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where ct (y) and cq (y) are the feature vectors from the unshifted template and the query iris patterns, respectively; St is the support of the template iris code and | St | is the size of the template. i is derived from classifying features that consist of (1) the mean intensity value in a small neighborhood of the pixel, (2) the standard deviation of the intensity values in the same neighborhood, (3) the percentage of pixels whose intensity is greater than one standard deviation above the mean of the entire iris plane, and (4) the shortest Euclidean distance to the centers of the upper and lower eyelids. This classi cation is done using FLDA.
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13.5 Recognition
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Figure 13.15. Probabilistic graphical model for modeling local deformation of iris patterns,
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Potential Functions and Density Estimation
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The next step is to de ne a potential function between each pair of connected nodes in the graphical model. The potential between two nodes di and dj should be higher if both vectors are closer in direction and should be lower if they are opposite in direction. Similarly, the potential between node i and j should be higher if their values are similar and should be lower if they are dissimilar. Furthermore, the potential functions between di and i should be independent from each other. Therefore, Eq. (13.20) is used to simplify the overall potential function: i,j (h i , h j ) = d,i,j (di , dj ) ,i,j ( i , j ), d,i,j (di , dj ) = e 2 (a di 0 , ,i,j ( i , j ) = 1 , 2 ,
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), (13.20)
i = j = 0, i = j = 1, i = j . /
Parameters a and b represent penalties on absolute and relative deformations, respectively, while parameters i correspond to priors on their corresponding occlusion con guration ( i , j ). The authors used a = 0.05, b = 0.1, 1 = 0.7, 2 = 0.14, and 3 = 0.08.
Iris Recognition
If we denote the random variable s to be the probability distribution of true match scores and denote to be the probability distribution of true occlusion metrics, then the distributions P(s) and P( ) are assumed to be normally distributed 2 2 with mean and variance s , s , , and , and we de ne F (S) = P(s < S) = F =P <
S 2 N(s; s , s )
ds, (13.21)
2 ) d
N( ; ,
to be the cumulative distribution functions (cdf) of s and . These parameters can be learned from the training data. We can compute the probabilities Fs (m(di )) and F ( i ) of having observed at least the true match score and occlusion metric for each pixel or region, respectively. We then achieve the monotonic potential function i (hi , Oi )by setting them equal to the corresponding probability for the believed state of i as follows: i (hi , Oi ) = Fs (m(di )), F ( i ), i = 0, i = 1. (13.22)
Loopy Belief Propagation (LBP)
Given a particular set of observations for nodes Oi , the structure in Figure 13.13 reduces to a Markov random eld (MRF) with potential functions described in the previous section [23, 24]. First we have to estimate the conditional distributions P(hi | O) for i = 1, . . . , 36 (O is the set of all observations O1 , . . ., O36 ) in order to compute the overall match score. One way to estimate the conditional distributions is to use loopy belief propagation [25]. It is an iterative optimization process over the joint distribution in a graphical model. In each iteration every unobserved node sends a message to each of its unobserved neighbors. The message i j k from node j to neighboring node k at iteration i is computed according to Eq. (13.23): i (hk ) = j k
j (hj , Oj ) j,k (hj , hk )