# New Evolutionary-Based Techniques for Image Registration
## 4 Evolution Strategies  
The evolution strategies (ESs) represent one of the major paradigms of the evolutionary algorithm, used in a wide variety of optimization tasks and typically continuous but also discrete, combinatorial search spaces with and without constraints [28]. One of the main features of ESs is self-adaptability, a mechanism that adjusts the parameters of the exploration distribution during the search process [29]. In most of the cases, the self-adaptation is strictly related to the mutation mechanism. In ESs the mutation process represents the main variation operator and usually implements the following strategies: uncorrelated mutation with one step, uncorrelated mutation with \(n\) steps and correlated mutations respectively. The first mutation procedure uses a single standard deviation parameter, or step size, to compute each allele of the resulted chromosome. The uncorrelated mutation with n steps deals with possible different step sizes to affect different alleles in a chromosome in order to treat dimensions differently. This way, the mutation procedure introduces different standard deviations for each axis, the resulted offspring being placed on an ellipse around the individual to be mutated, orthogonal to the axes. In case of the correlated mutation, the resulted offspring belong to an ellipse having any orientation by rotating it with a rotation matrix given by a covariance parameter. Mutation mainly depends on the mutation strength parameter [30]. At the beginning of the optimization process, the value of this parameter should be large to ensure the exploring of the search space and later, in the vicinity of the optimum point, the value of the parameter should decrease to ensure the exploitation phase. In the ES the phenotype space is typically \(\mathrm{R}^{\mathrm{n}}\). In this case, the genotype and the phenotype coincide and the representation of the chromosome is straightforward, with no codification being needed. If the search procedure implements self-adaptation, each candidate solution \(\mathrm{c}\) is defined in terms of solution part \(\mathrm{c}_{\mathrm{sol}}\) and parameter part \(\mathrm{c}_{\mathrm{part}}\)[29]. For instance, in the case of the standard uncorrelated multistep mutation, the chromosome representation is as follows  
\[\mathrm{c} = \left(\mathrm{c}_{\mathrm{sol}},\mathrm{c}_{\mathrm{part}}\right) \tag{20}\] \[\mathrm{c}_{\mathrm{sol}} = \left(\mathrm{x}_{1},\mathrm{x}_{2},\ldots\mathrm{x}_{\mathrm{n }}\right)\] (21) \[\mathrm{c}_{\mathrm{part}} = \left(\mathrm{\sigma}_{1},\mathrm{\sigma}_{2},\ldots\mathrm{ \sigma}_{\mathrm{n}}\right) \tag{22}\]  
where, for each \(\mathrm{i},\ 1\ \leq\mathrm{i}\leq\mathrm{n}\), \(\mathrm{\sigma}_{\mathrm{i}}\) is the step size of mutation. The algorithm uses a population \(\mathrm{P}\) of \(\mathrm{\mu}\) potential solutions named chromosomes, which are randomly generated in the first generation. Usually, the size of population is fixed. In the evolution strategies, the parent selection is not influenced by the fitness function, meaning that the entire population could generate offspring solutions. The size of offspring population should be larger than the parent set, the recommended size being \(\lambda\cong 7\cdot\mathrm{\mu}\). The crossover procedure creates new individuals, mixing the genetic material of parents. The recombination procedures are either local or global and use discrete and respectively intermediate crossover mechanisms. In case of the local recombination, a new offspring is generated using two parents randomly selected from population. In contrast to this, in case of the global recombination, each allele of the new offspring is obtained using a pair of parents. The resulted child has multiple parents. In our hybrid approach described in Section 5.2, the standard uncorrelated multistep mutation procedure is used to add some randomness to the solution.