#
An Improved Local Search Genetic Algorithm with a New Mapped Adaptive Operator Applied to Pseudo-Coloring Problem^{ †}

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Related Works

## 3. Mathematical Formulation of Pseudo-Coloring Problem

## 4. Methodology

- ${\mathit{M}}_{1}$
**Domain definition**: This phase consists of defining the image in grey-levels that must be pseudo-colored by our technique.- ${M}_{1.1}$
- However, all results can be easily extended to consider classes of images that have not passed any segmentation process, given the advances achieved in recent years in the area of image segmentation [42,43,44]. This type of consideration is outside the scope of the paper, so we will assume in this work that the method must work on an image in grey levels I pre-segmented into K sub-regions.
- ${M}_{1.2}$
- Once we have the pre-segmented image I, we need to define the neighborhood matrix $\Delta $ of this image so that the algorithm can perform the optimization having the information of which subregions of I are neighbors to each other and, consequently, they are assigned different colors.

- ${\mathit{M}}_{2}$
**Proposed optimization method**: In this step of the methodology, we use the proposed meta-heuristic, which consists of a genetic algorithm that was specifically designed to search in a color space for an optimal solution for PsCP. Our main contributions to the method are given in the form of two operators:- ${M}_{2.1}$
**Local search strategies**: As in our preliminary work [21], we will also use local search strategies in our GA in two operators: the mutation operator and an exclusive operator dedicated to conducting a local search with massive behavior.- ${M}_{2.2}$
**Adaptive rules**: In this text, we generalize the use of the rules for self-adaptation of parameters proposed in our preliminary work [21] having as inspiration ideas that have been successfully used in other classes of combinatorial optimization problems [20]. In this case, we make use of mapping functions to automate the way that mutation and crossing rates are updated in the course of GA iterations.

- ${\mathit{M}}_{3}$
**Algorithm responses**: After the optimization process carried out by our technique, we obtain two types of responses: a numerical and vector solution in the K-dimensional color space (${M}_{3.1}$) and the pseudo-colored image (${M}_{3.2}$). In the first case, we have numerical and quantitative information on how different are the colors attributed to I, calculated from the measures of Radlak and Smolka [14]. In the second case, the method must establish the pseudo-colored version of I from the obtained vector solution.- ${\mathit{M}}_{4}$
**Validation scenarios**: In our methodology, we will validate the solutions found by our meta-heuristics in a quantitative (${M}_{4.1}$) and in a qualitative (${M}_{4.2}$) ways through three different case studies (CS):- -
**CS I**: In the first CS, we will compare the quantitative and qualitative results obtained by our method that represent information on how distant the colors attributed to the I regions are in the CIELAB space and compare the results with the most recent color-coding techniques available in the specialized literature.- -
**CS II**: In the second CS, we will compare the quantitative results obtained by our method in 24 synthetic and abstract unreal images in comparison to the other existing techniques in this field of study.- -
**CS III**: In the third CS, we we will compare the quantitative results obtained by our method considering the Munsell atlas color space in comparison to the other existing techniques in the specialized literature.

## 5. Mapped Local Search Adaptive Genetic Algorithm for PsCP

- A new self-parameter adjustment operator [17] to avoid well-known problems in GA such as premature convergence and getting stuck in local optima. In addition, we present the generalization of our preliminary adaptive strategy from [21] using mapping functions, which proved to be favorable in studies related to other types of combinatorial optimization [19,20].
- The advancement of experimental results in benchmarks that define the state-of-the-art.

#### 5.1. Chromosome Decoding

#### 5.2. Fitness Function

- Both ${\overline{F}}_{\Delta}$ and ${F}_{\Delta}$ are defined taking into account the neighborhood matrix $\Delta $. That is, both functions are defined from the image structure that we intend to color.
- The main difference between ${\overline{F}}_{\Delta}$ and ${F}_{\Delta}$ is that the first one takes into account the order of the coordinates in the chromosome C, while the second one acts on a set $\mathcal{C}$. Didactically, it is simpler to reproduce ${\overline{F}}_{\Delta}$ and, therefore, we define the fitness function in this way.
- The meta-heuristic that finds the greatest values assumed by ${\overline{F}}_{\Delta}$ will find possible solutions to the problem presented in Equation (4), since these two situations consist of determining K colors with the greatest possible visual dissimilarity. In this way, the feasibility and the equivalence of the problem are maintained.

