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Returns a number between 0 and 1 based on seedNumber. The sole purpose of this function is to seed the Random Number Generator immediately before calling the Rand() function.

Randomize( seedNumber)

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During the k-means clustering iterations, if one of the clusters loses all its members, it is removed The k-means clustering is repeated ve times with a different set of initial cluster centroids in order to obtain a good set of clusters One impostor cluster and the entire authentic training set (consisting of 20 authentic images) are used to train a SVM classi er Three SVM classi ers, one for each impostor cluster, form the And rule ensemble The proposed And rule ensemble design is compared to Bagging with three Bootstrap SVM classi ers Adaboost [26] is not feasible for the NIST 24 plastic distortion data set, even for the SVM base classi er This is because there are no training errors made by an SVM classi er that uses all training images Hence no further classi ers are made in Adaboost.

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In other words, correlation can be incorporated into the analysis by using the same failure criterion but replacing the distance of Equation (16.25) with d = (x K 1 x )1/2

RandRange()

The diversity in the classi er decisions is measured using the pairwise classi er Q values The Q values are given in Figure 417 and Figure 418, for the proposed And.

RandRange( minimumValue, maximumValue)

Sgn()

plastic distortion database. These are shown as a function of the best threshold set for And fusion. Dashed (solid) lines are impostor (authentic) Q values.

Sgn( number)

(16.50)

Sqr()

4

Sqr( number)

0.5

BitAnd()

BitAnd( number1, number2)

Low et al. (1998) observe that, if m is the vector of means of the variables and C is the covariance matrix, Equation (16.50) can also be written d = [(x m)T C 1 (x m)]1/2 (16.51)

1 0

BitMaskClear()

Figure 4.18. Pairwise Q values of the Bagging ensemble of SVM classi ers on the NIST 24 plastic distortion database. An optimal set of thresholds are selected for And fusion, for which the Q values are shown. Dashed (solid) lines are impostor (authentic) Q values.

BitMaskClear( number, start, length)

BitMaskRead()

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rule ensemble and the Bagging ensemble, respectively The authentic and impostor pairwise classi er Q values are shown as a function of the optimal set of thresholds for the And rule The authentic and impostor Q values are positive in both gures The Bagging ensemble has authentic and impostor Q values close to 1, which implies similar decisions by the classi ers and low diversity The impostor Q values of the proposed ensemble are lower and close to 05, which signi es more diversity on the impostor decisions than the Bagging ensemble The most favorable conditional dependence for And fusion is negative impostor Q values and positive authentic Q values While this target has not been reached, more diversity than for the Bagging classi er ensemble has been achieved by our proposed design.

BitMaskRead( number, start, length)

In summary, de ning the constraint as before and minimizing the distance expressed by either Equation (16.50) or Equation (16.51) results in the correlated version of the Hasofer Lind analysis.

BitMaskSet()

The Cosine distance metric used in the proposed ensemble generaton does not capture the classi cation strategy of the SVM and is hence a poor metric for clustering images Due to the dif culty in obtaining good clusters of impostors, it is tough to obtain very diverse decisions on impostors The ROCs of 15 monotonic decision rules applied the three SVM classi ers in the proposed And rule ensemble are shown in Figure 419 The ROCs are obtained by nding a set of optimal thresholds on the three classi er scores for each decision fusion rule It is observed that the And rule is the best rule with an EER of 12% The performance of the monotonic fusion rules on the Bagging classi er ensemble is displayed in Figure 420 Here, the ROCs of many the decision fusion rules are comparable.

Returns an integer containing number bitwise masked with mask for length bits starting at position start.

The EERs of the Or rule and the Majority rule are 24% and 26% and hence are comparable Thus the better diversity on impostor decisions in the And rule ensemble have enabled a better fusion accuracy than the Bagging ensemble..

BitMaskSet( number, mask, start, length)

BitNot()

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