Multimodal biometric systems designed to manage access to sheltered assets and information have been found to offer a better deal of security and user-convenience. Multimodal biometric setups handle frameworks which pursue the organisation or synchronisation of the usage of different biometric traits in a way that advances the method of information fusion [29]. In addition, such systems also cater to situations where one or more of the nail plates might not be accessible, rendering the investigation feasible only if it can be carried out using the available traits.

Score-Level Fusion

For the work reported in this brief, four well-accepted score-level fusion rules have been used to check the efficacy of the proposed system in the verification mode, namely the sum rule, the product rule, the max rule, and the min rule.

If 5, and S2 are the scores from two different biometric traits, then their fused scores using the above mentioned rules are given as follows:

Rank-Level Fusion

Rank-level fusion amalgamates ranking lists procured from different biometric traits for deriving a final ranking list, in order to aid in the process of arriving at the final decision [30]. Also, certain systems provide scores or features, inappropriate for fusion [31]. In those cases as well, rank-level fusion is a very feasible choice for building multimodal systems [32]. Optimal performance accuracy in a multimodal system is achieved when the different traits under fusion are given appropriate weightage or importance. In this w'ork, the following linear and nonlinear fusion rules have been put to use for rank-level fusion:

Logistic Regression Method

The Logistic Regression method [33] may be considered as an important tool for combining classifiers having non-uniform performances. The Borda count method [30] evaluates the fused score as the sum of rank scores of all the considered classifiers. The final ranking list for this method is achieved by sorting the fused scores. The method works under the assumption that all classifiers perform equally well. Such an assumption makes the system extremely vulnerable to weak classifiers. The Borda count method requires substantial modification when a combination of classifiers having varying performances is considered. Such modification demands the assignment of weights to the rank scores of each classifier, where the assigned weights reflect the relative importance of each classifier from the perspective of their rank-level fusion. Let the probability of getting the true class be P(Y = 1 I x), and let it be denoted by л(х) where x = (a,, x2 ,..., x„,) corresponds to the rank scores assigned to that class by classifiers C,, C2,..., C,„. If it is assumed that jq has the largest value and if the class is ranked at the top by C, then

Here, cr, ft = (Д, Д>,..., Д,,) are constant parameters, log is called the

1 -л(х)

logit, and it is linearly related to x. These constants can be used as weights for the rank scores of each classifier. This is because the relative magnitudes of these constants signify the marginal contribution of the classifiers to the logit. As such, the fused rank RLr is given by Equation (9.6) as follows:

In this work, the aforementioned constants have been computed by two methods, viz. one by empirical computation and the other by Particle Swarm Optimisation (PSO) [34] technique.

Mixed Group Rank

The Mixed Group Rank method [35] makes use of the classical Highest Rank (minimum value amongst all rank scores) and the Logistic Regression method.

The Mixed Group Rank method makes a linear weighted combination of the minimum ranks of all the possible subgroups in the considered group of matchers. The final fused rank is given by

Here, G is the subgroup of matchers belonging to the entire group of matchers, is the rank assigned to user j, coG represents the weight assigned to the concerned subgroup of matchers, N is the total matchers used. For this study where three matchers have been used, i.e., the index, middle, and ring fingernail plates, Equation (9.7) shall be represented as follows:

For the Mixed Group Rank method, all concerned weights have been evaluated through PSO and empirical computation.

Inverse Rank Position

The Inverse Rank Position algorithm [36] uses the inverse of the sum of inverses of all rank scores for every matcher, and the final fused rank is given by

Here, rx (/) is the rank assigned to user / by the xth matcher, and N is the total number of matchers used.

Nonlinear Weighted Methods

Nonlinear methods [30] for rank-level fusion have been used in this work. The ranked list of user identities are nonlinearly weighted and consolidated as follows:

Here, rj(p) is the rank assigned to candidate p by the i',h matcher, ft), represents the weights assigned to the /th matcher, and Cp is the fused rank. For these non-linear matchers too, computation of weights has been performed via PSO, in addition to empirical computation [30].

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