# Active Contours in the Complex Domain for Salient Object Detection
## 3 Proposed Method
### Complex Force Function  
The signed pressure force function has an important role in expanding the dynamic contour [41] when the contour is inside the object boundary and shrinking when the dynamic contour is outside the object boundary. The function that is introduced here is different from ACSLGS and the design of complex signed pressure [41; 49]. We propose a pressure force, also called a complex force function, which utilizes the statistical information in the complex domain to change the intensities inside the contour to the foreground and those outside the contour to the background. The proposed complex force function can be written as  
\[C_{PF}=\Big{(}i\times I-\frac{c_{1}+c_{2}}{2}\Big{)} \tag{22}\]  
Figure 3: Geometrical representation of complex values.  
The complex force function dynamically increases the interior and exterior energies of the curves in order to reduce the chance of entrapment in a local minimum, while the curve is far from the boundary of the object. When the curve is close to the object boundary, then the energy is decreased, and the curve is automatically stopped near the boundary of the object. The introduced model reduces the computational cost efficiently. The same analogy is adopted for \(c_{1}\) and \(c_{2}\), defined in Equation (7) and (8).  
The global complex force function is created by multiplying the complex domain image per pixel with a low-pass filter and a complex force function. The proposed Algorithm 1 of active contour salience formulation is adopted according to Equation (15).  
```
Input: Reading in an image. Image transformation:
1: Transformation of original image into complex domain according to (16);
2: Initialization:
3: Initialize contour according to (23); \(\phi=1\) if \(\phi>0\); otherwise \(\phi=-1\);
4: Initialize the related parameters \(\alpha\) = 0.4, \(\sigma\) = 4 and \(\lambda=0.5\); set n = 0;
5: Repeat:
6: While n < iteration, do
7: Compute low-pass filter with Equation (21);
8: Compute complex force function with Equation (22);
9: Update the level-set function according to Equation (23);
10: n = n + 1;
11: end
12:
13: end
14:
15: If converge, end
16: Compute Gaussian filter for regularization according to Equation (2);
17: end
18:
19: Output: The resultant salient object \(\phi^{n}\).
```  
**Algorithm 1** Steps for our proposed algorithm active contours in the complex domain for salient object detection (ACCD_SOD)  
\[\frac{\partial\phi}{\partial t}=(i\times I).\lambda.(C_{PF}.H_{LP}) \tag{23}\]  
This algorithm maintains a fixed value 0.5 of \(\lambda\), which controls the evolution of the contour, while the ACSLGS model uses a parameter that is not fixed. During contour evolution, the proposed algorithm needs to initialize the level-set function \(\phi\) periodically for the signed distance function. We initialize the algorithm using a Gaussian filter to regularize the level-set function and avoid reinitialization. We define the initial level-set function \(\phi\) and the initialization of \(\phi\) for the proposed algorithm, as  
\[\phi(x,t=0)=\left\{\begin{array}{cc}i.p,&x\in\Omega_{0}-\partial\Omega_{0},\\ 0,&x\in\partial\Omega_{0},\\ -i.p,&x\in\Omega-\Omega_{0},\end{array}\right. \tag{24}\]  
where the representation of the image domain is \(\Omega\), \(\Omega_{0}\) is a subset of \(\Omega\) and \(\partial\Omega_{0}\) detects the boundary of the image domain; \(p\) is a constant, where \(p>0\), and \(p\) is important on the moving interface to control propagation on the specified interval.