## 2 Meta-Heuristics and Applications for Image Segmentation
### Novel Bat Algorithm  
The bat algorithm (BA) is an algorithm proposed by [PERSON] and [PERSON] in 2012[39] as an algorithm used to solve optimization problems. Later, [PERSON] et al.[40] transformed the algorithm to include the Doppler effect, giving this version the name of novel bat algorithm (NBA). This algorithm uses bats' behaviors and the echolocation ability to identify objects in a determinate space. The bats use echolocation to sense distance and change velocity and frequency to search for solutions for the fitness function that is being optimized.  
One of the novel features of the NBA algorithm is the consideration and modelling of the compensation of the Doppler effect on the pulses emitted and the environmental noise. Another aspect to highlight is the inclusion of the quantum behavior of bats, which allows these virtual bats to find food in different habitats to find the optimal overall solution to the problem.  
There are different variants of bat-inspired algorithms based on modelling some echolocation characteristics of bats. Therefore, the authors of the BA proposal in [41] provide a set of rules for optimization algorithms based on the behavior of bats, as described in [40]:  
* To measure distance, bats use echolocation, which also allows them to differentiate between their prey and obstacles.
* Bats fly randomly with a velocity \(q_{i}\) to a position \(z_{i}\) with a fixed frequency \(f_{init}\), varying the wavelength \(\lambda\) and volume \(V_{0}\) to look for their prey. Depending on the proximity of the target, they can automatically adjust the wavelength of their emitted pulses (frequency), and adjust the pulse emission rate \(l\,\epsilon\,[0,1]\).  
Figure 1: Flowchart of genetic algorithm.  
* Although the volume may vary in various ways in reality, it is important to consider that the noise varies from a positive number \(V_{0}\) to a constant minimum value \(V_{min}\).  
To this brief set of rules, we must add a couple more, which present a direct relationship with the new features proposed by the NBA algorithm [40]:  
* All bats can search for food in different habitats.
* All bats can compensate for the Doppler effect by echo.  
Finally, \(z_{i,j}^{G}\ (i\in\{1,\dots,n\},\ j\in\{1,\dots,\ d\})\) characterizes the position and \(q_{i,j}^{G}\) the velocities in an iteration \(G\), of \(n\) bats looking for food in a \(d\)-dimensional space.  
Updating the positions, velocities and other parameters of bats' behavior takes \(G\) number of iterations. Each one will perform the following steps explained in Figure 2:  
As seen, this algorithm can self-adapt throughout the optimization process, because of the compensation it makes for the Doppler effect in echoes and the adjustments it makes to the frequency according to the proximity of individuals to the solution. It is also worth mentioning the addition of habitat selection methods to BA, since these two types of behavior of bats, mechanical and quantum, allow the algorithm to have a better convergence and diversity of solutions.  
This algorithm was used for the segmentation of mammography images by [PERSON] et al. in 2016 [42], showing lower segmentation errors compared to the Otsu method.  
The NBA and GA algorithms are population-based algorithms so they perform a more extensive search throughout the search space, which provides an advantage compared to SA which is a single-trajectory algorithm.