# Suppression of the Block Effect while Enhancing Dark Images Using Asumu Derivative

Authors: Gustavo Asumu Mboro Nchama
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

Image contrast enhancement is one of the most important tasks in the field of image digital processing. With the help of computers and programming languages, many mathematical methods have been implemented to improve the visual quality of an image. However, the majority of these methods have a major drawback: they tend to make the image look blocked. This effect is visually unpleasant and is likely to cause a computer vision system to falsely recognize as edges the boundaries of different blocks that belong to the same smooth area in the original image. To avoid the block effect while enhancing the quality of dark images, we propose a new fractional differential mask based on the Asumu fractional derivative. Experiments with dark natural images show that, even though our method does not have a better performance in statistical quality, it achieves a solution to the problem that was the motivation to write this paper, suppressing efficiently the block effect while improving the visual qualities of dark images.


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## Body

## 1. Introduction

The need to improve the visual quality of images has been around for a long time. However, this need could not be materialized due to the absence of appropriate technology. With the creation of the first powerful computers and the invention of computerized axial tomography, there emerged the area of digital image processing, which is the set of techniques that are applied to digital images to improve their quality. These techniques include a series of operations where the initial data is an image and the final result is another image that has already been processed. The constant progress in this area has not been by itself, but in conjunction with other areas such as mathematics, computation, and probability. In addition to this, the development of hardware technologies and language programming has made it possible to carry out the creation and implementation of different techniques for contrast enhancement in different types of images. One of these techniques that has shown greater effectiveness in enhancing contrast is the use of masks. For example, in [1], the authors introduced a texture enhancement technique for medical images by using fractional differential masks based on the Srivastava–Owa fractional operator. In [2], there was proposed a new way of constructing a fractional-based convolution mask with an application to image edge analysis. In [3], a fractional differential mask based on the Riemann–Liouville fractional derivative was proposed. The experiments of the proposed mask showed a better visual effect for image enhancement with rich texture information than integral order masks. In the paper [4], the authors implemented a class of fractional differential masks with high precision. After discussing the capability of multiscale fractional differential masks for texture enhancement, experiments showed that, for rich-grained digital images, the capability of nonlinearly enhancing complex texture details in smooth areas by a fractional differential-based approach appears obvious and better than that achieved by traditional integral-based algorithms. In [5], the capabilities of the fractional differential approach for the detection of textural features in two-dimensional digital images were discussed. Numerical experiments showed that the textural details are enhanced better than other existing methods in the literature. In the paper [6], a novel fractional-order fusion model is presented to adequately extract contents from low-light areas, suppress noise, and preserve naturalness. Experimental results revealed that the proposed model achieves promising results and extracts more invisible contents in dark areas. In [7], the authors developed a fractional differential mask in the sense of the Riemann–Liouville definition for contrast enhancement images and demonstrated the capability of the fractional differential mask by varying the fractional differential order. In the paper [8], an algorithm based on the Caputo–Fabrizio fractional differential mask for image contrast enhancement is introduced. Experiments showed that the method can control the degree of contrast enhancement by varying the fractional differential order. In [9], the authors proposed a new way of constructing fractional differential masks based on the Caputo and Caputo–Fabrizio derivatives. The effectiveness of these methods was compared with the histogram equalization method and the multiplication of each pixel of an image by a constant. The experiment results showed the superiority of the proposed method, with better visual quality and higher gray-level co-occurrence matrix values in all four directions. In [10], the authors developed a fractional-based technique for contrast enhancement for grayscale images. Experiments showed that the proposed method has excellent feedback for enhancing the contrast of dark images; it enhances edge information effectively and reveals more detailed information than some traditional proposed methods. In [4], the authors implemented a class of fractional differential masks in the sense of Grümwald–Letnikov and Riemann–Liouville definitions to improve image quality. Experiments showed that the obtained masks could enhance complex texture details in smooth areas better than the traditional integral-based algorithms. In [3], there were analyzed the fractional differential masks in the Riemann–Liouville definition for the enhancement of digital images. Based on this mask, the experiments showed that this fractional differential calculation can not only maintain the low-frequency contour information in the smooth area but also nonlinear enhance the high-frequency edge and texture parts of the image. In the paper [6], the authors presented the construction of a fractional-order mask to enhance the low-light images and extract adequate contents from low-light areas, suppress noise, and preserve naturalness. In the paper [9], an algorithm based on the Caputo–Fabrizio fractional differential mask for image contrast enhancement is introduced. Experiments showed that the method can control the degree of contrast enhancement by varying the fractional differential order.The majority of the methods outlined above cannot avoid the block effect while enhancing the quality of dark images. One of the main reasons for this drawback is due to the way those fractional differential masks are constructed. To avoid the presence of the block effect while improving the quality of dark images, we develop a new way of constructing a fractional-order differential mask based on the Asumu fractional derivative. Moreover, we compare the effectiveness of the proposed algorithm with some others proposed in the literature. The outline of this paper is as follows: in Section2, we briefly review some well-known fractional differential masks; Section 3 presents the Asumu fractional differential filter; in Section 4, we give the experimental results of the proposed method on real images. A conclusion is considered in Section 5.


