## 3 Problem and Methods
### Sensitive Ant Colony Optimization Method  
The method used is an improved version of _Ant Colony Optimization (ACO)_[1] called the _Sensitive Ant Colony Optimization (SACO)_[24; 25]. The ant colony optimization sensitive approach for medical image edge detection is further presented. There is considered a colony of \(K\) ants engaged in a search within the graph space \(\mathcal{X}\), with \(M_{1}\times M_{2}\) nodes. SACO as well as ACO use artificial ants to move in a 2D image in order to build the pheromone matrix; each matrix element represents the edge information for every pixel in the image. In general, ACO and its versions builds a solution with the use of artificial ants; these agents search for the best path in a given space by depositing artificial pheromones [1; 23]. These pheromone trails are updated during the search process. The ACO and SACO general scheme includes an initialization process followed by \(N\) construction steps while creating and updating pheromone matrix, and, finally, performing the decision process to determine a beneficial solution. _Initialization process:_ In particular, for image edge detection with ACO and SACO during the initialization process, the entire ant colony (\(K\) ants) places ants randomly on the image matrix. Each image pixel is considered to be a node in a graph. Each initial pheromone matrix \(\tau_{(0)}\) has a constant \(\tau_{init}\) value. A constant value \(L\) defines the number of moves during the construction process. For SACO in particular, each \(PSL\) vector component is initialized with 1, starting as the original ACO (see Figure 1). _Construction phase:_ at the \(n\)-th construction step, a randomly chosen ant will move from node \(i\) to \(j\) according to the transition probability in (6) for \(L\) steps. \[p_{ij}^{n}=\frac{\left(\tau_{ij}^{(n-1)}\right)^{\alpha}\left(\eta_{ij}\right) ^{\beta}}{\sum_{j\in\Omega_{i}}\left(\tau_{ij}^{(n-1)}\right)^{\alpha}\left( \eta_{ij}\right)^{\beta}},\qquad\text{if }j\in\Omega_{i}, \tag{6}\]  
where \(\tau_{IJ}\) is the pheromone value on \((i,j)\); \(\eta_{IJ}\) the heuristic value connecting nodes \(i\) and \(j\) the same for all \(n\) construction steps; \(\alpha\) and \(\beta\) are the weighting factors for the pheromone and the heuristic; and \(\Omega_{i}\) includes the neighborhood nodes of node \(i\). The overall eight-connectivity neighborhood for each pixel \(I_{i,j}\) within the local configuration at the \(I_{i,j}\) pixel, \(cl_{i,j}\), for computing the variation value \(V_{c}(I_{i,j})\) defined by (8) is illustrated in [18]. Here, we propose the computation of \(\eta_{i,j}\) according to the local statistic of the pixel \((i,j)\) (Equation (7)). \[\eta_{i,j}=\frac{1}{Z}\cdot V_{c}(I_{i,j}), \tag{7}\]  
where \(Z=\sum\limits_{i=1}^{M_{1}}\sum\limits_{j=1}^{M_{2}}V_{c}(I_{i,j})\) is a normalization factor, \(I_{i,j}\) is the intensity value of the image pixel \((i,j)\); and the function \(V_{c}(I_{i,j})\) processes the \"clique\" \(cl_{i,j}\)[22]. The \(V_{c}(I_{i,j})\) value at pixel \(I_{i,j}\) is influenced by the image's intensity values for \(cl_{i,j}\), and its value is [22]:  
\[ \begin{array}{ll}V_{c}(I_{i,j})=&f\big{(}\big{|}I_{i-2,j-1}-I_{i+2,j+1}\big{|} +\big{|}I_{i-2,j+1}-I_{i+2,j-1}\big{|}\\ &+\big{|}I_{i-1,j-2}-I_{i+1,j+2}\big{|}+\big{|}I_{i-1,j-1}-I_{i+1,j+1}\big{|} \\ &+\big{|}I_{i-1,j}-I_{i+1,j}\big{|}+\big{|}I_{i-1,j+1}-I_{i+1,j-1}\big{|}\\ &+\big{|}I_{i-1,j+2}-I_{i+1,j-2}\big{|}+\big{|}I_{i,j-1}-I_{i,j+1}\big{|} \big{)}.\end{array}  \tag{8}\]  
In order to validate an edge within the solution, a decision is made for each image pixel by applying a threshold \(T\) (see [26]) to the final pheromone matrix \(\tau^{(N)}\). Figure 1: Symbolic illustration of the sensitive ant model showing the ants’ probability variation within the unit interval from the ACO probability when \(PSL=1\) to a random walk probability when \(PSL=0\). The artificial pheromone matrix values are updated both locally and globally. _Locally update the pheromone matrix_\(\tau\).