#### 5.3. Selection Process

#### 5.4. Crossover Operator

#### 5.5. Mutation Operator

- ${\mathrm{Mut}}_{1}$: In this routine, a mutation function is randomly taken into a set of mutation functions, and, using this function, ${N}_{{\mathrm{Mut}}_{1}}$ perturbations are performed on the chromosome, so that the good perturbations are maintained and the bad ones are ignored.
- ${\mathrm{Mut}}_{2}$: In this routine, a simple Gaussian mutation is applied to the chromosome.

Algorithm 1 Proposed mutation operator. | ||

Input: | ${P}_{\mathrm{kids}}$ | Offsprings generated in crossover |

${\overline{F}}_{\Delta}$ | Fitness function | |

${p}_{\mathrm{mut}}$ | Probability of mutation | |

${p}_{\mathrm{LS}}$ | Probability of ${\mathrm{Mut}}_{1}$ | |

${N}_{{\mathrm{Mut}}_{1}}$ | Number of mutation function applications for ${\mathrm{Mut}}_{1}$ | |

1:${f}_{\mathrm{mut}}:=\mathrm{rand}\_\mathrm{take}\left(\{{f}_{\mathrm{swap}},{f}_{\mathrm{invert}},{f}_{\mathrm{insert}}\}\right)$ ▹ The function $\mathrm{rand}\_\mathrm{take}\left(\xb7\right)$ randomly returns an element from a set. 2:${P}_{\mathrm{mut}}:=\left\{\right\}$ 3: for $C\in {P}_{\mathrm{kids}}$do4: if $\mathrm{rand}\left([0,1]\right)\ge {p}_{\mathrm{mut}}$ then ▹ Apply mutation in ${p}_{\mathrm{mut}}$ percent of kids.5: break 6: else if $\mathrm{rand}\left([0,1]\right)\le {p}_{\mathrm{LS}}$ then ▹ Apply ${\mathrm{Mut}}_{1}$ in ${p}_{\mathrm{LS}}$ percent of cases.7: ${F}_{C}:={\overline{F}}_{\Delta}\left(C\right)$ 8: for $k:=1$ to ${N}_{{\mathrm{Mut}}_{1}}$ do ▹ Apply ${f}_{\mathrm{mut}}$ ${N}_{{\mathrm{Mut}}_{1}}$ times.9: $i:=\mathrm{rand}\_\mathrm{take}\left(\{1,2,\dots ,3K\}\right)$ 10: $j:=\mathrm{rand}\_\mathrm{take}\left(\{1,2,\dots ,i-1,i+1,\dots ,3K\}\right)$ 11: $\widehat{C}:={f}_{\mathrm{mut}}\left(C,(i,j)\right)$ 12: ${F}_{\widehat{C}}:={\overline{F}}_{\Delta}\left(\widehat{C}\right)$ 13: if ${F}_{\widehat{C}}\ge {F}_{C}$then ▹ If the perturbation is beneficial, then it must be maintained.14: $C:=\widehat{C}$ 15: ${F}_{C}:={F}_{\widehat{C}}$ 16: end if17: end for18: else ▹ (${\mathrm{Mut}}_{2}$) Apply ${f}_{\mathrm{Gauss}}$ only once.19: $C:={f}_{\mathrm{Gauss}}\left(C\right)$ 20: end if21: ${P}_{\mathrm{mut}}:={P}_{\mathrm{mut}}\cup \left\{C\right\}$ 22: end for | ||

Output: | ${P}_{\mathrm{mut}}$ | Population of mutated individuals |

#### 5.6. Massive Local Search Operator

**Step 1**: The color ${c}_{i}$ receives a random increase, making it lighter.**Step 2**: The perturbation is maintained only if it is beneficial, increasing the fitness value of ${C}_{\mathrm{Best}}$.**Step 3**: If ${c}_{i}$ was not modified in the previous step, then a random decrease in ${c}_{i}$ is applied, making it darker.**Step 4**: The perturbation should be maintained if it is beneficial.

