# Optimal Multi-Level Thresholding Based on Maximum Tsallis Entropy via an Artificial Bee Colony Approach

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Nonextensive Entropy

_{i}denotes the probability of each state i. Therefore, the Shannon entropy can be described as:

_{q}(A + B) < S

_{q}(A) + S

_{q}(B); for q = 1, the Tsallis entropy reduces to an standard extensive entropy where S

_{q}(A + B) = S

_{q}(A) + S

_{q}(B); for q > 1, the Tsallis entropy becomes a superextensive entropy where S

_{q}(A + B) > S

_{q}(A) + S

_{q}(B).

## 3. Thresholding Model

_{i}; so p

_{i}≥ 0 and p

_{1}+ p

_{2}+ … p

_{L}= 1. If the image is divided into two classes, C

_{A}and C

_{B}by a threshold at level t, where class C

_{A}consists of gray levels from 1 to t and C

_{B}contains the rest gray levels from t + 1 to L, the cumulative probabilities can be defined as:

^{A}and p

^{B}can be defined as:

_{q}(t) of each individual class is dependent on the threshold t. The total Tsallis entropy of the image is written as follows [25]:

_{A}and C

_{B}. When the value of S

_{q}(t) is maximized, the corresponding gray-level t

^{*}is regarded as the optimum threshold value:

_{q}(t) calculated is equal to:

^{11}. Therefore, the exhaustive algorithm failed for multi-level thresholding, we will introduce the ABC method for fast computation in next section.

No of Classes m | 2 | 3 | 4 | 5 | 6 | 7 |

No of S_{q}(t) calculated | 255 | 32,385 | 2.73×10^{6} | 1.72×10^{8} | 8.64×10^{9} | 3.60×10^{11} |

## 4. Artificial Bee Colony

#### 4.1. Essential Components

- Food Resource: The value of a food source depends on different parameters, such as its proximity to the nest, richness of concentration of energy, and the ease of extracting this energy. For the simplicity, the profitability of a food source can be represented with a single quantity.
- Unemployed Foragers: If it is assumed that a bee has no knowledge about the food sources in the search field, it initializes its search as an unemployed forager looking for a food source to exploit. There are two types of unemployed forages: (I) Scouts: If the bee starts searching spontaneously without any knowledge, it will be a scout bee. The percentage of scout bees varies from 5% to 30% according to the information into the nest. The mean number of scouts averaged over conditions is about 10% in Nature [27]. (II) Onlookers: The onlookers wait in the nest and find a food source through the information shared by employed foragers. If the onlookers attend a waggle dance done by some other bee, the bees become recruits and start searching by using the knowledge from the waggle dance.
- Employed Foragers: Employed foragers are associated with a particular food source, which they are currently exploiting or are “employed” at. They carry with them information about this particular source, its distance and direction from the nest, and the profitability of the source. They share this information with a certain probability. After the employed foraging bee loads a portion of nectar from the food source, it returns to the hive and unloads the nectar to the food area in the hive. There are three possible options related to the residual amount of nectar for the foraging bee: (I) Unemployed Forager: If the nectar amount decreases to a low level or becomes exhausted, the foraging bee abandons the food source and become an unemployed bee. (II) Employed Forager 1: Otherwise, the foraging bee might dance and then recruit nest mates before returning to the same food source. (III) Employed Forager 2: Or it might continue to forage at the food source without recruiting other bees.

#### 4.2. Principles and Procedures

- Step 1. Initialize the population of solutions x
_{ij}(here i denotes the ith solution, and j denotes the jth epoch, i = 1, …, SN, here SN denotes the number of solutions) with j = 0: - Step 2. Repeat, and let j = j + 1;
- Step 3. Produce new solutions (food source positions) υ
_{ij}in the neighborhood of x_{ij}for the employed bees using the formula:_{kj}is a randomly chosen solution in the neighborhood of x_{ij}to produce a mutant of solution x_{ij}, Φ is a random number in the range [−1, 1]. Evaluate the new solutions; - Step 4. Apply the greedy selection process between the corresponding x
_{ij}and υ_{ij}; - Step 5. Calculate the probability values P
_{ij}for the solutions x_{ij}by means of their fitness values using the equation: - Step 6. Normalize P
_{ij}values into [0, 1]; - Step 7. Produce the new solutions (new positions) υ
_{ij}for the onlookers from the solutions x_{ij}using the same equation as in Step 3, selected depending on P_{ij}, and evaluate them; - Step 8. Apply the greedy selection process for the onlookers between x
_{ij}and υ_{ij}; - Step 9. Determine the abandoned solution (source), if it exists, and replace it with a new randomly produced solution x
_{ij}for the scout using the equation:_{ij}is a random number in [0, 1]. - Step 10. Memorize the best food source position (solution) achieved so far;
- Step 11. Go to Step 2 until termination criteria is met.

