The cell, which is the basic element of the CNN structure, is composed of structurally linear and non-linear circuit elements, such as capacitors, linear resistances, linear and non-linear controlled sources and independent sources. The first CNN cell structure in the literature proposed by Chua [1] is shown in Figure 2.

In Figure 2, E_ij is the independent voltage source, I_ij is the independent current source, C_x is the capacitor, R_x and R_y are linear resistors, I_xu(i,j;k,l) and I_xy(i,j;k,l) are linear voltage controlled current sources with the characteristics of I_xu(i,j;k,l) = B(i,j;k,l)v_ukl and I_xy(i,j;k,l) = A(i,j;k,l)v_ykl, respectively. i and j indicate inner-cell index and k and l are neighborhood indexes of M × N size network. v_xij and I_ij shows the state voltage and bias current values of inner-cell (C(i,j)), respectively. These templates are described more clearly in Section 2.2. The only nonlinear element is piecewise linear voltage controlled current source with the characteristics of I y x i j = 1 / 2 R y ( | v x i j + 1 | − | v x i j − 1 | ).

In Equation (1), the radius (r) neighborhood of a C(i,j) cell in the CNN is given: (1) N r ( i ,   j ) = { C ( k , l ) | max { | k − i | , | l − j | } ≤ r ,         1 ≤ k ≤ M ; 1 ≤ l ≤ N }

When the symmetry property is applied to all CNN, if C(i,j) ∈ N_r(k,l), then C(k,l) ∈ N_r(i,j), for all C(i,j) and C(k,l) cells. For a CNN, the time dependent (t) situations and output formulas are given in Equations (2) and (3), respectively:

State Equation: (2) C d v x i j ( t ) d t = − 1 R x v x i j ( t ) + ∑ C ( k , l ) ∈ N r ( i , j ) A ( i , j ; k , l ) v y k l ( t ) + ∑ C ( k , l ) ∈ N r ( i , j ) B ( i , j ; k , l ) v u k l ( t ) + I i j                 1 ≤ i ≤ M ; 1 ≤ j ≤ N

Output Equation: (3) v y i j ( t ) = 1 2 ( | v x i j ( t ) + 1 | − | v x i j ( t ) − 1 | ) ,                     1 ≤ i ≤ M ; 1 ≤ j ≤ N

In Equation (2), v_ukl(t) and v_ykl(t) are the input voltage and output voltage of the (k,l)th neighbor cell. v_xij(t) and I_ij as described above are the state and the bias current values of the cell C(i,j), respectively. v_yij(t) given in Equation (3) is the output voltage equation of piecewise linear function and it is called the output function. In image processing applications, the input and output voltage of any cell represent the values of same index number pixel of input image and output image, respectively. The operation limit of a CNN is between –1 and 1, while any pixel value of an image changes from 1 to 255. Therefore, pixel values of image are converted into operation limits of the CNN. Eventually, (–1) and (1) values indicate ‘black pixel’ and ‘white pixel’, and middle values represent ‘gray-level pixel’. In other words, v_uij(t) and v_ukl(t) indicates values of inner pixel and neighbor pixel in input image while v_yij(t) and v_ykl(t) indicates value of inner pixel and neighbor pixel in output image, respectively.

The dynamic behavior of a CNN is determined by the cloning template that can be considered as the indirect ratio of the inter-cell connections. The feedback cloning template A(i,j;k,l), the feed-forward cloning template B(i,j;k,l) and threshold cloning template I(i,j) given in Equation (2), form the cloning template for a cell. The feedback and feed-forward template are related with the outputs and inputs of neighbor cells. However, the threshold template is only related with the inner-cell. The C(i,j) cell is connected to neighboring cells at a ratio determined by these cloning template. Realizing the desired image processing can be done by arranging the values of this cloning template appropriately. For the r = 1 neighborhood of the cell, the matrix structure of the cloning template A(i,j;k,l), B(i,j;k,l) and I(i,j) are given in Equation (4). (4) A ( i , j ; k , l ) = [ a i − 1 , j − 1 a i − 1 , j a i − 1 , j + 1 a i , j − 1 a i , j a i , j + 1 a i + 1 , j − 1 a i + 1 , j a i + 1 , j + 1 ]         B ( i , j ; k , l ) = [ b i − 1 , j − 1 b i − 1 , j b i − 1 , j + 1 b i , j − 1 b i , j b i , j + 1 b i + 1 , j − 1 b i + 1 , j b i + 1 , j + 1 ]             I ( i , j ) = z ( i , j )

