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第554行： − − <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 − − − − <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. \\ − − & +\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)} - I_{(i+1,j)} \right\vert + \left\vert I_{(i-1,j+1)} - I_{(i-1,j-1)} \right\vert\\ − − & + \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> − − + − <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) = − − < 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) = − − <math>Z =\sum_{i=1:M_1} \sum_{j=1:M_2} Vc(I_{i,j})</math>, which is a normalization factor − − \begin{cases} − − 开始{ cases } − − − − \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\ − − 0, & \text{else} − − − \end{cases}</math><br /><math>f(x) = − − − \begin{cases} − − + − − + + + + + + + + + − <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) =+ − − \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> − − 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] − − \begin{cases} + + − − \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\ − − − 0, & \text{else} − − \end{cases}</math><br /><math>f(x) = − − \begin{cases} 第639行：
第597行： − \pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\ − − 0, & \text{else} − + + − + − + − thumb − − 拇指 − − − − − − ''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]]. − − − − 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>
→Image processing
''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.
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:
the local statistics at the pixel position (i,j).
the local statistics at the pixel position (i,j).
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math>
<math>\eta_{(i,j)}= \tfrac{1}{Z}*Vc*I_{(i,j)}</math>
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 />
<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
<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. \\
& +\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\\
& +\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\\
& + \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>
\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\
<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) =
< 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) =\begin{cases}
\sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (3)} \\
0, & \text{else}
0, & \text{else}
\end{cases}</math><br />
<math>f(x) =\begin{cases}\pi x \sin(\frac{\pi x}{2 \lambda}), & \text{for 0 ≤ x ≤} \lambda \text{; (4)} \\
0, & \text{else}\end{cases}</math>
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>
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。
<br /><math>
\tau_{new} \leftarrow
\tau_{new} \leftarrow
(1-\psi)\tau_{old} + \psi \tau_{0}
(1-\psi)\tau_{old} + \psi \tau_{0}
</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。
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.
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。
Image Edge detected using ACO:<br />The images below are generated using different functions given by the equation (1) to (4).
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>
[[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。
\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>
\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>
\tau_{new} \leftarrow
\tau_{new} \leftarrow
(1-\psi)\tau_{old} + \psi \tau_{0}
(1-\psi)\tau_{old} + \psi \tau_{0}
</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>
[[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]]