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蚁群优化算法 (查看源代码)

2020年12月8日 (二) 14:14的版本

删除1,309字节 、 2020年12月8日 (二) 14:14
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''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.
 
''Step1: Initialization:<br />''Randomly place <math>K</math> ants on the image <math>I_{M_1 M_2}</math> where <math>K= (M_1*M_2)^\tfrac{1}{2}</math> . Pheromone matrix <math>\tau_{(i,j)}</math> are initialized with a random value. The major challenge in the initialization process is determining the heuristic matrix.
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<math>Z =\sum_{i=1:M_1}  \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor
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是一个归一化因子
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There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:
 
There are various methods to determine the heuristic matrix. For the below example the heuristic matrix was calculated based on the local statistics:
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<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\
      
the local statistics at the pixel position (i,j).
 
the local statistics at the pixel position (i,j).
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\
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& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\
      
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math>
 
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math>
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& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>
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Where <math>I</math> is the image of size <math>M_1*M_2</math><br />
 
Where <math>I</math> is the image of size <math>M_1*M_2</math><br />
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<math>Z =\sum_{i=1:M_1}  \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor 是一个归一化因子
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0;  (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0;  (2)} </math><br /><math>f(x) =
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< math > f (cdot) </math > 可以用以下函数计算: < br/> < math > f (x) = lambda x，quad text { for x ≥0; (1)} </math > < br/> < math > f (x) = lambda x ^ 2，quad text { for x ≥0; (2)} </math > < br/> < math > f (x) =
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<math>Z =\sum_{i=1:M_1}  \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor
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\begin{cases}
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开始{ cases }
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\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (3)} \\
   
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\
 
<math>\begin{align}Vc(I_{i,j}) = &f \left( \left\vert I_{(i-2,j-1)} - I_{(i+2,j+1)} \right\vert + \left\vert I_{(i-2,j+1)} - I_{(i+2,j-1)} \right\vert \right. \\
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0, & \text{else}
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& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\
 
& +\left\vert I_{(i-1,j-2)} - I_{(i+1,j+2)} \right\vert + \left\vert I_{(i-1,j-1)} - I_{(i+1,j+1)} \right\vert\\
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\end{cases}</math><br /><math>f(x) =
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& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\
 
& +\left\vert I_{(i-1,j)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\
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\begin{cases}
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& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>
 
& + \left. \left\vert I_{(i-1,j+2)} - I_{(i-1,j-2)} \right\vert + \left\vert I_{(i,j-1)} - I_{(i,j+1)} \right\vert \right) \end{align}</math>
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (4)} \\
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0;  (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0;  (2)} </math><br /><math>f(x) =
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< math > f (cdot) </math > 可以用以下函数计算:
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<br />
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<math>f(x) = \lambda x, \quad \text{for x ≥ 0;  (1)} </math><br />
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<math>f(x) = \lambda x^2, \quad \text{for x ≥ 0;  (2)}</math><br />
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<math>f(x) =\begin{cases}
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\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (3)} \\
 
0, & \text{else}
 
0, & \text{else}
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\end{cases}</math><br />
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<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (4)} \\
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0, & \text{else}\end{cases}</math>
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<math>f(\cdot)</math> can be calculated using the following functions:<br /><math>f(x) = \lambda x, \quad \text{for x ≥ 0;  (1)} </math><br /><math>f(x) = \lambda x^2, \quad \text{for x ≥ 0;  (2)} </math><br /><math>f(x) =
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The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math>
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />Step 2 Construction process:<br />The ant's movement is based on 4-connected pixels or 8-connected pixels. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />Step 3 and Step 5 Update process:<br />The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math>
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上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。[数学]
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\begin{cases}
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上面每个函数中的参数 < math > lambda </math > 用来调整函数各自的形状。< br/> 步骤2构建过程: <br/> 蚂蚁的移动是在4个连接的像素或8个连接的像素进行的。根据概率方程< math > p _ { x，y } </math > 给出蚂蚁移动的概率< br/> 步骤3与步骤5更新过程 < br/> 信息素矩阵更新两次。在步骤3中，蚂蚁(由 < math > tau _ {(x，y)} </math > 给出)的踪迹被更新，就像在步骤5中，蚂蚁的踪迹蒸发率由下面的方程进行更新。
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<br /><math>
 
\tau_{new} \leftarrow
 
\tau_{new} \leftarrow
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\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (3)} \\
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(1-\psi)\tau_{old} + \psi \tau_{0}
 
(1-\psi)\tau_{old} + \psi \tau_{0}
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0, & \text{else}
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</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
 
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
    
这里是信息素衰减系数。
 
这里是信息素衰减系数。
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\end{cases}</math><br /><math>f(x) =
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\begin{cases}
      
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.
 
Step 7 Decision Process:<br />Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrix τ. Threshold for the below example is calculated based on Otsu's method.
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步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。
 
步骤7决策过程: 一旦 k 只蚂蚁在 n 次迭代中移动了一个固定的距离 l，可以基于信息素矩阵 τ 上的阈值 T判断它是否是一个边缘。下面例子的阈值是根据 Otsu 的方法计算的。
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\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{;  (4)} \\
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0, & \text{else}
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref>
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[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]
    
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。
 
使用 ACO 检测图像边缘: < br/> 下面的图像是使用方程(1)至(4)给出的不同函数生成的。
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.<br /><math>
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\end{cases}</math><br />The parameter <math>\lambda</math> in each of above functions adjusts the functions’ respective shapes.<br />''Step 2 Construction process:<br />''The ant's movement is based on [[4-connected neighborhood|4-connected]] [[pixel]]s or [[8-connected]] [[pixel]]s. The probability with which the ant moves is given by the probability equation <math>P_{x,y}</math><br />''Step 3 and Step 5 Update process:<br />''The pheromone matrix is updated twice. in step 3 the trail of the ant (given by <math>\tau_{(x,y)}</math> ) is updated where as in step 5 the evaporation rate of the trail is updated which is given by the below equation.
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<br /><math>
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thumb
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拇指
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\tau_{new} \leftarrow
 
\tau_{new} \leftarrow
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(1-\psi)\tau_{old} + \psi \tau_{0}
 
(1-\psi)\tau_{old} + \psi \tau_{0}
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</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
 
</math>, where <math>\psi</math> is the pheromone decay coefficient <math>0< \tau <1</math>
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''Step 7 Decision Process:<br />''Once the K ants have moved a fixed distance L for N iteration, the decision whether it is an edge or not is based on the threshold T on the pheromone matrixτ. Threshold for the below example is calculated based on [[Otsu's method]].
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Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).<ref>{{cite web|title=File Exchange {{ndash}} Ant Colony Optimization (ACO)|website=[[MATLAB]] Central|url=http://www.mathworks.com/matlabcentral/fileexchange/32009-ant-colony-optimization--aco-}}</ref>
      
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]
 
[[File:(a)Original Image (b)Image Generated using equation(1) (c)Image generated using equation(2) (d) Image generated using equation(3) (e)Image generated using equation(4).jpg|none|thumb]]
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