Algorithm 2 Proposed massive search operator. | ||

Input: | ${C}_{\mathrm{Best}}=({c}_{1},{c}_{2},\dots ,{c}_{K})$ | Best chromosome in the population |

${\overline{F}}_{\Delta}$ | Fitness function | |

1: for $i=1$ to K do2: ${\widehat{c}}_{i}:={c}_{i}$ ▹ Initially, ${\widehat{c}}_{i}$ is equal to ${c}_{i}$. 3: $\gamma :=\mathrm{rand}\left((0,1]\right)$ 4: ${c}_{i}^{\prime}:=(1+\gamma ){c}_{i}$ ▹ ( Step 1) Random increase.5: ${C}^{\prime}:={\mathrm{proj}}_{{\mathbb{Y}}^{K}}\left({c}_{1},{c}_{2},\dots ,{c}_{i-1},{c}_{i}^{\prime},{c}_{i+1},\dots ,{c}_{K}\right)$ 6: if ${\overline{F}}_{\Delta}\left({C}^{\prime}\right)>{\overline{F}}_{\Delta}\left({C}_{\mathrm{Best}}\right)$ then ▹ (Step 2) If the perturbation is beneficial, then update ${\widehat{c}}_{i}$ to ${c}_{i}^{\prime}$.7: ${\widehat{c}}_{i}:={c}_{i}^{\prime}$ 8: else ▹ If the increase wasn’t beneficial, then try apply a decrease on ${c}_{i}$.9: ${c}_{i}^{\prime}:=(1-\gamma ){c}_{i}$ ▹ ( Step 3) Random decrease.10: ${C}^{\prime}:={\mathrm{proj}}_{{\mathbb{Y}}^{K}}\left({c}_{1},{c}_{2},\dots ,{c}_{i-1},{c}_{i}^{\prime},{c}_{i+1},\dots ,{c}_{K}\right)$ 11: if ${\overline{F}}_{\Delta}\left({C}^{\prime}\right)>{\overline{F}}_{\Delta}\left({C}_{\mathrm{Best}}\right)$ then ▹ (Step 4) If the perturbation is beneficial, then update ${\widehat{c}}_{i}$ to ${c}_{i}^{\prime}$.12: ${\widehat{c}}_{i}:={c}_{i}^{\prime}$ 13: end if14: end if15: end for16: $\widehat{C}:={\mathrm{proj}}_{{\mathbb{Y}}^{K}}\left(\left({\widehat{c}}_{1},{\widehat{c}}_{2},\dots ,{\widehat{c}}_{K}\right)\right)$ ▹ The new chromosome is updated considering only the perturbations that were beneficial and contributed to improving the fitness value of ${C}_{\mathrm{Best}}$ | ||

Output: | $\widehat{C}$ | Best individual in the neighborhood of ${C}_{\mathrm{Best}}$ |

#### 5.7. Mapped Adaptive Rules

**Fact 1**: The statistical measures used are calculated using the vectors $\frac{1}{M}{\mathcal{F}}_{\mathrm{it}}$ and $\frac{1}{M}{\mathcal{F}}_{\mathrm{it}+1}$. Since the coordinates of these vectors are contained in the interval $[0,1]$ due to the normalization caused by the multiplication of the factor $\frac{1}{M}$, then all the statistical measures are contained in the interval $[0,1]$. In this way, each one of the four addends of ${f}_{\mathrm{improvement}}\left(\xb7,\xb7\right)$ can account for a maximum of $\frac{1}{4}\left|0-1\right|=0.25$ and, therefore, the values assumed by the function ${f}_{\mathrm{improvement}}\left(\xb7,\xb7\right)$ are contained in the range $[0,1]$.**Fact 2**: If there are no changes between two consecutive generations, then there will be no differences between ${\mathcal{F}}_{\mathrm{it}}$ and ${\mathcal{F}}_{\mathrm{it}+1}$. Thus, ${\mathcal{F}}_{\mathrm{it}}={\mathcal{F}}_{\mathrm{it}+1}$ and, therefore, ${f}_{\mathrm{improvement}}\left({\mathcal{F}}_{it},{\mathcal{F}}_{\mathrm{it}+1}\right)=0$.**Fact 3**: From the two previous facts, we can easily deduce that the greater the differences between the fitness of two consecutive generations, the closer to 1 will be the value of ${f}_{\mathrm{improvement}}\left(\xb7,\xb7\right)$ associated with these values of fitness. At the same time, the more similar the fitness values of two consecutive generations, the closer to zero will be the respective value of ${f}_{\mathrm{improvement}}\left(\xb7,\xb7\right)$. This means that the function defined in Equation (12) is able to establish a good numerical correspondence between changes in fitness values between two consecutive populations and can be used as an indicator of population stagnation.