The statement of Equation (4) can be stated as a general definition as in Equation (5): (5) A = [ a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 ]                             B = [ b 1 b 2 b 3 b 4 b 5 b 6 b 7 b 8 b 9 ]                 I = zwhere A, B and I are represented as the template set, and this set, which is designed for the desired application, is given in Equation (6) as a vector: (6) x i = [ a 1 ,   a 2 ,   a 3 ,   a 4 ,   a 5 ,   a 6 ,   a 7 ,   a 8 ,   a 9 ,   b 1 ,   b 2 ,   b 3 ,   b 4 ,   b 5 ,   b 6 ,   b 7 ,   b 8 ,   b 9 ,   z ]

If the network cloning template of CNN can be arranged according to its purpose, the CNN can realize the desired simultaneous image processing successfully due to its ability to perform parallel processing. In the following section, the desired x_i template set is obtained by using the ABC optimization algorithm for realizing edge detection processes on a CNN structure. The optimization structure and architecture of ABC algorithm are defined the following section.

This algorithm is developed based on the foraging principle of honeybees. There are three types of foraging bees in a honey bee colony; employed bees, onlooker bees and the scout bees. At the first stage, half of the colony is composed as employed bees (n_e) and the other half is composed as onlooker bees (n_o). There is only one employed bee for each nectar source. Nectar sources symbolize possible solutions. That is, the number of employed bees is equal to the number of nectar sources. The basic steps of the algorithm can be stated as follows:

Step 1: Assign the control parameter values

Each cycle is mainly composed of three phases. For the first phase, the employed bees are sent to the sources and the nectar amounts of the sources visited are calculated. For the second phase, onlooker bees are sent to their sources and their nectar amounts are determined. For the third phase, it is ensured that the scout bee is located on a randomly selected new source. A food source corresponds to a possible solution to the problem where optimization is being attempted. The nectar amount of any source represents the quality level of the solution represented by that source. There are scout bees searching randomly in each colony. These bees do not use any kind of preliminary knowledge while searching for food and the search procedure is completely random. In the ABC algorithm, one of the employed bees is selected and made scout bee. This selection process is made based on the “limit” parameter. If a solution representing a source couldn’t be developed with a certain number of trials, this source is abandoned and the employed bee visiting this source becomes the scout bee. The number of trials determined for abandoning the source is identified by the “limit” parameter. In a robust search process, both exploration and exploitation occur simultaneously. The employed and onlooker bees are in charge of exploiting sources while scout bees are responsible for the exploration process. The bees work to maximize the energy function, stating the amount of food brought to the hive at any given unit time. The selection possibility is based on a probability function. After watching the dance of the employed bees and selecting the source at its position with the calculated probability value, the onlooker bee determines a source within the neighborhood of this source and starts collecting the nectar of this source. The position of the selected neighbor is calculated. If the nectar amount of the source at the new position is more than old one, then the bee visits the hive and shares her information with others and new position is memorized. Otherwise, old position is still kept in memory. If the nectar source at the a position remained constant without improving through cycles defined by the number of the “limit” parameter, the source at this position is abandoned and the employed bee of that source becomes the scout bee, doing random research. The newly found source is assigned instead of abandoned position [36–38]. The mathematical description of the ABC algorithm is explained exhaustively in the following section.

