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where f and fi are the feature vectors of an unknown sample and the ith class, respectively; f j and fi j are the jth component of the feature vector of the unknown sample and that of the ith class, repectively; and c is the total number of classes, i ndicates the Euclidean norm, and dn (f,fi ) denotes a similarity measure. d1 , d2 , and d3 are L1 distance, L2 distance, and cosine similarity measure, respectively.
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Wildes proposed a classi cation scheme based on normalized correlation between two iris patterns. Let p1 [i, j] and p2 [i, j] be two image arrays of size n by m. De ne 1 = (1/nm) 1 = 1 nm
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where 1 and 1 are, respectively, the mean and the standard deviation of the intensities of p1 . Let 2 and 2 be similarly de ned for p2 . Then the normalized correlation between p1 and p2 can be de ned as
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The advantage of the normalized correlation method is that it measures the correlation in a global sense and simultaneously accounts for the local variations in the image intensity that corrupt the standard correlation. In implementation, Wildes performs normalized correlation in local blocks of size 8 8 in each of the four spatial frequency bands derived from the Laplacian pyramid. This will result in multiple correlation values for each band. Subsequently, he chooses the median of the normalized correlation values for each of the pyramid layer, which gives four goodness-of-match values when comparing two iris patterns. These four values can be treated as a feature vector of length four to indicate whether these two irises are from the same eye or not. In the end, Wildes performs FLDA to reduce the dimensionality of the feature vector and then performs classi cation.
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Probabilistic Graphical Model Approach
Kerekes et al. [24] proposed a novel matching algorithm based on modeling of the local deformation of iris patterns with a probabilistic distribution on a lattice-type undirected graphical model, and they used Gabor wavelet-based similarity scores and intensity statistics as observations in the model. They reported a signi cant improvement over Daugman s algorithm. Model Description
The rst assumption behind their model is that iris patterns, even from the same eye, suffer from local deformations due to pupil dilations or contractions that ultimately lead to segmentation errors. Local deformations mean that when an iris is divided into several small patches, every patch may be translated in different directions and different distances. Such deformations are not constant and therefore cannot be recovered by a global transformation. Local shifts of iris pattern are shown in Figure 13.14.
Iris Recognition
Figure 13.14. (a) Reference iris pattern. (b) iris pattern from the same eye, but with local
deformation. Shifts of white boxes between (a) and (b) indicates how many local shifts are in each small window.
This type of local shifts can be modeled with an undirected lattice-type graphical model, depicted in Figure 13.15. Suppose the entire iris map is divided into 36 small patches. Each node di in the model (i = 1, . . . , 36) represents a 2D discrete-valued shift vector for each local patch, and the true values of the shift vectors are hidden and cannot be observed from outside. The components of di are the vertical and horizontal shifts (in pixels) of the template region relative to the corresponding query region. The nodes i (i = 1, . . . , 36) are hidden binary-valued occlusion variables, where i = 0 represents an occluded region and i = 1 represents a valid unoccluded iris region. Nodes Oi represent the observations, which include the match score array mi (x) and the occlusion statistic i . The de nition of mi (x) is given in Eq. (13.19): m(x) = 1 |St | ct (y)T cq (y x),