .

To create a mask of size m × m, the initial m coefficients from the Eqs. (1) and (2) are considered. Table 1 presents the 5 × 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 − α 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 α ( α − 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 − α ( α − 1 ) ( α − 2 ) 3 and α ( α − 1 ) ( α − 2 ) ( α − 3 ) 4 are equally distributed among them, respectively.

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 α j = α ( α − 1 ) … ( α − j ) , j = 1 , 2 , 3.

Once the masks in the eight directions k π 4 , k = 0 , 1 , 2 , … , 7, have been acquired, we combine them to create a 9 × 9 mask as shown in Table 3. Next, we divide each element by the total sum of the coefficients ( 8 − 12 α + 4 α 2 ) to create a 9 × 9 differential mask, ensuring that the resulting image’s intensity values fall within the range of [ 0 , 255 ].

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: W ( p ) = ξ + θ β ( 1 − ( 1 − p ) β ) − η ν ( 1 − ( 1 − p ) − ν ) and its density function: d d p W ( p ) = w ( p ) = θ ( 1 − p ) β − 1 + η ( 1 − p ) − ν − 1 . In these equations, 0 ⩽ p ⩽ 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: s = n . q . Let’s assume s 1 = [ s ] , and s 2 = s − s 1 . Then the qth quantile can be found as: Q ( q ) = X s 1 + s 2 ( X s 1 + 1 − X s 1 ) . So the fifth and ninety-fifth percentiles can be obtained by setting q = 0.05 and q = 0.95 respectively.

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 (3) t 1 = Q ( 0.05 ) , t 2 = Q ( 0.95 ) 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: (4) α = 0.1 , G ( i , j ) ⩽ t 1 , κ , t 1 < G ( i , j ) < t 2 , 0.9 , G ( i , j ) ⩾ t 2 , where (5) κ = [ G ( i , j ) − min ( G ) max ( G ) − min ( G ) × λ ] + 0.1. 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 λ = 0.5 in relation (5).

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.

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: PSNR = 10 · log 10 ( MAX 2 MSE ) , where 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: MSE = 1 N ∑ j ∑ k | I ( j , k ) − I ¯ ( j , k ) | 2 , where N is the number of pixels in the image and I ( j , k ) and 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.

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.

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: H ( I ) = − ∑ k = 1 n P ( x k ) log 2 P ( x k ) , 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.

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.

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.

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.

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.

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.

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.