Kriging-Weighted Laplacian Kernels for Grayscale Image Sharpening

Sharpening filters are used to highlight fine image details, including object edges. However, sharpening filters are very specific to different types of images as they may create undesired edge effects, over-highlight fine details, or emphasize noise. Laplacian, Laplacian of Gaussian, high-boost, unsharp masking filters, and their extended algorithms are among most widely used sharpening spatial filters. This paper introduces a method that integrates anisotropic averaging with the Laplacian kernels for grayscale image sharpening. The proposed methodology is based on the concept of kriging computation in geostatistics for determining optimal interpolation weights in spatial domain. The convolution of kriging and Laplacian kernels is then carried out for image sharpening. Experimental results suggest certain advantages of the proposed linear convolution model for image sharpening over the Laplacian, Laplacian of Gaussian, high-boost, unsharp masking, and anisotropic diffusion methods in terms of the balance of sharpness and natural visualization. Another advantage of the proposed method is that it does not require any input statistical parameters.


I. INTRODUCTION
Sharpened images produced by Laplacian, Laplacian of Gaussian (LoG), unsharp masking, and high-boost filters are based on linear convolution kernels. Image convolution is the process of adding values of a pixel and its neighbors linearly weighted by a filter kernel to every pixel. The purpose of convolution is to modify the spatial characteristics of an image, resulting in, such as smoothing, sharpening, enhancing, or edge highlighting to suit various applications [1]- [4]. In general, the convolution of a kernel k and an image f (x, y) can be expressed as g(x, y) = k⊛f (x, y) = where g(x, y) is the output image, and ⊛ denotes the convolution operator.
Laplacian filter-based kernels [5], which are edgesharpening filters, have been utilized in several applications [6]- [15]. The Laplacian operator takes partial derivatives along the two spatial axes of an image f (x, y) and is defined as [5] ▽ 2 f (x, y) = ∂ 2 f (x, y) ∂x 2 + ∂ 2 f (x, y) ∂y 2 = f (x + 1, y) + f (x − 1, y) + f (x, y + 1) +f (x, y − 1) − 4f (x, y), which highlights sharp intensity transitions and reduces the effect of regions having slowly varying gray levels, resulting in the following 3 × 3 Laplacian filters where the left matrix is based on Eq. 2, and the right is an extension of Eq. 2 to include the diagonal elements.
To rectify the effect of featureless image background while sharpness is still reserved can be obtained by subtracting the Laplacian image from the original, giving [5] f * (x, y) = f (x, y) − ▽ 2 f (x, y) = f (x, y) − [f (x + 1, y) + f (x − 1, y) +f (x, y + 1) + f (x, y − 1) − 4f (x, y)], = 5f (x, y) − [f (x + 1, y) + f (x − 1, y) +f (x, y + 1) + f (x, y − 1)], which results in a 3 × 3 composite sharpening kernel and its extended version, respectively, as Because L 1 and L 2 are derived from the second derivative of an image, which highlights regions of rapid intensity change, both kernels are sensitive to noise. To address this issue, it is suggested to Gaussian-smooth the image before applying the Laplacian. This procedure is known as the LoG, which is defined as [16], [17] ▽ 2 G(x, y) = 1 πσ 4 1 − where σ is the standard deviation, All L 1 , L 2 , and LoG kernels yield isotropic results. Because an isotropic operator applies the rate of change equally in all directions of an image, it carries no particular sensitivity or bias toward different spatial orientations. An anisotropic topography has directional attributes that may be different at all points or directions of an image. Effort was spent on addressing this issue by minimizing the overall anisotropy produced by Laplacian kernels [18], or using local Laplacian kernels [6], [15].
A high-pass filtered (sharpened) image, denoted as f HP (x, y), can be obtained as where f (x, y) and f LP (x, y) are original and low-pass filtered (blurred or unsharp) images, respectively. Because of the use of an unsharp image, this process is also known as unsharp masking (UM) [5]. A high-boost filtered image, denoted as f HB (x, y), is defined as wkere α ≥ 1 is a scaling factor. If the Laplacian image is used as f HP (x, y), then f HB (x, y) is defined as which gives the following sharpening second-order derivative kernels: If α = 1, both high-boost kernels H 1 and H 2 become Laplacian L 1 and L 2 , respectively.