## 2. A Short Review of Some of the Well-Known Fractional Differential Masks

As we outlined above, many efficient algorithms have been proposed in the literature for image contrast enhancement. The construction of many of these algorithms is based on fractional differential order operators. Some of the fractional differential-order derivatives used for the construction of such masks are Riemann–Liouville, Caputo, and Caputo–Fabrizio fractional differential derivatives. For example, in [7], authors used the following Riemann–Liouville fractional differential mask:(1)c2sum0c2sum0c2sum0c1sumc1sumc1sum0c2sumc1sum8c0sumc1sumc2sum0c1sumc1sumc1sum0c2sum0c2sum0c2sum,where sum=8∗c0+c1+c2 and the coefficients c0, c1 and c2 given by(2)c0=1Γ2−α,c1=21−α−2Γ2−α,c2=31−α−2⋅21−α+1Γ2−α.In [9], authors used the following Caputo-Fabrizio fractional differential mask:(3)c20c20c20c1c1c10c2c1sumc1c20c1c1c10c20c20c2,where sum=c0+c1+c2/8 and the coefficients c0, c1, and c2 given by(4)c0=1α1−e−α/1−α,c1=1α2e−α/1−α−e−2α/1−α−1,c2=1α2e−2α/1−α−e−3α/1−α−e−α/1−α.In [9], authors used the Caputo fractional derivative to obtain the following mask:(5)c20c20c20c1c1c10c2c1sumc1c20c1c1c10c20c20c2,where sum=c0+c1+c2/8 and the coefficients c0, c1, and c2 given by(6)c0=11−αΓ2−α,c1=21−α−2⋅11−αΓ2−α,c2=2⋅21−α−31−α−11−αΓ2−α.