## 5. Experiments

#### 5.1. Effectiveness of Tsallis Entropy

^{A}and P

^{B}denote the probability of class C

_{A}and C

_{B}, respectively, ω

_{A}and ω

_{B}denote the corresponding the cumulative probabilities, μ

_{A}and μ

_{B}denote the corresponding mean value, the formalism of objective function of different criteria are shown in Table 2. Here, we add a minus symbol to the objective function of MCET to transform the minimization to maximization.

Criteria | Objective | Type |
---|---|---|

MET | S(A) + S(B) | Maximize |

MBCVT | ω_{A} × ω_{B} × (μ_{A} − μ_{B})^{2} | Maximize |

MCET | −μ_{A} × ω_{A} × log(μ_{A}) − μ_{B} × ω_{B} × log(μ_{B}) | Minimize |

μ_{A} × ω_{A} × log(μ_{A}) + μ_{B} × ω_{B} × log(μ_{B}) | Maximize | |

MTT | S_{q}(A) + S_{q}(B) + (1 − q) × S_{q}(A) × S_{q}(B) | Maximize |

Image | MET | MBCVT | MCET | MTT |
---|---|---|---|---|

Two equal Gaussian peaks | √ | √ | √ | |

One peak is higher | √ | √ | ||

Larger distance between two peaks | √ | √ |

#### 5.2. Rapidness of the ABC

m | GA | PSO | ABC |
---|---|---|---|

3 | |||

89 172 | 85 171 | 85 170 | |

4 | |||

63 127 188 | 64 126 185 | 62 128 189 | |

5 | |||

52 99 157 205 | 49 100 155 205 | 50 101 155 204 | |

6 | |||

40 85 128 169 213 | 42 86 127 172 215 | 42 85 128 170 214 |

m | GA | PSO | ABC |
---|---|---|---|

3 | |||

68 180 | 66 188 | 70 185 | |

4 | |||

73 112 192 | 67 113 187 | 71 113 188 | |

5 | |||

70 110 136 192 | 68 109 139 194 | 68 108 138 192 | |

6 | |||

68 111 136 168 187 | 69 112 134 169 188 | 69 110 135 169 189 |

m | GA | PSO | ABC |
---|---|---|---|

3 | |||

102 199 | 95 199 | 99 195 | |

4 | |||

29 85 190 | 28 92 196 | 28 88 192 | |

5 | |||

27 93 144 195 | 31 88 141 197 | 29 90 141 195 | |

6 | |||

33 93 121 138 193 | 34 92 121 141 192 | 32 91 120 140 192 |

m | GA | PSO | ABC |
---|---|---|---|

3 | |||

29 70 | 25 61 | 25 65 | |

4 | |||

23 64 111 | 24 69 106 | 26 66 108 | |

5 | |||

31 67 103 198 | 29 68 104 201 | 28 68 105 199 | |

6 | |||

30 70 106 147 204 | 31 70 104 146 201 | 30 70 104 148 203 |

Image | m | GA | PSO | ABC |
---|---|---|---|---|

Synthetic | 3 | 1.8849 | 1.8479 | 1.8210 |

4 | 1.9487 | 1.8930 | 1.8316 | |

5 | 1.9780 | 1.9507 | 1.8709 | |

6 | 2.0453 | 2.0367 | 1.9807 | |

Lena | 3 | 1.9143 | 1.8931 | 1.6611 |

4 | 1.9036 | 1.9115 | 1.6594 | |

5 | 2.0176 | 1.9500 | 1.7121 | |

6 | 2.1598 | 2.0577 | 1.7740 | |

Cameraman | 3 | 1.8755 | 1.7708 | 1.7083 |

4 | 1.9168 | 1.8558 | 1.7003 | |

5 | 1.9686 | 1.8722 | 1.7975 | |

6 | 2.0175 | 1.9933 | 1.9191 | |

Peppers | 3 | 1.8890 | 1.8540 | 1.5321 |

4 | 1.9636 | 1.8495 | 1.5527 | |

5 | 1.9985 | 1.9350 | 1.6924 | |

6 | 2.1395 | 2.0122 | 1.6484 |

#### 5.3. Influence of the Parameter q

## 6. Conclusions

## Acknowledgment

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Zhang, Y.; Wu, L.
Optimal Multi-Level Thresholding Based on Maximum Tsallis Entropy via an Artificial Bee Colony Approach. *Entropy* **2011**, *13*, 841-859.
https://doi.org/10.3390/e13040841

**AMA Style**

Zhang Y, Wu L.
Optimal Multi-Level Thresholding Based on Maximum Tsallis Entropy via an Artificial Bee Colony Approach. *Entropy*. 2011; 13(4):841-859.
https://doi.org/10.3390/e13040841

**Chicago/Turabian Style**

Zhang, Yudong, and Lenan Wu.
2011. "Optimal Multi-Level Thresholding Based on Maximum Tsallis Entropy via an Artificial Bee Colony Approach" *Entropy* 13, no. 4: 841-859.
https://doi.org/10.3390/e13040841