Laplacian, LoG, unsharp masking, and high-boost filters, which are benchmark methods for comparing imagesharpening algorithms [19], have been presented herein. Several other methods developed for image sharpening are such as the anisotropic diffusion (AD) filter [20], [21], which iteratively selected the AD coefficient to allow various degrees of smoothing within different image regions and maintain sharpness between region boundaries; the class of iterative morphological operators, which were based on a Laplacian function, was proposed for grayscale image sharpening [22]; fuzzy morphology for enhancing details in historical images [23]. Because wavelet coefficients can reveal high frequency components of an image at multiple resolutions, a waveletbased algorithm was developed for image sharpening [24]. Histogram equalization [25], which aimed to improve contrast by considering changes in pixel values according to the intensity distribution of the input image, was utilized for sharpening images. The Sobolev isotropic diffusion [26] was developed for image sharpening. This method applied the Sobolev gradients to compute the partial differential equations for image diffusion and sharpening.
Some other methods for image sharpening developed in the past decade include a method using a multi-scale scheme and wavelet discrete transform, where the initial result was processed by unsharp masking filters and the final image produced using a wavelet fusion scheme [27]. Locally adaptive techniques for image sharpening were introduced to achieve "pleasant" visualization when smoothing parameters can be appropriately chosen [28]. An approach for adjusting image sharpness was recently proposed by constructing the binding filter, which is based on the Gaussian distribution, to define the kernel weights for combining with the high-boost filter, and the results achieved having properties of both Gaussian smoothing and high-boost filtering [29]. Another recent work [30] implemented the anisotropic diffusion filter [20] with twenty iterations for image smoothing and then used the unsharp masking for image sharpening.
As an attempt to consider anisotropic properties of an image to be convolved with Laplacian kernels, this paper introduces the use of kriging weights as a secondary kernel whose derivation is based on statistical properties of experimental anisotropic semivariograms. Unlike having fixed values of kernel coefficients, the kriging weights vary according to changes in image semivariograms, giving different convolution results in images of different spatial statistics.
The rest of this paper is organized as follows. Section II presents mathematical derivations of the proposed krigingweighted Laplacian operators. Section III shows several examples to illustrate, compare, and discuss the results obtained from the kriging-weighted Laplacian method with those produced from other image-sharpening models. Finally, Section IV is the conclusion of findings, including remarks on some limitations of the proposed method and suggestions of issues for future research.

II. KRIGING-WEIGHTED LAPLACIAN OPERATORS
Let z(a) be a random function of spatial location a, the ordinary kriging holds the following intrinsic hypothesis [31]: where E[·], µ, and γ(·) denote expected value, mean, and semivariogram of the random function, respectively; and a and a + h are any two spatial locations separated by distance h.
Let z(a i ) and z(a j ) be the values of two data points at two spatial locations a i and a j , respectively. It was shown that [32] To derive kriging weights in the setting of the Laplacian operators, let a 0 be the location of f (x, y), and a 8 of f (x + 1, y + 1), which are shown in Fig. 1. The value of f (a 0 ) can be predicted using the best linear unbiased estimator of ordinary kriging as [32] f where w i , i = 1, . . . , 8 are kriging weights that can be optimally determined by minimizing the estimation variance σ 2 (a 0 ): Using Eq. 12, the estimate is done via the following constrained optimization which is subject to Based on the following proof [32] Var Eq. 14 can be rewritten as (16) Using Eq. 11, The Lagrangian function for ordinary kriging estimate of f (a 0 ) can be constructed as where λ is a Lagrange multiplier.
Let γ(a i − a j ) now be denoted as γ i,j . To minimize the variance of estimation, all first derivatives of the Lagrangian function with respect to w i , i = 1, . . . , 8, and λ must be set to zero, which are The above Lagrangian method results in the determination of unique kriging weights by solving the following set of simultaneous equations: To consider anisotropy in the semivariogram model, the experimental semivariogram measured in the horizontal direction of an image can be expressed as VOLUME x, 2022 Graphical procedure for grayscale image sharpening using kriging-weighted Laplacian method.
where N (∆x) is the number of pixel pairs separated by ∆x.
Likewise, for the vertical direction, the experimental semivariogram can be defined as where N (∆y) is the number of pixel pairs separated by ∆y.