## 3. Asumu Fractional Differential Mask

In [11], G. Asumu suggested new fractional derivatives to theoretically improve the Riemann–Liouville and Caputo fractional derivatives. One of the derivatives is based upon the Riemann–Liouville viewpoint and the other one is on the Caputo approach. The Asumu fractional-order operator in the Caputo sense is defined as follows:Definition 1.
Leta,b∈R, a<b, n−1<α≤n∈N. The new fractional derivative of a function ux,y in the Caputo sense of order α, with respect to the variable x, is defined by(7)Daxα,nACux,y≔1Γn−α⋅dndxn∫axx−sn+α−1⋅dndsnus,yds,where dn/dxn represents the ordinary derivative of order n with respect to the variable x.
In [11], the formula (7) is rewritten in another way.(8)D0xα,nACux,y=∏k=1nn+α−kΓn−α∫0xx−sα−1uns,yds.
Forn=1, 0<α<1, the formula (8) becomes(9)D0,xα,1ACux,y=αΓ1−α∫0xx−sα−1⋅u′s,yds.
Taking a partition of the interval0,x in N equal parts with step Δx=x/N, the numerical approximation of formula (9) takes the form(10)D0,xα,1ACux,y≈αΓ1−α⋅∑k=0N−1∫k⋅x/Nk+1⋅x/Nx−ξα−1⋅u′ξk,ydξ=αΓ1−α⋅∑k=0N−1u′ξk,y∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ.
Since the functionux,y with respect to the variable x (analogously, with respect to the variable y), is continuous in 0,x and derivable in 0,x, then, from the Lagrange theorem, there exists a point ξk∈kx/N,k+1x/N such that(11)ukx+x/N,y−ukx/N,yΔx=u′ξk,y.
Combining (10) with (11), we obtain(12)D0,xα,1ACux,y≈αΓ1−α⋅∑k=0N−1u′ξk,y∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ.=αΓ1−α∑k=0N−1ukx+x/N,y−ukx/N,yΔx∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ.
The result of the integral∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ is(13)∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ=x−ξααk⋅x/Nk+1⋅x/N=x−k+1x/Nα−x−kx/Nαα.
Therefore,(14)ukx+x/N,y−ukx/N,yΔx∫k⋅x/Nk+1⋅x/Nx−ξα−1dξ=ukx+x/N,y−ukx/N,yα⋅Δx⋅x−k+1xNα−x−kxNα=ukx+x/N,y−ukx/N,yα⋅Δx⋅x⋅N−k+1⋅xNα−N⋅x−k⋅xNα=ukx+x/N,y−ukx/N,yα⋅Δx⋅N−k+1⋅xNα−N−k⋅xNα=ukx+x/N,y−ukx/Nα⋅Δx⋅N−k−1α−N−kα⋅xNα=ukx+x/N,y−ukx/N,yα⋅Δx1−α⋅N−k−1α−N−kα.
Replacing (14) with (12), we have(15)D0,xα,1ACux,y≈1Γ1−α⋅Δx1−α⋅∑k=0N−1ukx+xN,y−ukxN,y⋅N−k−1α−N−kα=1Γ1−α⋅Δx1−α,Nα−N−1αu0+2⋅N−1α−N−2α−Nαu1+2N−2α−N−3α−N−1αu2+⋯+2N−jα−N−j+1α−N−j−1αuj+2α−1αuN−2+2⋅1α−2αuN−1−1α⋅uN,where the N+1 casual pixels u0,u1,⋯,uN corresponding to N+1 nodes are obtained after taking a partition of the interval 0,x in N equal parts with step Δx=x/N, and are given by(16)u0=u0,y,u1=uxN,y,⋮uk=ukxN,y,⋮uN=ux,y.