**Situation 1**: There are signs of stagnation in relation to the populations ${\mathcal{F}}_{\mathrm{it}}$ and ${\mathcal{F}}_{\mathrm{it}+1}$: In this case, we will have ${f}_{\mathrm{improvement}}\left({\mathcal{F}}_{\mathrm{it}},{\mathcal{F}}_{\mathrm{it}+1}\right)\to 0$ and, to reverse this situation we are going to increase the mutation rate and decrease the crossover rate.**Situation 2**: There is a significant improvement in the population ${\mathcal{F}}_{\mathrm{it}+1}$ compared to the population ${\mathcal{F}}_{\mathrm{it}}$. In this case, we will have ${f}_{\mathrm{improvement}}\left({\mathcal{F}}_{\mathrm{it}},{\mathcal{F}}_{\mathrm{it}+1}\right)\to 1$ and, therefore, we can start to return the mutation and crossover rates to their initially defined values. For that, we must decrease the mutation rate and increase the crossover rate.

**Situation 1**, then our mapping function must represent a negative value and if we come across

**Situation 2**, then the mapping function must represent a positive value. Following these requirements, a possible structure for the mapping functions used for adjusting the crossover rate and for adjusting the mutation rate is presented in Equation (13).

#### 5.8. Proposed Algorithm

- Step 1
- A population of chromosomes must be generated, each of which must be defined by K colors taken randomly in space $\mathbb{Y}$;
- Step 2
- Select individuals taking into account their fitness values for the crossover operator;
- Step 3
- Evaluate these individuals;
- Step 4
- Randomly select individuals to undergo mutation;
- Step 5
- Select the best individual among all those considered so far;
- Step 6
- Massively search its neighborhood for individuals with a better fitness value;
- Step 7
- Adjust the method parameters with the adaptation operator;
- Step 8
- Create a new population and evaluate it;
- Step 9
- If the maximum number of iterations has not been reached, then you need to go back to step 2.

## 6. Experiments and Results

#### 6.1. Setup and Implementation

#### 6.2. Case Study I: Qualitative and Quantitative Analysis Considering Real World Images and $\mathbb{Y}=sRGB$

**Brains**: This image is not very complex, as it has only six sub-regions. However, we can see that, in the case of the pseudo-colorization obtained by GA (Figure 9b), the colors used in the central part of the brain are two different shades of green, which can cause some visual confusion. This does not occur with the pseudo-colorizations obtained by the other methods.**Two Brains**: Concerning the images of the Two Brains, we can see that basic GA (Figure 10b) obtains a reasonably good visual separation between the sub-regions, presenting some confusion only with the colored regions in shades of red and pink in the central-left region of the image. In the pseudo-colorized image calculated by the LSGA method, there is also a confusion problem as well as in the basic GA. When analyzing Figure 10c, the central right region has a dark blue color very close to a region that was colored with black. These problems do not occur in the pseudo-colorization performed by the proposed technique with the addition of adaptive rules.**Maps**: All methods used many shades of blue to pseudo-color this image, which can cause some visual confusion. For example, in the top-right section of the pseudo-colored image using the LSAGA technique (Figure 11d), we can see three colored regions with similar shades of blue. In addition, the pseudo-colored image with GA (Figure 11b) presents more serious issues. In detail, note that there is a small circular subregion in the right section of all pseudo-colored images, however, this subregion is almost visually imperceptible in the pseudo-colored image with GA.**Mosaic**: That is an image in which the methods have great difficulty for pseudo-coloring due to their high number of sub-regions. However, even though all pseudo-colored images have a high degree of visual separability between their regions, it is possible to see that the pseudo-colorized version by GA (Figure 12b) has in its top-left part several shades of yellow in neighboring areas and has in its bottom-right section neighboring regions colored with similar shades of red and pink. Something similar also occurs with the pseudo-colorized image with the LSGA (Figure 12c).**Bicycle**: The image has $K=6$ sub-regions and a small number of connections are defined by the neighborhood matrix. Therefore, all techniques have good visual results. However, all images show shades of blue in some pair of neighboring sub-regions, but this does not compromise the visual distinction of each sub-region, since the shades of blue used are very dissimilar.**Horses**: Analyzing the image, we can see that all methods present good pseudo-coloring results, with dissimilar color assignments in neighboring areas. However, this fact does not happen, for example, in the GA pseudo-colored image (Figure 14c), in which the blue horse in the central part of the image is pseudo-colored with a color visually similar to the background color.**Motorcycle**: This image has a reasonably large number of sub-regions ($K=18$), yet all the techniques were able to associate a good pseudo-coloring for each sub-region, we highlight some negative points, such as in the case of the pseudo-colorized image by LSAGA (Figure 15e), which contains two people in the top-left part of the background that represent adjacent areas and that have been colored with shades visually similar in blue.**Lunch**: That image is also highly complex, due to the number of sub-regions ($K=20$) that define a connection in the neighborhood matrix. Thus, not all techniques perform well. For example, in the pseudo-colorized image by our LSGA (Figure 16d), the table in the top-right part of the image receives a color similar to the background color. Something similar occurs with the color of the chair in the top-left section. Another example is presented by the pseudo-coloring obtained by our LSAGA (Figure 16e), which contains shades of blue in very close areas, such as the person in blue, in the central part of the image, sitting in a blue chair, in the case of the pseudo-colorized image by mapped methods this does not occur.