In this section, the working mechanism of ABC is presented mathematically. The position of a food source (x_i) represents a feasible solution of problem and the amount of nectar of the food source indicates the fitness value of the associated solution in the ABC algorithm. The colony size of employed bees (n_e) is equal to the colony size of onlooker bees (n_o) in the population. A set of food sources positions (x₁,…, x_ne) is produced randomly: (7) x i = x j min + r a n d ( 0 , 1 ) ( x j max − x j min )where i = 1,…,SN and j = 1,..,D. SN is the number of food sources and D is number of optimization parameters. Also, it must be noted that SN = n_e = n_o. All counters associated with solutions are reset to 0 in this phase. The colony of employed bees can be expressed by n_e dimension vector x⃗(n) = (x₁ (n), ..., x_ne (n)), where n is cycle value of ABC algorithm. In additional, individual search space can be signified as S. x_i ∈ S and i ≤ n_e in x⃗. After initialization, the fitness value of each solution is calculated and x⃗(0) is obtained. To evolve quality of solutions, employed bees change their position from the current position to neighboring source positions by using the following equation: (8) v i j = x i j + ϕ i j ( x i j − x k j )where j ∈ {1, ..., D}, k ∈ {1, ..., SN}, k ≠ j, j and k are randomly chosen index. Ø_ij is a real number produced randomly in the range [–1,1]. Values of parameters produced in this process are within in determined boundaries (v ∈ S and S → S). After producing v_i, to select better solution for next generation, greedy selection operator is applied. Probability distribution of this operator can be given as follows: (9) P { x i ,   v i } = { 1 , f ( v i ) ≥ f ( x i ) 0 , f ( v i ) < f ( x i )where f(v_i) and f(x_i) are nectar amounts of food sources at v_i and x_i, respectively. If v_i is better solution than x_i, the employed bee memorizes position of v_i, otherwise position of x_i is retained. In the end of this process, if a better solution cannot be obtained, the trials counter associated with the solution is incremented by 1, and otherwise it is reset to 0.

After the employed bees phase is completed, the phase of onlooker bees is started by selecting an employed bee from the colony. The probability of the selection process depends on the fitness values of the solutions and many selection schemas such as roulette wheel and stochastic universal sampling can be used. Probabilistic function of roulette wheel mostly used ABC can be described as follows: (10) P { x i } = f ( x i ) ∑ i = 1 S N f ( x i )

This function means that, if the fitness value of a solution increases, the visiting number of onlooker bees to that source increases too. To decide if a modification will be made on an onlooker bee position, a random real number within the range [0,1] is generated for each source. If the generated number is less than the probability value in Equation (10), then the onlooker bee changes position by using Equation (8) to find new solutions. Greedy selection is applied to the modified source and then if the new position is better than the old position, the memory of the onlooker bee is updated, otherwise the old position is kept. According to the result of this process, the counter associated with onlooker bees is incremented by 1 or reset to 0 similar to the operation in the employed bee phase.

To end a cycle, if a counter value of employed bees and onlooker bees reaches its “limit” value, the source of this counter is abandoned. A new food source is discovered by the scout bee and it replaces the abandoned source. This operation can be defined as follows: (11) x i ( n + 1 ) = { x min + rand ( 0 , 1 ) ( x max − x min ) , counter ≥ limit x i ( n ) , counter < limit

This operation is a noticeable property of ABC algorithms because this operation is different from other algorithms and it can improve the search efficiency of the ABC. A detail theoretical explanation about convergence and complexity analysis of ABC can be found in [39]. ABC optimization study is explained exhaustively in order to realize proposed application by using ABC in the following section.

Ideal edge detection of a noise free image which has homogenous objects or regions is achieved by using the pseudo code given below. The image obtained by using this method is used as the desired output image of the proposed method in this paper:

start

for i=1:m

  for j=1:n

    if eP_i,j < eP_i+1,j then dP_i,j=1

    if eP_i,j < eP_i,j+1 then dP_i,j=1

    if eP_i,j > eP_i+1,j then dP_i+1,j=1

    if eP_i,j > eP_i,j+1 then dP_i,j+1=1

  end

end

Here, i and j indicate the coordinate index of the pixels in the image, eP indicates the pixel value of the training input image, dP indicates the pixel value of the training output image. In order to explain how the Pseudo code works, a part of the pixel values of an artificial image given in Figure 3(a) is used. The lines with pixel changes is detected using the given pseudo code and the ideal edge detection in Figure 3(b) is made and the result is obtained. It is important that the lines be determined in one pixel thinness and with continuous lines for the edge detection quality.

The image of 374 × 374 pixel size given in Figure 4(a) is used as the training input image. The training input image is an artificial image that is formed in a way that it covered 62% of the grayscale. Each octahedral part has a homogeneous area with the same pixel value. As explained in above, the ideal edge detection process is performed by determining the edge lines where homogeneous octahedral areas intersect. Hence, the desired output image is obtained, which is given in Figure 4(b). Moreover, the image histograms of input and desired images are demonstrated in Figure 4(c,d), respectively.

