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Indeed, all the subspace methods proposed in the literature are very sensitive to the misalignment of the facial images In most cases, the images are normalized according to the eyes coordinates that are located manually This fact renders the subspace methods unusable with automatic face detectors that are not perfect, since their veri cation performance greatly deteriorates under such circumstances On the other hand, EGM has the ability to correct misalignments due to the elastic matching step However, the computational cost of EGM is bigger that the computational cost of subspace techniques This fact places EGM among the techniques that cannot be easily used for face recognition task where a large number of test images should be examined in a limited time period The outcome of the comparative study in reference 27 is given in Table 61 The experiments were performed in the M2VTS.

Table Equality, Relational, and Logical Operators Operator ! < <= > >= == != && Function logical NOT less than less than or equal greater than greater than or equal equality inequality logical AND logical OR Use !expr expr < expr expr <= expr expr > expr expr >= expr expr == expr expr != expr expr && expr expr expr

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The equality, relational, and logical operators evaluate to either of the bool constants true or false If such an operator is used within a context requiring an integer value, the result is promoted to a value of either 1 (true) or 0 (false) For example, in the following code fragment, we wish to count all the elements of a vector less than a user-specified value To accomplish this, we add the result of the less-than operator to a count of the elements (The += operator is a shorthand notation; it means that we are adding the right-hand expression to the current value of the left-hand operand We discuss the compound assignment operators in Section 44)

int elem_cnt = 0; vector<int>::iterator iter = ivecbegin(); while ( iter != ivecend() ) {

Table 6.1. Comparison of Equal Error Rates for Subspace and Elastic Graph Matching Veri cation Techniques in the Aligned (A) and Attacked (by Scale (S) and Rotation (R)) M2VTS Database Veri cation Technique EGM Eigenfaces Fisherfaces EER(A) (%) 6.05 10 40 8.3 26 EER(R) (%) 6.65 13.1 38 9.5 26 EER(S) (%) 7.4 13 39 11 31

file:///F|/WinDDK/resources/CPPPrimer/c++primerhtm (144 / 1065) [2001-3-29 11:32:04]

// same as: elem_cnt = elem_cnt + (*iter < some_value) // true/false value of *iter < some_value // is promoted to either 1 or 0 elem_cnt += *iter < some_value; ++iter; }

(1.30)

The logical AND (&&) operator evaluates to true only if both its operands evaluate to true The logical OR (||) operator evaluates to true if either of its operands evaluates to true The operands are guaranteed to be evaluated from left to right Evaluation stops as soon as the truth or falsity of the expression is determined Given the forms

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expr2 is guaranteed not to be evaluated if either of the following is true:

The discussion so far focuses on linear dimensionality reduction and regression. It has been shown that both discriminant analysis and regression can be adapted to nonlinear models by using the kernel trick [46 48]. Mika et al. [49] extended the Fisher discriminant analysis to its kernel version in the binary-class case. Following the work in reference 50, Baudat and Anouar [51] proposed the generalized discriminant analysis (GDA) algorithm for multiclass problems. The equivalence relationship between kernel discriminant analysis (KDA) and kernel regression has been studied in reference 35 for binary-class problems. The analysis presented in this chapter can be applied to extend this equivalence result to multiclass problems. A symmetric function : X X R, where X denotes the input space, is called a kernel function if it satis es the nitely positive semide nite property [46]. That is, for any x1 , . . . , xn X , the kernel Gram matrix K IRn n , de ned by Kij = (xi , xj ), is positive semide nite. Any kernel function implicitly maps the input set X to a high-dimensional (possibly in nite) Hilbert space H equipped with the inner product ( , )H through a mapping from X to H : (x, z) = ( (x), (z))H . In KDA, three scatter matrices are de ned in the feature space H as follows:

In a logical AND expression, expr1 evaluates to false In a logical OR expression, expr1 evaluates to true

A valuable use of the logical AND operator is to have expr1 evaluate to false in the presence of some boundary condition that would make the evaluation of expr2 dangerous For example:

(1.31)

while ( ptr != 0 && ptr->value < upperBound && ptr->value >= 0 && notFound( ia[ ptr->value ] )) { }

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