Analogously, with respect to the variabley, we obtain(17)D0,yα,1ACux,y≈1Γ1−α⋅Δy1−α⋅∑k=0N−1ux,ky+yN−ux,kyN⋅N−k−1α−N−kα=1Γ1−α⋅Δy1−α,Nα−N−1αu0+2⋅N−1α−N−2α−Nαu1+2N−2α−N−3α−N−1αu2+⋯+2N−jα−N−j+1α−N−j−1αuj+2α−1αuN−2+2⋅1α−2αuN−1−1α⋅uN.
Since in a digital2D gray image ux;y, the shortest distance in x and y coordinates is considered one pixel, then we put Δy=Δx=1. As a consequence, the approximate differences (15) and (17) can then be expressed in terms of the sum as(18)D0,xα,1ACux,y≈1Γ1−α⋅∑k=0NcN−k⋅uk≈D0,yα,1ACux,y,where the N+1 coefficients ckk=0,N, which depend on the parameter α, are given as(19)c0=−1αΓ1−α,c1=2⋅1α−2αΓ1−α,c2=2α−1αΓ1−α,⋮cj=2⋅N−jα−N−j+1α−N−j−1αΓ1−α,⋮cN−1=2⋅N−1α−N−2α−NαΓ1−α,⋮cN=Nα−N−1αΓ1−α.
A pixel in an image has eight neighboring pixels associated with the directions0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315°. So, in the construction of fractional masks, it is convenient to take into account these eight associated symmetrical directions. The eight masks associated with these directions are obtained by placing the coefficients given by (19) in each of these directions, as can be seen in Figure 1. The effectiveness of a mask in the improvement of the visual quality of an image lies in the way of choosing the coefficient of the pixel of the central position of the mask. For example, in [7] this value was taken as 8c0/sum, while in our proposed mask (see (22)) this constant has been taken as sum/8, where(20)sum=c0+c1+c2.
To obtain the fractional differential on the eight symmetrical directions and make the fractional differential masks have antirotation ability, eight fractional differential masks on the directions of0°, 45°, 90°, 135°, 180°, 225°, 270°, and 315° are implemented in Figure 1. For the implementation of a practical application, we take only the following three coefficients of the equation given above:(21)c0=−1αΓ1−α,c1=2⋅1α−2αΓ1−α,c2=2α−1αΓ1−α,which are used to construct the following proposed fractional differential mask:(22)R=I3+I5−I7−I9,where(23)I=c20c20c20c1c1c10c2c1sum8c1c20c1c1c10c20c20c2,sum=c0+c1+c2.Figure 1
Eight masks based on the eight symmetric directions.
(a)(b)(c)(d)(e)(f)(g)(h)