#### 6.3. Case Study II: Quantitative Analysis Considering a Synthetic Benchmark of Images and $\mathbb{Y}=sRGB$

#### 6.4. Case Study III: Quantitative Analysis Considering Real World Images and $\mathbb{Y}$ as Munsell Atlas Color Set

**Basketball**: All the proposed methods have the same best value (Max) considering the 50 iterations. However, our cub-LSAGA method presents superior results considering the other statistical measures: worst value (Min), mean and standard deviation (STD). The LS method is deterministic and, therefore, presents the same solution in every evaluation. However, all of our methods have the worst values (Min), results higher than the result obtained by LS.**Texture**: In this case, LS presents competitive results to our techniques. However, except for our cub-LSAGA, all the techniques proposed in this text have the best fitness value greater than the value obtained by LS. In addition, all techniques that make use of adaptation strategies presented a value of fitness as the worst value (Min), better than the best value presented by the Random technique. Likewise, our lin-LSAGA had the worst value greater than the best value presented by GA. This technique also presented the best average among all the compared techniques.**Oscar**: In this case, except for cub-LSAGA, all of our techniques showed the highest best value. In addition, except for computational time, our sig-LSAGA presents the best statistical measures. It is worth mentioning that, all of the proposed techniques obtain a worse value for fitness than a better result than the other techniques with more than 10 units of difference.**Webpage**: For this image, our sig-LSAGA presents the best fitness and average value. Meanwhile, our cub-LSAGA stands out considering the worst fitness value and the standard deviation. However, all the proposed techniques have higher average values than the best values presented by the other techniques.

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**A grey-level image I; its sub-regions ${I}_{1},{I}_{2},{I}_{3}$ and ${I}_{4}$; the graph representing the connections between the sub-regions; and its associated neighborhood matrix $\Delta $.

**Figure 3.**Example of pseudo-coloring of a grey-level image I, pre-segmented in $K=4$ regions, according to a chromosome C in the color space $\mathbb{Y}=$ sRGB.

**Figure 4.**Schematic of the proposed crossover operator. Such that ${\alpha}_{i}$ is a random value taken in $[0,1]$ for all i.

**Figure 5.**Schematic diagram of the functioning of three mutation functions on the same chromosome $\mathcal{C}$.

**Figure 6.**The plot of the graph of the mapping functions considered for the proposed mapped adaptive operator.

**Figure 8.**Graph of the neighborhood matrix between the sub-regions of each grey-level image. The number in each node is the associated index of the sub-region.

**Figure 17.**Numerical results (fitness values) displayed in boxplots of 50 independent runs of each technique.

**Figure 19.**Graph of the neighborhood matrix between the sub-regions of each grey-level image used in the work of Bianco and Schettini [39]

Image | Number of Subregions (K) | Nonzero Values in $\Delta $ | Number of Connections in the Graph |
---|---|---|---|

Brains | 6 | 24 | 12 |

Two Brains | 31 | 132 | 66 |

Maps | 38 | 174 | 87 |

Mosaic | 100 | 502 | 251 |

Bicycle | 7 | 22 | 11 |

Horses | 13 | 58 | 29 |

Motorcycle | 18 | 96 | 48 |

Lunch | 20 | 110 | 55 |

**Table 2.**Methods used for comparison. The symbol “-” means that the method does not use a mapping function.

Method | Local Search | Simple Adaptive Rules | Mapped Adaptive Rules | Mapping Function |
---|---|---|---|---|

GA | No | No | No | - |

LSGA | Yes | No | No | - |

LSAGA | Yes | Yes | No | - |

lin-LSAGA | Yes | No | Yes | Yes (Equation (16a)) |

sig-LSAGA | Yes | No | Yes | Yes (Equation (16b)) |

cub-LSAGA | Yes | No | Yes | Yes (Equation (16c)) |

**Table 3.**Statistical measurements of the fitness values presented in 50 runs of each method for the case study (CS) I images. The values highlighted in bold represent the best value of the evaluated measure available in the line.