For a matrix, the main diagonal is the line of elements running from the top-left to bottom-right corners; whereas the antidiagonal is the line of elements running from the topright to bottom-left corners. Let ∆D + and ∆D − indicate (∆x, ∆y) measured in the directions of the main diagonal and antidiagonal of an image, respectively. The experimental semivariograms measured in the main diagonal and antidiagonal directions can be obtained, respectively, as where N (∆D + ) is the number of pixel pairs separated by (∆x, ∆y) in the main-diagonal direction, and where N (∆D − ) is the number of pixel pairs separated by (∆x, ∆y) in the antidiagonal direction.
Using the form of Eq. 20, the kriging weights can be determined by solving for w in the following ordinary kriging system represented in matrix form as where A, w, and b are given in Table 1. It can be seen that A and b consist of statistical distances that incorporate anisotropy, spatial correlation, and clustering information; and therefore provide the set of weights in terms of spatial continuity [33]. As a result, the 3 × 3 kriging-weighted kernel for Laplacian filter L 2 is of variable coefficients, depending on the anisotropic semivariogram of an image, where, being similar to the Laplacian kernel, the weight at the kernel center is set to -1 to impose that the sum of all elements of the kernel is zero so that the convolution of a homogeneous 2: Measures of image sharpness and visual quality of "Moon", "Lena", "Bridge", and "Nature" images.

Attributes
Original region is zero. Likewise, the kriging-weighted kernel for Laplacian filter L 1 can be constructed by considering only f (a 2 ), f (a 4 ), f (a 5 ), and f (a 7 ) for the estimate of f (a 0 ) to compute corresponding w * 1 , w * 2 , w * 3 , and w * 4 , respectively, 4 i=1 w * i = 1; whereas weights for the diagonal elements are set to zero. The corresponding kriging system yields A * , w * , and b * of size 5 × 5, 5 × 1, and 5 × 1, respectively. Thus, kriging-weighted kernels for L 1 and L 2 are defined, respectively, as Because convolution is commutative: α ⊛ β = β ⊛ α, and associative: α⊛(β ⊛κ) = (α⊛β)⊛κ, where α, β, and κ are kernels, multistage filtering therefore can be performed in a single operation. Instead of having an image convolved with a Laplacian kernel first, then the result convolved with the kriging-weighted kernel, the convolution of an image with the kriging-weighted Laplacian kernel can be carried out as where ω i is a hybrid filter or kriging-weighted Laplacian kernel that is defined as Fig. 2 graphically shows the procedure for sharpening a grayscale image using the proposed kriging-weighted Laplacian method. Fig. 3 shows the "Moon" image of size 537 × 358, and its sharpened versions obtained from the Laplacian (L 1 and L 2 ), LoG, high-boost (H 1 and H 2 ), kriging-weighted Laplacian (ω 1 and ω 2 ), unsharp marking (UM1 and UM2), and anisotropic diffusion-based unsharp masking or, for short notation, anisotropic diffusion (AD1 and AD2) filters. For the LoG, σ = 0.5 and the filter size is 5 × 5. For the high-boost filtering, α = 1.5 was selected. For the unsharp marking, the standard deviation of the Gaussian low-pass filter was chosen as 1 (UM1) and 3 (UM2). For the anisotropic diffusion, AD1 and AD2 indicate the implementation of the unsharp masking with the standard deviation of the Gaussian low-pass filter = 1 and 3, respectively. The Laplacian, LoG, high-boost, and unsharp masking filters were computed using the R2022a Matlab Image Processing Toolbox. The anisotropic diffusion filter was computed using public Matlab codes [34] provided by authors of the work [30].

III. RESULTS
Here, a measure of sharpness in grayscale images using gradients [35] was adopted to evaluate the performance of different methods for image sharpening. The absolute relative difference in sharpness between the original and a filtered image is defined as |∆ r | = |f − f * | /f , where f and f * are original and filtered images, respectively. While the sharpened images given by the Laplacian, LoG, kriging-weighted Laplacian, unsharp masking, and anisotropic diffusion filters have similar visualizations (measures of sharpness were between 3.68 (L 1 ) and 7.50 (L 2 )); the high-boost operators amplified the average gray levels of the original image, resulting in significantly brightened images (measures of sharpness obtained by H 1 = 8.82 and H 2 = 14.30).