In this section, the assumptions and the optimization structure that are used to design the CNN’s cloning template are explained. In order to guarantee the stability of the CNN, the symmetry assumption given in Equation (12) is applied to the cloning template defined in Equation (5) and eventually Equations (13,14) are obtained [3]. As a result, the duration of the optimization is reduced by decreasing the template parameters to be optimized and additionally it is ensured that stable outputs can be acquired for the CNN: (12) A ( i , j ; k , l ) = A ( k , l ; i , j ) B ( i , j ; k , l ) = B ( k , l ; i , j ) } | x i j ( 0 ) | ≤ 1 ,         | u i j | ≤ 1 ,             1 ≤ i ≤ M ; 1 ≤ j ≤ N (13) A = [ a 1 a 2 a 3 a 4 a 5 a 4 a 3 a 2 a 1 ]           B = [ b 1 b 2 b 3 b 4 b 5 b 4 b 3 b 2 b 1 ]       I = z (14) x i = [ a 1 ,   a 2 ,   a 3 ,   a 4 ,   a 5 ,   b 1 ,   b 2 ,   b 3 ,   b 4 ,   b 5 ,   z ]

The ABC design mechanism of the cloning template is shown in Figure 5. In this mechanism, the output image of the CNN converges to the desired image by adjusting the cloning template of the CNN by means of the ABC algorithm. The ABC algorithm produces template set and sends this set to the the CNN. CNN runs by using this set and training image, and it generates an output image. The fitness value of the objective function is calculated by comparing the output image of the CNN and the ideal edge detected image. The ABC algorithm evaluates with this fitness value in order to obtain better results.

Running of ABC optimization can be explained as follows: a D × SN size matrix is randomly selected as the initial colony and each row of the matrix represents a cloning template set, a possible solution to the problem, in the ABC optimization. D is the dimension of the problem and SN is the size of the colony. There are three main phases in the ABC algorithm; the employed bees phase, the onlooker bees phase and the scout bees phase.

The fitness value of each bee in the employed bee’s colony (n_e) is calculated by using Equation (16) which is based on the correlation between output image of the CNN mechanism and the desired output Figure 4(b). Local search is applied by Equation (10) at neighbors of all existing cloning templates. If better fitness values are obtained after the local search process, these templates are included in the population and the employed bees phase is completed. The onlooker bees phase is similar to the previous phase. The only difference from the other is the operation of the local search mechanism. This operation is not applied to all template sets, it is applied to some template set which is the probability selected with the roulette wheel. Thus, the selection of the worse template is as much possible as selection of a better template and therefore diversity in the population is ensured. If a solution is not improved in both phases, the testing counter that is related with the solution is incremented by 1, otherwise the counter reset to 0. In the scout bees phase, which template is removed from population is arranged by controlling the testing counters by comparing these counters with the parameter named “limit”. If the testing counter value of a template set reaches the limit value, this set is removed from the population and a new cloning template set produced randomly is replaced instead of the removed templates. All the explained processes are repeated as many times as indicated by the as number of cycles of the ABC: (15) C = ∑ i m ∑ j n ( z P i j − z P ¯ ) ( t P i j − t P ¯ ) ( ∑ i m ∑ j n ( z P i j − z P ¯ ) 2 ) ( ∑ i m ∑ j n ( t P i j − t P ¯ ) 2 ) (16) f ( x i ) = 1 C + ɛ ,                   0 ≤ r ≤ 1where i and j are pixel index on the image, zP is pixel value of the CNN output image, tP is pixel value of the edge detected training image, z P ¯ is mean of the output image, t P ¯ is mean of the edge detected training image, ɛ is a small positive constant, and f is objective function. The C value in Equation (16) is the correlation between the two images and this is given in Equation (15).

By calculating the similarity between the output images of the CNN and the ideal edge detected image, the optimization process is continued to prove better results by the ABC algorithm. Simulations are developed on a MATLAB platform and the processing of the CNN structure is realized using the MATCNN library [40,41].

The performance and robustness of ABC algorithm in determining the edge detecting CNN cloning template are assessed statistically. The optimization results are obtained with multiple runs on different control parameters of ABC; “limit” and the “size of colony” (NP), which are the control parameters of ABC, are considered in the simulations [36].