Cub-LSAGA | Sig-LSAGA | Lin-LSAGA | LSAGA | LSGA | GA | |
---|---|---|---|---|---|---|

Image | Brain | |||||

Max | 111.5897 | 111.5897 | 111.5685 | 111.5897 | 111.5897 | 110.4434 |

Min | 103.4240 | 103.4240 | 103.4240 | 103.4240 | 93.8159 | 88.6637 |

Mean | 109.2113 | 109.9077 | 109.7841 | 109.5214 | 107.9427 | 100.2385 |

STD | 2.9257 | 2.3071 | 2.4198 | 2.3563 | 3.9305 | 5.2523 |

AT (s) | 39.9599 | 46.4401 | 45.1142 | 44.4965 | 33.9584 | 66.9531 |

Image | Two Brains | |||||

Max | 103.8950 | 95.5579 | 103.2217 | 105.3690 | 103.2437 | 92.4752 |

Min | 65.8645 | 80.4354 | 76.6066 | 74.1934 | 74.9277 | 62.6070 |

Mean | 87.0920 | 87.4156 | 85.8846 | 87.0389 | 85.7157 | 78.3665 |

STD | 9.4722 | 4.3749 | 6.8571 | 6.2291 | 6.9172 | 6.9151 |

AT (s) | 160.4555 | 157.1275 | 163.7412 | 158.64375 | 144.0959 | 97.7062 |

Image | Maps | |||||

Max | 93.7054 | 94.5564 | 91.4757 | 93.5399 | 97.0676 | 85.1661 |

Min | 64.4656 | 66.1475 | 71.2507 | 66.8662 | 64.9360 | 55.9863 |

Mean | 79.1163 | 83.6823 | 79.2163 | 81.2258 | 80.9922 | 72.4412 |

STD | 8.0998 | 7.6433 | 6.1562 | 6.6415 | 6.4602 | 6.2322 |

AT (s) | 222.9510 | 226.4532 | 226.4238 | 224.1458 | 203.8438 | 130.4597 |

Image | Mosaic | |||||

Max | 80.4838 | 82.2895 | 80.6392 | 83.7880 | 81.5133 | 53.8912 |

Min | 69.8773 | 63.7500 | 64.4427 | 61.2706 | 56.4168 | 43.3865 |

Mean | 75.1833 | 73.5342 | 72.8356 | 73.5591 | 72.1155 | 48.7392 |

STD | 3.4294 | 4.9239 | 5.1849 | 6.2869 | 5.7767 | 2.4751 |

AT (s) | 990.0295 | 992.4713 | 991.2589 | 989.3925 | 936.8443 | 331.4825 |

Image | Bicycle | |||||

Max | 130.4377 | 129.6426 | 129.6426 | 130.5154 | 130.2436 | 129.3809 |

Min | 129.1017 | 129.3329 | 128.6836 | 123.3457 | 123.3457 | 93.9292 |

Mean | 129.5630 | 129.5257 | 129.4862 | 129.3585 | 129.2352 | 123.2097 |

STD | 0.2439 | 0.1219 | 0.2094 | 1.2629 | 1.2606 | 8.5688 |

AT (s) | 40.3541 | 42.4896 | 48.4984 | 45.5781 | 34.0573 | 68.9844 |

Image | Horse | |||||

Max | 111.4548 | 110.3349 | 111.0386 | 110.5609 | 108.4003 | 104.8786 |

Min | 87.9770 | 89.0085 | 85.7987 | 85.3845 | 87.4883 | 79.3059 |

Mean | 100.5157 | 100.6137 | 102.0270 | 97.4747 | 98.1981 | 90.5370 |

STD | 7.6174 | 5.6810 | 6.5707 | 7.2756 | 5.8545 | 6.0897 |

AT (s) | 72.5547 | 73.9412 | 75.2496 | 69.1302 | 53.5729 | 74.4583 |

Image | Motorcycle | |||||

Max | 105.8383 | 101.0411 | 101.9775 | 103.5309 | 97.8846 | 101.8992 |

Min | 76.7733 | 77.0004 | 80.2057 | 72.0469 | 74.3101 | 68.4586 |

Mean | 90.3223 | 88.7403 | 89.1553 | 87.0939 | 87.0660 | 82.5259 |

STD | 7.1476 | 6.4653 | 6.1515 | 6.6437 | 6.0711 | 8.9604 |

AT (s) | 89.7412 | 85.9214 | 86.6647 | 89.6563 | 73.5052 | 85.3021 |