To observe more distinct differences in image sharpening using the six filters, Figs. 4, 5, and 6 show the 225 × 225 "Lena", 195 × 259 "Bridge", and 182 × 278 "Nature" images, respectively. Figs. 4, 5, and 6 also show images filtered by the Laplacian, LoG, high-boost, unsharp masking, anisotropic diffusion, and kriging-weighted Laplacian models, where the same filter parameters described earlier were used for the four applicable models.
For the "Lena" image, it can be seen that while L 1 , L 2 , LoG, H 1 and H 2 sharpened the image (measures of sharpness were between 15.96 given by L 1 and 43.81 given by H 2 ) but also highlighted pixels of high changes in intensity, yielding undesirable visual effects such as pixels of the eyes and hair; on the other hand, both unsharp masking (sharpness measures of UM1 and UM2 = 15.90 and 15.81, respectively) and anisotropic diffusion (sharpness measures of AD1 and AD2 = 16.53 and 16.89, respectively) filters tended to suppress some highlights of the outer region of the hair; however, ω 1 (sharpness = 20.45) and ω 2 (sharpness = 20.11) kernels provided a sharpened image with a much better visual appearance.
For the "Bridge" image, the high-boost filters amplified the image contrast, yielding almost a white background for the sky and brightened water; L 1 , L 2 and LoG produced VOLUME x, 2022 the least sharpened versions, in which the suspension cable was invisible; UM1, UM2, AD1, AD2, ω 1 , and ω 2 provided similar sharpened versions of the original image (measures of sharpness were between 7.56 (AD1) and 9.77 (ω 2 )). AD2 can be noticed to yield some highlighted distortion of the vertical cables. The sharpened outputs of ω 1 and ω 2 were balanced between those of the other 5 methods.
For the "Nature" image, L 2 (measure of sharpness = 13.72), and H 2 (measure of sharpness = 50.44) kernels yielded most dominantly noise-sensitive results; LoG (measure of sharpness = 15.58) and ω 1 (measure of sharpness = 15.60) provided similar visualizations. UM1, UM2, AD1, AD2, ω 1 , and ω 2 provided similar sharpened images (measures of sharpness were between 11.12 and 15.60). While some noise effect was emphasized by AD2, it can be observed that the ω 2 kernel could balance both sharpness and noise suppression. Similar to the visual effect of the "Moon" image sharpened by H 1 and H 2 , two high-boost filtered "Nature" images show much higher contrasts, where dark pixels of low frequencies were significantly lightened. Table 2 shows the measures of sharpness and relative sharpness of the "Moon", "Lena", "Bridge", and "Nature" images obtained from the Laplacian, LoG, high boost, unsharp masking, anisotropic diffusion, and kriging-weighted Laplacian methods. It was apparent that the high-boost filters provided the sharpest visualization by highlighting the bright pixels of the images, whereas the kriging-weighted Laplacian filters gave results balanced between the Laplacian and LoG filters. All filtered images increased the sharpness of the original images. While being able to increase sharpness in images, both ω 1 and ω 2 provided results that have the most similar naturally visual appearance to the original images by avoiding over-highlighting fine details with a sharp discontinuity. H 2 yielded highest measures of image sharpness but resulted in the most unnatural visual expression. The LoG filter highlighted the fine details with a lesser degree than the  .
: "Albert Einstein" image and its sharpened versions provided by Laplacian (L 1 and L 2 ), LoG, high-boost (H 1 and H 2 ), anisotropic diffusion (AD1 and AD2), unsharp marking (UM1 and UM2), and combined kriging-weighted Laplacian kernel and unsharp masking (ω 1 -UM1 and ω 1 -UM2).   AD2) is based on the use of the unsharp masking, the kriging-weighted Laplacian kernel ω 1 was also applied to be further filtered by unsharp masking operators UM1 and UM2, denoted as ω 1 -UM1 and ω 1 -UM2, for the purpose of comparison of the performance between the two anisotropic methods. Filters L 1 , L 2 , LoG, H 1 , and H 2 produced similar sharpened "Albert Einstein" and "Bow" results as previously discussed for the other four images. In particular, amplification of the image contrast yielded by H 1 and H 2 , and unnatural hair as well as facial effects of the "Albert Einstein" output from AD2 can be clearly observed. Combined filters ω 1 -UM1 and ω 1 -UM2 resulted in the highest measures of sharpness (20.63 and 22.71 respectively for the "Albert Einstein" image, and 28.04 and 30.41 respectively for the "Bow" image) among other filters UM1, UM2, AD1, and AD2. Table 3 shows the measures of sharpness and relative sharpness of these two images obtained from the Laplacian, LoG, high boost, unsharp masking, anisotropic diffusion, and kriging-weighted Laplacian methods.
