# New Evolutionary-Based Techniques for Image Registration
## 2 The Proposed Evolutionary Computing General Framework for Image Registration  
The main idea of performing image registration task using a particular EC approach is quite simple and can be described as follows. Let us denote by \(I_{1}\) the reference image and we assume that the sensed image \(I_{2}\) is defined by  
\[I_{2}(x,y)=\ I_{1}(x_{1},y_{1}) \tag{1}\]  
\[(x_{1},y_{1})=\ T_{SP}(x,y) \tag{2}\]  
where \(T_{SP}\) is the geometric transformation that depends on a particular set of parameters, SP. We assume that \(T_{SP}\) is invertible and the analytic expressions of \(T_{SP}\) and \(T_{SP}^{-1}\) are known. The attempt to align \(I_{2}\) to \(I_{1}\) consists of computing the transformation parameters SP and applying the inverse \(T_{SP}^{-1}\),  
\[(x,y)=\ T_{SP}^{-1}(x_{1},y_{1}) \tag{3}\]The computation of the transformation parameters can be carried out by an evolutionary algorithm (EA), where the search space is established based on SP, the fitness is defined in terms of a similarity/dissimilarity function computed either for the pair \(\left(\mathrm{I}_{1},\mathrm{I}_{2}\left(\mathrm{\tilde{T}}_{\mathrm{SP}}^{-1} \right)\right)\) or for \(\left(\mathrm{I}_{2},\mathrm{I}_{1}\left(\mathrm{\tilde{T}}_{\mathrm{SP}} \right)\right)\), and \(\mathrm{\tilde{T}}_{\mathrm{SP}}\) corresponds to the computed values of the parameters. One of the most commonly used degradation models corresponds to the affine transformation. The main geometric transformations involved with an affine function are rigid spatial transformations, the shear effect acting along axes and changes in the aspect ratio. The rigid transformation can be described in terms of translation, rotation and scale changes. Its main property is that the objects in images preserve their relative shape and dimensions. The 2D rigid transformation having as inputs the image \(\mathrm{I}_{1}\) and the parameters \(\left(\mathrm{a},\mathrm{b},\mathrm{s},\mathrm{\theta}\right)\) produces the output \(\mathrm{I}_{2}\) defined by  
\[\mathrm{I}_{2}\left(\mathrm{x},\mathrm{y}\right)\;=\;\mathrm{I}_{1}\left( \mathrm{x}_{1},\mathrm{y}_{1}\right) \tag{4}\]  
\[\left[ \begin{array}{c}\mathrm{x}_{1}\\ \mathrm{y}_{1}\end{array} \right]=\left[ \begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array} \right]+\mathrm{s}\cdot\left[ \begin{array}{cc}\cos \theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array} \right]\cdot\left[ \begin{array}{c}\mathrm{x} \\ \mathrm{y}\end{array} \right] \tag{5}\]  
where \(\left[ \begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array} \right]\) defines the translation, s stands for the scale factor and \(\mathrm{R}\;=\left[ \begin{array}{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array} \right]\) is the rotation matrix. The shear effect acting along axes is defined by  
\[\mathrm{F}_{\mathrm{x}}=\left[ \begin{array}{cc}1&\mathrm{d}\\ 0&1\end{array} \right],\;\mathrm{F}_{\mathrm{y}}=\left[ \begin{array}{cc}1&0 \\ \mathrm{h}&1\end{array} \right] \tag{6}\]  
while the changes in aspect ratio are given by the matrix  
\[\mathrm{S}\;=\left[ \begin{array}{cc}\mathrm{s}_{\mathrm{x}}&0\\ 0&\mathrm{s}_{\mathrm{y}}\end{array} \right] \tag{7}\]  
The affine transformations are obtained by applying any sequence of rigid transformations, shears and changes in the aspect ratio [15]. In our work we use the perturbation model given by a sequence of a shear transformation, acting along the \(x\) axis, and a rigid function  
\[\left[ \begin{array}{c}\mathrm{x}_{1}\\ \mathrm{y}_{1}\end{array} \right]=\left[ \begin{array}{cc}1&\mathrm{d}\\ 0&1\end{array} \right]\cdot\left(\left[ \begin{array}{c}\mathrm{a}\\ \mathrm{b}\end{array} \right]+\mathrm{s}\cdot\left[ \begin{array}{cc}\cos \theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array} \right]\cdot\left[ \begin{array}{c}\mathrm{x} \\ \mathrm{y}\end{array} \right]\right) \tag{8}\]  
In this case, the set of transformation parameters is \(\mathrm{SP}=\left\{\mathrm{a},\mathrm{b},\mathrm{s},\mathrm{\theta},\mathrm{d}\right\}\).