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x[i]x[i + m] , (16.1)
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where x[i] is the windowed ECG, for i = 0, 1 . . . (N |m|1), and x[i + m] is the time-shifted version of the windowed ECG, with a time lag of m = 0, 1, . . . , (M 1); M << N. Even though the major contributors to the AC are the three characteristic waves, normalization is required because large variations in amplitudes appear, even among the windows of the same subject. The fact that the AC embeds distinctive characteristics for every subject, and thus can be used to capture similarities between signals recorded at different times, can be con rmed by the AC plots in Figure 16.4.
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Figure 16.4. Normalized autocorrelation of ECG windows from six subjects of the PTB database.
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Two records are available for every subject, recorded at different times. Sequences from the same record are shown in the same shade.
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Electrocardiogram (ECG) Biometric for Robust Identi cation
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This observation implies that an AC vector can be used directly for classi cation. However, depending on the sampling frequency of the ECGs, the dimensionality of an AC window can be considerably high, and dimensionality reduction is required. Discrete Cosine Transform for Dimensionality Reduction. The discrete cosine transform (DCT) is applied to the normalized autocorrelation coef cients for dimensionality reduction. This methodology is referred to as AC/DCT. The DCT frequency coef cients are estimated as
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(2i + 1) u , 2N
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(16.2)
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where N is the length of the signal y[i] for i = 0, 1, . . . , (N |m| 1). For the AC/DCT method, y[i] is the autocorrelated ECG obtained from Eq. (16.1). G[u] is given by 1, k = 0, N (16.3) G(k) = 2 , 1 k N 1. N Due to the energy compaction property of DCT, a lower-dimension representation is obtained. Near-zero frequency components of the spectrum can be discarded. Assuming we take an M-point DCT of the autocorrelated signal, only C << M nonzero DCT coef cients will contain signi cant information for identi cation. From a frequency domain perspective, the C nonzero coef cients correspond to the frequencies between the cutoffs of the bandpass lter that is used in preprocessing. This is because after the AC operation, the bandwidth of the signal is kept. The DCT coef cients retain the discriminative properties of the AC samples among different subjects as depicted in Figure 16.5. The re ned feature space is propagated to the classi cation step, where every compressed input DCT vector is compared to the ones stored in the gallery set. Linear Discriminant Analysis for Dimensionality Reduction. Another option to reduce the dimensionality of the feature space is the linear discriminant analysis (LDA). Supervised learning is carried out in the transformed domain, so that eventually feature dimensionality is reduced and the classes are better distinguished. This scheme is referred to as AC/LDA. Given a training set Z = {Zi }U , with U classes, where each class Zi = {zij }Ui i=1 j=1 contains a number of autocorrelated windows zij , a set of K feature basis vectors { m }K can be estimated by maximizing Fisher s ratio. Maximizing this ratio is m=1 equivalent to solving the following eigenvalue problem: = arg max
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| T S b | , | T S w |
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(16.4)
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16.4 The ECG Biometric for Robust Identi cation
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Figure 16.5. Discrete cosine transform coef cients of autocorrelated ECG windows from six
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subjects of the PTB database. Two records are available for every subject, recorded at different times. Sequences from the same record are shown in the same shade.
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where = [ 1 , . . . , K ], and Sb and Sw are the between and within-class scatter matrices respectively. These matrices are de ned as Sb = 1 N
Ui (zi z)(zi z)T ,
i=1 U Ui
(16.5)
Sw = where zi =
(zij zi )(zij zi )T ,
i=1 j=1
(16.6)
and N = and positive semide nite. Linear discriminant analysis nds as the K most signi cant eigenvectors of (S W ) 1 S b , which correspond to the rst K largest eigenvalues. A test input window z is subjected to the linear projection y = T z, prior to classi cation [36]. 16.4.1.3 Classi cation
Ui 1 j=1 zij is the mean of class Zi , N is the total number of windows, Ui U i=1 Ui . Both the between- and within-class scatter matrices are symmetric
Classi cation represents the last step of the identi cation procedure. For this step, every input feature vector is compared to the ones stored in the gallery set in order to