There are two types of approaches to measure the quality of an image: no-reference quality metrics and full-reference quality metrics. No-reference quality methods compute statistical features of an input (either original or filtered) image to assess the image quality. Full-reference algorithms compare the quality between the original and a filtered image. The perception-based image quality evaluator (PIQE) [36], which is a no-reference image-quality metric, were used to evaluate quality of the images filtered by the Laplacian, LoG, high-boost, and kriging-weighted Laplacian filters. The PIQE score is inversely correlated to the perceptual quality of an image in the range [0, 100]. A lower score indicates higher perceptual image quality.
The mean-squared error (MSE), peak signal-to-noise ratio (PSNR), and structural similarity (SSIM) were used as fullreference quality metrics to evaluate the sharpened images. Because the MSE computes the average squared difference between original and filtered pixel values and the PSNR is derived from the MSE, these two measures may not agree well with the human perception of quality. Because the SSIM metric [37] considers local pixel intensities, luminance, and contrast to compute a single local quality score, the SSIM measure is more in alignment with human-based quality scoring. For the SSIM measure, a value closer to 1 indicates better image quality. Table 2 shows the no-reference and full-reference quality metrics for the "Moon", "Lena", "Bridge", and "Nature" images, and Table 3 shows the no-reference and full-reference quality metrics for the "Albert Einstein" and "Bow" images.
For the "Moon" image, AD2 has the highest (63.67) and L 1 has the lowest (19.90) PIQE scores. The PIQE scores of both ω 1 and ω 2 are between the highest and lowest values, suggesting the balance of visual effect in image sharpness VOLUME x, 2022 provided by the kriging-weighted Laplacian kernels. For the "Lena", L 2 and AD2 obtained the highest (59.50) and lowest (46.33) PIQE scores, respectively. For the "Bridge", L 2 and L 1 have the highest (72.00) and lowest (43.52) PIQE scores, respectively. For the "Nature", both H 1 and H 2 have the highest PIQE score (67.10), and L 1 has the lowest PIQE score (41.10). Similarly, for the "Albert Einstein" and "Bow" images, both H 1 and H 2 have the highest PIQE score and L 1 has the lowest. Similar observations about the two krigingweighted Laplacian kernels apply to the other 5 images ("Lena", "Bridge", "Nature", "Albert Einstein" and "Bow" images).
Regarding the MSE and PSNR, both H 1 and H 2 have the highest values, whereas ω 1 , ω 2 , UM1, UM2, AD1, and AD2 likely have the lowest values for all six images as shown in Figs. 3 -8. By means of the SSIM, UM1 has the highest score (0.98) for the "Moon", whereas the score for ω 1 (0.91) is also close to 1; both UM1 and ω 1 have the highest score (0.96) for the "Lena"; ω 1 have the highest score (0.98) for the "Bridge", whereas ω 2 and UM1 have the second highest value (0.96); for the "Nature", ω 2 has the highest score (0.98); UM1 and ω 1 -UM1 have the highest scores for both "Albert Einstein" (0.98) and "Bow" (0.97) images. Both H 1 and H 2 have the lowest SSIM scores (0.00), except for the "Moon" image, H 1 scores 0.67. Such results suggest the combination of the kriging-weighted Laplacian and unsharp marking is more favorable than the anisotropic diffusion-based unsharp masking.
The main computational cost for the proposed method is the solving of a system of simultaneous linear equations with either 5 or 9 variables and their associated semivariogram values using either L 1 or L 2 , respectively. Such a cost is not high because the computation is done only once, and the times taken for the kriging-weighted Laplacian filter were about 0.03 sec, 0.02 sec, and 0.01 sec longer than the conventional Laplacian, HBF, and LoG filters, respectively, for the images shown in Figs. 4-6, obtained from the central processing unit (CPU) of Intel(R) Core(TM) i7-6500U CPU@2.50GHz. Fig. 9 displays the plot of computational times required by the Laplacian (L 1 ), HBF (H 1 ), LoG, unsharp masking (UM1), anisotropic diffusion (AD1), and kriging-weighted Laplacian (ω 1 ) methods against different resolutions of the "Bow" image performed on the same mentioned computer hardware, showing the anisotropic diffusion required longest computational time, and the speed for the kriging-weighted Laplacian slowly increased with higher image resolutions.
