# From Iris Image to Embedded Code: System of Methods
## 5 Feature Extraction and Matching  
We use the standard approach [3] here, which first transforms the iris ring to a so-called _normalized_ image. This image is a rectangular raster, it is obtained from the iris ring by the polar transformation, analogous to (19), (20), where \(r_{1}\) and \(r_{2}\) are set to the radius of pupil and iris, respectively. In fact, more elaborate version of (19) is used:  
\[ \begin{split}& x(\phi,\rho)=\Big{(}1-\frac{\rho}{H}\Big{)}x_{1} \bigg{(}\frac{2\pi\phi}{W}\bigg{)}+\frac{\rho}{H}x_{2}\bigg{(}\frac{2\pi\phi}{W }\bigg{)}\,\\ & y(\phi,\rho)=\Big{(}1-\frac{\rho}{H}\Big{)}y_{1}\bigg{(}\frac{2 \pi\phi}{W}\bigg{)}+\frac{\rho}{H}y_{2}\bigg{(}\frac{2\pi\phi}{W}\bigg{)}\,\\ & x_{1}(\phi)=x_{p}+r_{p}\cos(\phi)\,y_{1}(\phi)=y_{p}+r_{p}\sin( \phi)\,\\ & x_{2}(\phi)=x_{1}+r_{l}\cos(\phi)\,y_{2}(\phi)=y_{l}+r_{l}\sin( \phi)\,\end{split}  \tag{24}\]  
where \(x_{p}\), \(y_{p}\), \(r_{p}\) are the position and radius of pupil and \(x_{l}\), \(y_{l}\), \(r_{l}\) are the position and radius of iris. In comparison to (19) this variant accounts for the difference of pupil and iris centres. The key idea of standard iris feature extraction is to convolve the normalized iris image with a filter, calculating the most informative features of the texture. Earlier Gabor wavelet was used for feature extraction. In one-dimensional space, it is represented as  
\[g_{\sigma\lambda}(x)=\exp\!\left(-\frac{x^{2}}{2\sigma^{2}}\right)\exp\!\left( -i\frac{x}{\lambda}\right)\,\ \ G_{\sigma\lambda}(u)=\exp\!\left(-\frac{(u-\lambda^{-1})^{2} \sigma^{2}}{2}\right)\, \tag{25}\]  
where \(\sigma\) defines the width of the wavelet in the spatial domain, \(\lambda\) is the wavelength of modulation of the Gaussian by a harmonic function. By introducing inverse values \(S=1/\sigma\) and \(W=1/\lambda\), a simplified representation in the frequency domain can be obtained:  
\[G_{SW}(u)=\exp\!\left(-\frac{(u-W)^{2}}{2S^{2}}\right). \tag{26}\]  
It turned out that the modification of the Gabor wavelet called Log-Gabor function is better for feature extraction. Log-Gabor is given in the frequency domain as:  
\[G_{SW}(u)=\exp\!\left(\frac{-\log^{2}(u/W)}{2\log^{2}(S/W)}\right)=\exp\!\left( -\frac{(\log u-\log W)^{2}}{2\log^{2}L}\right). \tag{27}\]  
This is equivalent to (26), in which each variable is replaced by its logarithmic counterpart. \(L=S/W=\lambda/\sigma\) represents the ratio of the modulation wavelength to the width of the Gaussian. Research has shown that Log-Gabor wavelets are most likely optimal for the template generation problem. Therefore, we use this type of filter. The parameter \(\lambda\) is essentially the characteristic size of the objects in the image extracted by this filter, and \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Database**} & \multirow{2}{*}{**Error,** **Pixels**} & \multicolumn{6}{c}{**Method**} \\ \cline{3-8}  & & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON] et al.** & **[PERSON]** & **Presented** \\ \hline BATH & \(\varepsilon_{P,abs}\) & 3.44 & 3.73 & 5.32 & 4.29 & 3.27 & 0.52 \\  & \(\varepsilon_{P,abs}\) & 4.38 & 4.54 & 6.72 & 4.65 & 3.19 & 1.42 \\ \hline CASIA & \(\varepsilon_{P,abs}\) & 5.37 & 2.15 & 3.67 & 4.79 & 1.19 & 2.44 \\  & \(\varepsilon_{P,abs}\) & 6.12 & 4.39 & 5.15 & 5.39 & 3.02 & 1.62 \\ \hline MMU & \(\varepsilon_{P,abs}\) & 3.15 & 2.61 & 4.98 & 3.92 & 1.14 & 0.97 \\  & \(\varepsilon_{P,abs}\) & 3.96 & 4.18 & 5.78 & 4.67 & 3.76 & 1.77 \\ \hline NDIRIS & \(\varepsilon_{P,abs}\) & 6.37 & 2.13 & 5.59 & 5.92 & 1.79 & 0.83 \\  & \(\varepsilon_{P,abs}\) & 7.51 & 3.53 & 7.23 & 7.38 & 3.11 & 1.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Matching against other methods. is the number of periods of the harmonic function in Equation (25) which have sufficient amplitude and influence to the result. Optimal values of \(\lambda\) and \(L\) are selected according to [40].