In order to show the effect of the scout production mechanism on the performance of the algorithm, the average of the best function values found for the different “limit” values (1 × n_e × D, 2 × n_e × D, 4 × n_e × D and “without scout”) and colony sizes (20, 40 and 80) is given in Table 1. In the simulations, the cycle and time step constant of the CNN are selected as 30 and 0.3, respectively. Three thousand cycles are set for each combination of NP, and “limit” value and ABC is run independently 30 times and the results obtained (f values in Equation (16)) are given in Table 1 as mean (MEAN) and standard deviation (STD).

In Table 1, it is seen that STD values are very low in all combinations of NP and “limit” values. Therefore, it can be stated that the ABC optimization technique for this problem is quite a robust algorithm. Moreover, the fact that the MEAN value gets smaller as the n value gets bigger supports the assumption that choosing a smaller n value can lead to better results. Beside the results of Table 1, in order to present a stable evaluation of the effect of control parameters of the ABC on its performance, ANalysis Of the VAriance (ANOVA) based on the mean variance is used. ANOVA examines the statistical difference of one, two or more groups over one quantitative one. The null hypothesis we established for ANOVA is that all population means are equal and the other hypothesis is that at least one mean is different from the others. Calculated F value is compared to the value coming from the F distribution associated with the degrees of freedom and significance value. If this value is lower than 0.05, the effect of the variable is considered to be statistically significant. ANOVA statistics are calculated by using 12 different values of control parameters on MATLAB platform. Each data group has 30 f values coming from each run. Results of ANOVA test for NP and limit control parameters are presented in Table 2. In the table, each column presents values of the sum of squares (SS), the degrees of freedom (df), Mean Squares (MS), which is the ratio SS/df and F statistics, which is the ratio of the mean squares. From the results in Table 2, it can be said that f function is influenced significantly by the values of NP and limit.

At the values of MEAN = 0.7935 and STD = 0.0521 where optimization is at its best and most effective, the population size and limit values are 80 and 440, respectively. The best C value obtained so far is C = 0.9331 and the cloning template obtained that correspond to this value are presented in Equation (17):

The cloning template given in Equation (17) is the optimal template designed by the ABC algorithm in this study for the purpose of edge detection.

In this section, cloning templates of previous studies are presented and used to compare with the ABC results. Designing the cloning template of CNN is an important task to achieve the desired results and several technique-based analytic methods, local learning algorithms and heuristic methods are proposed. Some of heuristic methods also used are Differential Evolution (DE) [27], Evolution Strategies (ES) [28] techniques and Genetic Algorithm (GA) [22,32]. The first studies in the CNN area were concentrated around binary images and later, some studies including gray-level images are presented. Most template design studies do not include performance analyses of heuristic methods for edge detection application and they are only introduced for design studies. In addition, templates presented in previous studies are not compared subjectively and quantitatively with other templates determined in the same area. Furthermore, edge detection studies based on CNN introduced without applying threshold process with grayscale real images first are very limited. Therefore, there are doubts about the reliability of heuristic methods in this kind of applications. In order to alleviate these deficiencies, a performance analysis of the ABC is performed with multiple runs using its different control parameters. The cloning template obtained with the ABC algorithm is compared subjectively and quantitatively with other well-known techniques and previous studies on designing CNN cloning templates. In the following tables, cloning templates determined in previous studies for edge detection applications are presented. One of the presented cloning templates is from the MATCNN Template Library (EDT-ML) [41]. The other cloning template is obtained by using the Differential Evolution method (DE-CNN) [27]. Another cloning template is introduced by using Evolution Strategies (ES-CNN) [28] and Xavier [29]. In the following sections, comparisons are made using these templates.

The edge detection quality of the templates in Equation (17), determined as a result of optimization, is assessed by using both artificial binary and grayscale images, and real images. Edge detection is performed on all images by using the well-known classical edge detector and the previously proposed CNN based edge detectors and the novel CNN templates in Equation (17).

Firstly, subjective comparisons are presented in Figures 6–9. For better evaluation of these subjective comparisons, the proposed method is given at the end of the rows of figures. The study is continued with objective evaluation based on results of accepted comparison metrics.

Subjective comparisons of obtained cloning template and well-known edge detection methods by use of the binary/grayscale artificial images are given in Figure 6.

