error rate value, the higher the accuracy of the biometric system.
T SR= (1−number of accepted imposter+ number of rejected genuine
total number of accesses )×100%
(3.5) EER= F AR+ F RR
2 (3.6)
In signal detection theory, a Receiver Operating Curve (ROC) is a graph-ical plot of the sensitivity vs. (1-specificity) for a binary classifier system as its discrimination threshold is varied. It can also be represented equivalently by plotting (1-FRR) vs. FAR. The best possible prediction method would yield a point in the upper left corner of the ROC space. The less the tradeoff between the sensitivity and specificity, the better the performance is. This is tantamount to saying: the closer the area under the ROC is to one, the bet-ter the performance is. Fig. 3.14 represents two typical ROC curves. Curve B has an area under the curve closer to one than curve A and then curve B is the better performance.
3.8 Wavelet Determination of wBNMF
In the field of face recognition using the AR database, we try to determine the best wavelet filter for wBNMF. Given a digitized image containing the neutral facial expression of a person, the facial features in the image can be extracted and then be used to identify the facial image of the same person2 from a testing set of smile facial expression images. The main steps involve projecting the facial images into a low dimensional feature space through the basis matrix W , and the Riemannian metric is used to define the similarity measure between two faces.
3.8. WAVELET DETERMINATION OF WBNMF
Figure 3.14: Receiver Operating Curve.
First, an experiment is carried out to determine the most appropriate basis number for BNMF. Under consideration of computational efficiency, we take the well-known AR database with reduced dimension of 60 × 60 and 1000 iterations are used to update W and H. In addition, the distance of the same person’s facial images in the training and testing sets respectively are measured to learn the possible thresholds. Different pairs of False Acceptance Rate (FAR) and False Rejection Rate (FRR) at various ascending thresholds are created to draw a Receiver Operating Curve (ROC), and the best thresh-old with the lowest Equal Error Rate (EER) can be detected. The resulting receiver operating curves with different basis numbers, r = 25, 75, 125, 175, are drawn in Fig. 3.15. Note that the closer the receiver operating curve
3.8. WAVELET DETERMINATION OF WBNMF
basis number (r) threshold FAR (%) FRR (%) EER (%) 25 2.3242e + 006 12.9697 15.2500 14.1098 75 2.9913e + 006 9.6843 20.5000 15.0922 125 3.5046e + 006 12.8131 18.0000 15.4066 175 3.6686e + 006 12.8131 18.2500 15.5316
Table 3.1: Error measures of BNMF using different basis numbers for the AR database. The lowest EER means the best recognition accuracy.
for each receiver operating curve, the most up-left point containing the best threshold can produce the best face recognition rate.
According to Table 3.1, the number of basis components are chosen to verify the best performance with the corresponding threshold. The optimum verification rate for the AR database using BNMF is EER = 14.1098 with FAR = 12.9697, FRR = 15.2500 when r = 25 at threshold = 2.3242e+006.
It can be observed that the optimum r is not the biggest one. This result indicates that moderately large r of BNMF is sufficient to discriminate dif-ferent faces. To sum up, the performance of a facial recognition system is measured by previously-mentioned error rates. Good performance can bal-ance the tradeoff between the verification rates and the false accept rates depending on application needs. An ideal system would have a verification rate of 100% and a false accept rate of 0%. But such systems do not exist.
Another experiment is accomplished by using a similar set of r = 25 to determine the optimum verification rate when wBNMF is integrated with multiple wavelet filters using decomposition level 3. In addition, we adopt five types of orthogonal wavelet filters: Haar, Discrete Meyer, Daubechies, Symlets and Coiflets. The last three ones have two different orders
respec-3.8. WAVELET DETERMINATION OF WBNMF
(a)
(b)
Figure 3.15: Comparison of BNMF using various basis numbers for the AR database. Since the upper left point in the original receiver operating curve is not distinct, we magnify the region between 70 and 90 of (1-FRR) in the form of percentage as shown in the lower figure.
3.8. WAVELET DETERMINATION OF WBNMF
wavelet filter threshold FAR (%) FRR (%) EER (%) Haar 2.3215e + 006 13.5076 15.0000 14.2538 Discrete Meyer 2.2844e + 006 12.7146 16.7500 14.7323 Daubechies 5 2.3022e + 006 12.7652 15.5000 14.1326 Daubechies 10 2.1606e + 006 10.5682 18.0000 14.2841 Symlets 5 2.3024e + 006 12.4672 15.5000 13.9836 Symlets 10 2.3329e + 006 13.2096 15.2500 14.2298 Coiflets 1 2.2746e + 006 12.5076 16.5000 14.5038 Coiflets 5 2.3076e + 006 12.7955 15.5000 14.1477
Table 3.2: Error measures of wBNMF using different wavelet filters at the most appropriate threshold. An appending number to the wavelet filter is its corresponding order.
tively. Fig. 3.16 illustrates the receiver operating curves for the eight wavelet filters using wBNMF and the AR database. As shown in Fig. 3.16, it reveals that the Symlets filter of order 5 performs the least Equal Error Rate when the best chosen wavelet filter is integrated with wBNMF. The optimum error measures using the most appropriate threshold are recorded in Table 3.2. As a result, integration of Discrete Meyer filter and wBNMF achieve the best performance when r = 25 for AR database.
As a classic method for face recognition, Principle Component Analysis (PCA) is usually applied for matrix factorization. The result eigenface bases are arbitrary signs and resemble the distorted versions of whole faces. Similar to the PCA-based method, NMF does not provide any information for class discrimination but it finds the parts-based representation by matrix factoriza-tion. The primary objective of this paper is to present a framework for face recognition by combining Wavelet Transform (WT) and Basis-emphasized Non-negative Matrix Factorization (BNMF). The novel method, wBNMF, is
3.8. WAVELET DETERMINATION OF WBNMF
(a)
(b)
Figure 3.16: Comparison of wBNMF using different wavelet filters at various thresholds. Note that the most upper left point in Fig. 3.16(b) is produced by wBNMF using the Symlets wavelet filter of order 5.
3.8. WAVELET DETERMINATION OF WBNMF
method threshold FAR (%) FRR (%) EER (%) TSR (%) PCA 2.3325e + 006 12.0631 16.5000 14.2816 87.89 NMF 2.3246e + 006 12.9874 15.2500 14.1187 89.72 wBNMF 2.3024e + 006 12.4672 15.5000 13.9836 90.86
Table 3.3: Error measures of PCA, NMF and wBNMF using the Symlets wavelet filter of order 5.
applied to find an optimal subspace in which the ratio of the between-class scatter and the within-class scatter is maximized, and it is essential to classify the face images of an individual.
Given the AR face database, all these three algorithms decompose the original images into basis images and corresponding coefficients. This de-composition is computed with 25 basis components. The experimental result demonstrates the superiority of the proposed algorithm, wBNMF, in human face recognition compared with PCA and NMF. The experiment also shows that the wBNMF algorithm is less sensitive than the PCA algorithm to the selection of training and testing patterns.
3.8. WAVELET DETERMINATION OF WBNMF
(a)
(b)
Figure 3.17: Comparison of PCA, NMF and wBNMF approaches at various thresholds. Note that the most upper left point, the lowest EER, is produced by wBNMF.