Image | Lunch | |||||

Max | 98.6577 | 103.8848 | 99.3338 | 105.9357 | 100.8967 | 95.3289 |

Min | 75.2950 | 75.2677 | 73.4639 | 70.43786 | 70.1481 | 65.2178 |

Mean | 84.7076 | 88.4268 | 87.1569 | 82.81681 | 85.0893 | 80.1618 |

STD | 6.0211 | 7.3464 | 6.1305 | 7.3561 | 7.2866 | 7.7565 |

AT (s) | 103.2497 | 99.1463 | 98.8976 | 100.8177 | 82.4479 | 87.6302 |

**Table 4.**Max distance of colors in a fully-connected images. Green numbers are the max (best) values in line and red numbers are the min (worst) values in line.

Regions (K) | Cub-LSAGA | Sig-LSAGA | Lin-LSAGA | LSAGA | LSGA | GA [14] | Greedy [14] |
---|---|---|---|---|---|---|---|

2 | 249.2 | 249.2 | 249.2 | 249.2 | 249.2 | 249.2 | 233.85 |

3 | 166.11 | 166.11 | 166.11 | 166.11 | 166.11 | 166.11 | 164.64 |

4 | 130.64 | 130.64 | 130.64 | 130.64 | 129.64 | 130.21 | 129.64 |

5 | 111.59 | 111.59 | 111.59 | 111.59 | 111.59 | 111.43 | 108.81 |

6 | 102.58 | 102.58 | 102.58 | 102.58 | 102.58 | 102.48 | 93.78 |

7 | 94.7 | 94.7 | 94.7 | 94.7 | 93.75 | 93.04 | 86.95 |

8 | 86.15 | 86.15 | 86.15 | 86.15 | 86.13 | 84.78 | 80.03 |

9 | 81.49 | 81.49 | 81.49 | 81.49 | 80.43 | 78.68 | 74.45 |

10 | 77.8 | 77.8 | 77.8 | 77.8 | 74.9 | 74.65 | 71.92 |

11 | 68.1 | 69.43 | 69.43 | 69.43 | 68.1 | 66.71 | 65.77 |

12 | 65.61 | 65.61 | 65.61 | 65.61 | 64.65 | 64.84 | 61.86 |

13 | 64.26 | 64.26 | 64.26 | 64.26 | 62.5 | 63.13 | 57.79 |

14 | 60.89 | 60.89 | 60.89 | 60.89 | 59.1 | 58.8 | 57.32 |

15 | 57.16 | 57.16 | 57.16 | 57.16 | 56.7 | 53.52 | 55.27 |

16 | 51.53 | 55.82 | 55.82 | 55.82 | 51.53 | 51.01 | 53.4 |

17 | 53.56 | 53.56 | 53.56 | 53.56 | 52.55 | 49.67 | 51.32 |

18 | 50.56 | 50.56 | 50.56 | 50.56 | 50.47 | 48.17 | 49.42 |

19 | 45.08 | 50.5 | 47.9 | 50.5 | 48.24 | 45.08 | 47.9 |

20 | 44.67 | 49.26 | 49.26 | 49.26 | 45.83 | 44.67 | 47.57 |

21 | 46.54 | 45.68 | 45.68 | 45.68 | 44.78 | 42.66 | 46.54 |

22 | 46.36 | 46.36 | 44.23 | 46.36 | 44.87 | 41.63 | 44.23 |

23 | 44.74 | 44.74 | 44.74 | 43.62 | 43.28 | 41.3 | 44.74 |

24 | 39.77 | 43.86 | 39.77 | 43.86 | 42.22 | 39.77 | 43.61 |

25 | 38.55 | 43.09 | 38.55 | 43.09 | 41.82 | 38.55 | 41.98 |

Image | Number of Subregions (K) | Nonzero Values in $\Delta $ | Number of Connections in the Graph |
---|---|---|---|

Basketball | 4 | 10 | 5 |

Texture | 8 | 38 | 19 |

Oscar | 5 | 14 | 7 |

Webpage | 8 | 24 | 12 |

**Table 6.**Optimization in Munsell Atlas color space with 1269 colors. The symbol “-” means that the respective measurement is not available in the original work. The values highlighted in bold represent the best value of the evaluated measure available in the table line.

