# From Iris Image to Embedded Code: System of Methods
## 5 Feature Extraction and Matching
Iris features \(V(\phi,\rho)\) are calculated by convolution of the normalized image (20) with a Gabor or Log-Gabor filter, the transformation is performed in the spectral domain:  
\[ \begin{split} V(\phi,\rho)=& N(\phi,\rho)*g_{\sigma \lambda}(\phi)=\\ =&\mathcal{F}^{-1}\{\mathcal{F}\{N(\phi,\rho)\} \mathcal{F}\{g_{\sigma\lambda}(\phi)\}\}=\\ =&\mathcal{F}^{-1}\{\mathcal{F}\{N(\phi,\rho)\}G_{ \lambda L}(u)\}\.\end{split}  \tag{28}\]  
where \(\sigma\) and \(\lambda\) define the width of the wavelet along the angular axis and the modulation frequency, \(s\) is the width along the radial axis, \(\mathcal{F}\) is the Fourier transform. The features used to form the patterns are computed as binary values of real and imaginary parts of the array \(V(\phi,\rho)\):  
\[ \begin{split} T_{Re}(\phi,\rho)&=\mathcal{H}[ \Re(V(\phi,\rho))]\,\\ T_{Im}(\phi,\rho)&=\mathcal{H}[\Im(V(\phi,\rho))]\, \end{split}  \tag{29}\]  
where \(\mathcal{H}[\cdot]\) is the Heavyside function. So, eye image \(b(x,y)\) produces a template \(T(\phi,\rho)\), and each element of the template contains two bits. We use features raster of 13 pixels in radial direction \(r\) and 256 pixels in tangential direction \(\phi\). Since each pixel produces two bits in (29) the total size of the template is 6656 bit [40]. Although here we do not build a classification system, which calculates a distance between templates and compares it against a classification threshold, template matching is implicitly present, as it will be shown below. Thus, we need to describe the matching method. In a standard iris recognition approach templates \(T_{1}\) and \(T_{2}\) are matched by normalized Hamming distance:  
\[\rho_{0}(T_{1},T_{2})=\frac{1}{|\Omega|}|\{T_{1}(\phi,\rho)\
eq T_{2}(\phi, \rho),(\phi,\rho)\in\Omega\}|\, \tag{30}\]  
where \(\Omega=M_{1}\cap M_{2}\) is the intersection of the visible areas (presenting true data) of the two irises. Because of the uncertainty of the iris rotation angle, a more complex distance formula is used. The rotation of the original image of the eye is equivalent to a cyclic shift of the normalized image along the \(O\phi\) axis. Therefore, one of the templates (together with the mask) is subjected to several shift and compare operations:  
\[ \begin{split}\rho(T_{1},T_{2})&=\min_{\psi}\rho_{ \psi}(T_{1},T_{2})\,\\ \rho_{\psi}(T_{1},T_{2})&=\frac{1}{\Omega(\psi)}|\{ T_{1}(\phi+\psi,\rho)\
eq T_{2}(\phi,\rho),(\phi,\rho)\in\Omega(\psi)\}|\\ \Omega(\psi)&=M_{1}(\phi+\psi)\cap M_{2}(\phi)\, \end{split}  \tag{31}\]  
where \(\psi\in[-S;S]\) is the image rotation angle. Here things may be simplified. For the embedding method, only irises with low occlusion levels are acceptable. Thus, it is supposed that masks \(M_{1}\) and \(M_{2}\) cover all of the iris area, and \(\Omega\) set spans all templates. Omitting mask, rewriting \(|\{T_{1}\
eq T_{2}\}|\) as \(\sum T_{1}\oplus T_{2}\) and using single order index \(i\) instead of coordinates \((\phi,\rho)\) put (30) as:  
\[\rho_{0}(T_{1},T_{2})=\frac{1}{N}\sum_{i=1}^{N}T_{1}(i)\oplus T_{2}(i)\, \tag{32}\]where \(T(i)\) is the \(i\)-th bit of the template, operation \(\oplus\) is the sum modulo 2, \(N\) is the size of the template. Furthermore, (31) transforms to  
\[ \begin{split}\rho(T_{1},T_{2})&=\min_{\psi}\rho_{ \psi}(T_{1},T_{2})\,\\ \rho_{\psi}(T_{1},T_{2})&=\frac{1}{N}\sum_{i=1}^{N}T _{1}(i(\psi))\oplus T_{2}(i)\,\end{split}  \tag{33}\]  
where \(i(\psi)\) index is recalculated accordingly. The recognition system is designed to supply the following conditions with the lowest possible errors:  
\[ \begin{split} T_{1},T_{2}\text{ taken from same person}& \implies\rho(T_{1},T_{2})\leqslant\theta\,\\ T_{1},T_{2}\text{ taken from different persons}& \implies\rho(T_{1},T_{2})>\theta\.\end{split}  \tag{34}\]  
Violation of the first condition in (34) is called _false reject_ and its probability is referred to as _false reject rate_ (FRR). FRR of the system is estimated in tests as the ratio of the number of false rejects to the number of all matches of biometric traits of the same persons.