## 4. Experimental Results

The aim of this section is to demonstrate that the fractional differential mask based on the Asumu (FDMA) definition can avoid the block effect while enhancing the quality of dark images. To this end, we analyze the texture-enhancing ability of the proposed mask and discuss the relationship between the fractional power parameterα and texture-enhancing details, as well as compare it with histogram equalization (HE) and the methods proposed in [7, 9]. In Figures 2–5, we can see the images obtained by the proposed method and by the compared methods. For comparison purposes, we have used the contrast of the gray-level co-occurrence matrix (GLCM) in four directions as the capability of contrast enhancement techniques. And Tables 1–4 are the contrasts of GLCM in four directions. Even though the statistical results of the proposed method are poor in comparison with the compared methods (as we can see in Tables 1–4), the proposed method achieves overcoming the block effect while improving the quality of dark images. The other methods used in the comparison, despite the fact that they offer better statistical results (higher GLCM values in four directions), cannot overcome the block effect in the process of improving the contrast of dark images. To observe this difference with respect to the block effect, it is enough to zoom in on the processed images. Once the zoom is done, it will be seen that the images obtained by the proposed method are free of the block effect, while the images obtained by using other techniques have the block effect. This proves that the proposed method achieve to solve the problem of the block effect, which cannot be solved by other methods. Unfortunately, the price paid for using our method in order to overcome this effect is the low contrast in the obtained images, as it is reflected in the tables of statistical values (see Tables 1–4). An example of the proposed algorithm in the MATLAB code is shown in Figure 6; while the MATLAB algorithm of our proposed method was done as follows:(1)
Reading the image by the use of the function imread ()(2)
Converting the image into a grayscale image by using the function rgb2gray ()(3)
Establishing theα value(4)
Reading the coefficientsc0, c1, and c2(5)
Reading the term sum, given bysum=c0+c1+c2/8(6)
Reading the matrixI(7)
Reading the matrixR(8)
Filtering the image with the maskR with the help of the function conv2 ()(9)
Showing the filtered image by using the function imshow ()Figure 2
Dark image contrast enhancement by using different methods: (a) is the original image of a man in a dark room; (b) is the obtained image by processing the image of (a) through the HE method; (c) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.0051; (d) is the obtained image by processing the image of (a) using our proposed method for α=0.799; (e) is the obtained image by processing the image of (a) using the method proposed in [7] for α=0.006; (f) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.003.
(a)(b)(c)(d)(e)(f)Figure 3
Dark image contrast enhancement by using different methods: (a) is the original image of a dark office; (b) is the obtained image by processing the image of (a) through the HE method; (c) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.0053; (d) is the obtained image by processing the image of (a) using our proposed method for α=0.8; (e) is the obtained image by processing the image of (a) using the method proposed in [7] for α=0.006; (f) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.012.
(a)(b)(c)(d)(e)(f)Figure 4
Dark image contrast enhancement by using different methods: (a) is a dark girl's original image; (b) is the obtained image by processing the image of (a) through the HE method; (c) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.0053; (d) is the obtained image by processing the image of (a) using our proposed method for α=0.8; (e) is the obtained image by processing the image of (a) using the method proposed in [7] for α=0.006; (f) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.003.
(a)(b)(c)(d)(e)(f)Figure 5
Image contrast enhancement by using different methods: (a) is the original dark building image; (b) is the obtained image by processing the image of (a) through the HE method; (c) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.0053; (d) is the obtained image by processing the image of (a) using our proposed method for α=0.8; (e) is the obtained image by processing the image of (a) using the method proposed in [7] for α=0.006; (f) is the obtained image by processing the image of (a) using one of the methods proposed in [9] for α=0.012.
(a)(b)(c)(d)(e)(f)Table 1
Contrast of GLCM in 4 directions of a man in a dark room image.
Angle0°45°90°135°Original image0.00070.08880.08840.0888MethodHE0.25940.46940.30380.4799CF method [9] for α=0.01650.67081.19250.80981.2050Proposed method forα=0.7990.16520.41580.29940.4179Method [7] for α=0.0061.06701.77141.32351.8091Caputo method [9] α=0.0030.55761.29921.05001.3155Table 2
Contrast of GLCM in 4 directions of a dark office image.
Angle0°45°90°135°Original image0.11480.11480.00010.1148MethodHE0.58631.25040.75301.2734CF method [9] for α=0.00531.19262.19711.29132.2879Proposed method forα=0.80.27760.47910.23940.4848Method [7] for α=0.0062.16483.18561.91333.3170Caputo method [9] α=0.0121.59682.28301.31092.3450Table 3
Contrast of GLCM in 4 directions of the image of a woman in a dark room.
Angle0°45°90°135°Original image0.00030.00040.00030.0004MethodHE0.28150.54100.38490.5315CF method [9] for α=0.00530.64530.91960.63710.9073Proposed method forα=0.80.16870.27950.14330.2749Method [7] for α=0.0060.98531.47621.20101.4555Caputo method [9] α=0.0030.79411.38361.06331.2999Table 4
Contrast of GLCM in 4 directions of the image of a dark building.
Angle0°45°90°135°Original image0.17490.38540.21640.3845MethodHE0.74931.50590.82641.4670CF method [9] for α=0.00532.05463.86862.21863.8428Proposed method forα=0.80.31430.54360.26920.5310Method [7] for α=0.0063.75346.30733.97506.2433Caputo method [9] α=0.0032.65480.42472.80754.4615Figure 6
The MATLAB code of the proposed algorithm.


## 5. Conclusions

The goal of this paper is to overcome the block effect while improving the visual quality of dark images. For this purpose, we have established a new fractional differential mask based on the Asumu fractional differential operator. Even though the statistical results of the proposed method are poor in comparison with the compared methods, we observed that the proposed method has avoided the presence of the block effect in the obtained images, which did not occur in the images obtained by the compared methods. This means that grace to the proposed method, we have solved the block effect problem. Unfortunately, the price paid for using our method in order to overcome this effect is the low contrast in the images obtained by this method, which is reflected in the tables of statistical values (see Tables1–4). The experimental analysis of the proposed mask has shown that images corresponding to values of α belonging to the interval 0.79,0.81 look better. In future work, we plan to improve the algorithm presented in this paper to identify objects in dark images.



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