Figure 6(a) shows that all classic edge methods determined the ideal edges of the Text image successfully. For the Shape image, as can be seen in Figure 6(b), it is seen that compared with the proposed method the other methods are unsuccessful as indicated by the duplicate lines that occur in the output edge image. It is obvious that for the gray-level artificial images, the Check, Honeycomb and Ledge, as shown in Figure 6(c–e), respectively, the proposed method detected the edges as well as the ideal edge detection, and, in fact is considerably more successful than well-known edge detection methods. Subjective comparisons of obtained cloning template and previous studies based on CNN by use of the binary/grayscale artificial images are given in Figure 7.

Figure 7(a) shows that methods based on CNN, except for DE-CNN, determined the ideal edges of the Text image successfully. For the Shape image, as can be seen in Figure 7(b), ES-CNN templates, Xavier templates and ABC-CNN templates detects all boundaries as single lines. However the DT-CNN method is unsuccessful as indicated by the duplicate lines that occur in the output edge image and the results of the ED-TML method are unclear. It is obvious that for the gray-level artificial images, the Check, Honeycomb and Ledge, as shown in Figure 7(c–e), respectively, the proposed method detected the edges as well as the ideal and noiseless edge detection, and, in fact is considerably more successful than other methods. Subjective comparisons of obtained cloning template and well-known edge detection methods by use of the binary/grayscale real images are given in Figure 8.

For real images, Coin, Wheel, Flowers and Church, the edge detection results are given in Figure 8(a–d), respectively. As can be seen from Figure 8, the results of Roberts, Prewitt and Sobel are unsatisfactory. The results obtained by the Canny methods are a bit better than the others; however, the proposed ABC-based CNN method has the best results, with more details and less noise in the edge images. For example, the results of the Canny method have duplicate edge lines for the outer borders of the coins in Figure 8(a) and loss of details for Figure 8(b–d). Subjective comparisons of obtained cloning template and previous studies based on CNN by use of the binary/grayscale real images are given in Figure 9.

As can be seen the results of ED-TML are unsatisfactory. In the Xavier and ES-CNN method results, the output images have too much noise, which reduces the perception of the edge detection output. There are many undetected objects in the DE-CNN results, especially for Figure 9(b,c). It is obvious that the results of the proposed ABC-based CNN method have more distinctive details than other methods and less noise in the edge images.

Besides visual evaluation of the edge detection quality on binary and grayscale artificial images, it is also the aim of the study to assess the edge detection quality quantitatively. For this reason, the similarity between the results shown in Figures 6 and 7 and the ideal edge detection results given in Figure 10 are compared.

Correlation (C), structural similarity (SSIM) [43] and mean squared error (MSE) values are used as the comparison operators and results are presented in Table 4. To calculate the correlation, the equation given in Equation (15) is used. Mean Squared Error (MSE) is given in Equation (18): (18) M S E = 1 m n ∑ i = 0 m − 1 ∑ j = 0 n − 1 ( z P ( i , j ) − t P ( i , j ) ) 2where m × n is size of image, i and j are coordinate index of the pixels in the image. SSIM, which is another similarity operator used for comparison, is given in Equation (19) [43]: (19) S S I M = ( 2 μ z R μ t R + c 1 )   ( 2 σ z R t R + c 2 ) ( μ z R 2 + μ t R 2 + c 1 )   ( σ z R 2 + σ t R 2 + c 2 ) (20) c 1 = ( k 1 L ) 2 ,     c 2 = ( k 2 L ) 2

Here, μ_zR, μ_tR are the mean gray-level intensity values of zR and tR images, respectively. σ_zR ² and σ_tR ² are the variance of zR and tR images, respectively. σ_zRtR is the covariance of zR and tR images. L is the dynamic interval of the pixel value and k₁ and k₂ are the constant numbers that are 0.01 and 0.03 [43].

The higher values for C and SSIM, and the lower values for MSE in Table 4 imply better results. As previously mentioned for the visual evaluation of the Text image results, all methods give better results. However, the quantitative results given in Table 4 show that some of the methods are unsuccessful in terms of C and MSE. This is because the output edge images have drifts along the edge borders in comparison to the ideal edge image. A similar situation can be seen on the results of other images. As a result, it is clearly seen from Table 4 that the ABC-based CNN method performs better than others in terms of C, MSE and SSIM. Also, these numerical results confirm the previously mentioned visual evaluations.