Edge-sharpening kernels aim to increase the contrast on image edges, that is the darker side of the edge becomes darker and the brighter side brighter; therefore, contrast of the whole image is not affected. As a result, the filtering may lead to edge degradation and other traces such as halos or white artifacts [38] because of sharp transitions in signals. The high-boost filtering raises the high-frequency components of an image by adding more weight to the masked pixels, and therefore can effectively enhance image-shading effects; however, in many cases, it can be very sensitive to noise and magnify unwanted image details. While the Laplacian filtering can restore fine image details, its second-order derivative tend to amplify noise. The LoG is a combination of Laplacian and Gaussian filters, which tries to smooth the original image before applying the Laplacian filter for image sharpening; thus, in some cases, it fails to pick up fine characteristics of an image to a satisfactory level. Results obtained from the unsharp masking and anisotropic diffusion-based unsharp masking can preserve the sharpness of edges better than the Laplacian and LoG methods, but largely depend on the selection of the standard deviation of the Gaussian low-pass filter that may adversely affect the image sharpening if not appropriately selected.
Based on the spatial-statistics properties of an image on its vertical, horizontal, and diagonal directions, the krigingweighted Laplacian filter derives and convolves the kriging weights with a kernel of second-order derivative for image sharpening. For an isotropic and homogeneous image, it may provide an output similar to the LoG (as in the case of the "Moon" image). For an image of anisotropic content, the kriging-weighted Laplacian filter is able to suppress unwanted details (Lena's hair streaks and "Nature" image), or provide a balanced result between Laplacian, LoG, and highboost filtering methods ("Bridge" image). For sharpening images of textural content such as the "Lena" and "Albert Einstein" images, the kriging-weighted Laplacian method can provide sharper and more natural visualization than the unsharp masking and anisotropic diffusion-based unsharp masking. Another advantage of the kriging-weighted Laplacian method over the LoG, high-boost, unsharp masking, and anisotropic diffusion filters is that the proposed method is free from statistical parameter specification, which plays a sensitive role to the other models.

IV. CONCLUDING REMARKS
Kriging, which is a geostatistical approach, has been applied to image and signal processing [39]- [44]. An image sharpening filter that incorporates anisotropic information captured by ordinary kriging into the Laplacian filters has been presented. Values of the kriging-weighted kernels are estimated from a spatial variance of an image under study, whose directions of intensity distribution can be of either isotropy or anisotropy.
Results obtained from the foregoing examples have illustrated certain advantages of the kriging-weighted Laplacian method: It is capable of producing filtered images that have sharper visualization than the Laplacian, LoG, unsharp masking, and anisotropic diffusion filters, and less effect of unwanted-detail highlights than the Laplacian, LoG, and high-boost filters. The proposed method can provide results that are of better balance between image sharpness and subjective image quality than the Laplacian, LoG, highboost, unsharp masking, and anisotropic diffusion filters. In addition, kriging-based kernels of difference sizes can be derived for convolving with other types of filters to suit various applications in image processing. Kriging-based filtering can also be used as adaptive kernels that consider local semivariograms of an image to result in different levels of filtering for modeling structural variability of the image continuity.
Some current limitations of the kriging-weighted Laplacian method are that the computation of the kriging system to determine the optimal kriging weights is based on the experimental anisotropic semi-variograms and applies to only grayscale images. A feasible future work for improving the proposed method would be the consideration of minimization of errors between the experimental anisotropic semivariogram and several theoretical anisotropic semivariogram models for kriging interpolation developed in geostatistics [45] to construct better spatial correlations among intensities and spatial patterns of pixels. Another future work is the extension of the kriging-weighted Laplacian kernels for color-image sharpening, which can be readily derived by the incorporation of the multivariable semi-variogram [46] to compute kriging weights of color images. Furthermore, for processing a large number of high-resolution images, the computing speed of the proposed method can be potentially enhanced by considering a parallel strategy for optimizing the running time of the Laplacian-based sharpening approach using graphic processing units [47].