Cub-LSAGA | Sig-LSAGA | Lin-LSAGA | LSAGA | LSGA | GA [14,39] | Random [39] | LS [39] | |
---|---|---|---|---|---|---|---|---|

Image | Basketball | |||||||

Max | 83.8193 | 83.8193 | 83.8193 | 83.8193 | 83.8193 | 70.1 | 60.95 | 70.1 |

Min | 80.9203 | 80.2733 | 80.9203 | 78.1239 | 74.6126 | - | - | - |

Mean | 82.5140 | 82.2057 | 82.2179 | 82.0358 | 81.8048 | 61.68 | 17.76 | - |

STD | 0.7331 | 0.8182 | 0.7500 | 1.2027 | 1.4551 | - | - | - |

AT (s) | 38.3354 | 40.4001 | 38.7352 | 39.6563 | 32.3594 | 15.08 | 0.0013 | 3.03 |

Image | Texture | |||||||

Max | 58.7465 | 63.5075 | 63.0651 | 66.3612 | 63.0651 | 47.45 | 45.15 | 60.63 |

Min | 46.7823 | 46.0836 | 47.7114 | 45.8120 | 36.5557 | - | - | - |

Mean | 53.0731 | 54.4095 | 55.4383 | 54.6570 | 52.4303 | 43.75 | 17.78 | - |

STD | 3.3279 | 4.0777 | 3.4021 | 4.6342 | 5.7014 | - | - | - |

AT (s) | 58.1205 | 59.9801 | 59.2552 | 50.5365 | 35.8229 | 29.48 | 0.0021 | 18.87 |

Image | Oscar | |||||||

Max | 82.4165 | 83.8193 | 83.8193 | 83.8193 | 83.8193 | 59.76 | 56.44 | 63.5 |

Min | 78.1239 | 79.5652 | 78.1239 | 77.3148 | 76.4953 | - | - | - |

Mean | 81.8081 | 82.0605 | 81.9420 | 81.7832 | 81.2568 | 54.74 | 18.74 | - |

STD | 1.3161 | 0.9360 | 1.0757 | 1.4110 | 1.6553 | - | - | - |

AT (s) | 39.8854 | 42.1464 | 40.7552 | 39.0000 | 28.5573 | 17.93 | 0.0015 | 5.56 |

Image | Webpage | |||||||

Max | 80.9203 | 82.3743 | 82.2811 | 82.1702 | 80.9203 | 50.13 | 42.93 | 62.33 |

Min | 67.2430 | 62.8774 | 64.4095 | 62.8774 | 62.2342 | - | - | - |

Mean | 75.7342 | 76.9152 | 76.1263 | 76.0651 | 73.6511 | 44.15 | 14.89 | - |

STD | 4.0863 | 4.7845 | 4.6217 | 4.6063 | 5.5967 | - | - | - |

AT (s) | 50.1254 | 47.8946 | 47.7740 | 48.6667 | 37.0156 | 30.36 | 0.002 | 18.99 |

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**MDPI and ACS Style**

Viana, M.S.; Morandin Junior, O.; Contreras, R.C.
An Improved Local Search Genetic Algorithm with a New Mapped Adaptive Operator Applied to Pseudo-Coloring Problem. *Symmetry* **2020**, *12*, 1684.
https://doi.org/10.3390/sym12101684

**AMA Style**

Viana MS, Morandin Junior O, Contreras RC.
An Improved Local Search Genetic Algorithm with a New Mapped Adaptive Operator Applied to Pseudo-Coloring Problem. *Symmetry*. 2020; 12(10):1684.
https://doi.org/10.3390/sym12101684

**Chicago/Turabian Style**

Viana, Monique Simplicio, Orides Morandin Junior, and Rodrigo Colnago Contreras.
2020. "An Improved Local Search Genetic Algorithm with a New Mapped Adaptive Operator Applied to Pseudo-Coloring Problem" *Symmetry* 12, no. 10: 1684.
https://doi.org/10.3390/sym12101684