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The verification problem may be formally posed as follows: given an input featurevector XQ and a claimed identity I, determine if (I,XQ) belongs to 1 or 2, where1 indicates that the claim is true (a genuine user) and 2 indicates that the claimis false (an impostor). Typically, XQ is matched against XI , the biometric templatecorresponding to user I, to determine its category. Thus,

1 if S(XQ,XI) ,(I,XQ)

2 otherwise,

where S is the function that measures the similarity between XQ and XI, and is apredefined threshold. Therefore, every claimed identity is classified as 1 or 2 basedon the variables XQ, I, XI and , and the function S.

These sensors when embedded in compact systems like laptops,mouse, and cellular phones provide a small contact area (e.g., 0 .60.6 ina Veridicom sensor) for the fingertip and, therefore, sense only a limited portionof the fingerprint. This complicates the problem of matching impressions due2The earliest published work on automatic fingerprint matching and classification can be tracedback to the 1960s [26, 27, 28].

11to the lack of sufficient minutiae information. It is, therefore, essential to augmentminutiae information with alternate information available in fingerprintsin order to deal with the issues introduced by partial fingerprint images.

Image Filtering using Gabor FiltersA 2D Gabor filter can be thought of as a complex plane wave modulated by a 2DGaussian envelope. These filters optimally capture both local orientation and frequencyinformation1 and their development was motivated by observing the linearresponse of the receptive field in simple striate cortex cells. By tuning a Gabor filterto a specific frequency and direction, the local frequency and orientation informationcan be obtained. Thus, they are suited for extracting texture information from images.Daugman has successfully used these filters to extract salient features from thehuman iris [11].An even symmetric Gabor filter has the following general form in the spatialdomain:

-1 x

f2

y

f2

G,f(x, y) = exp + Cos(2fx),2 z

2y2

x_= xsin + ycos,y_= xcos ysin,

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where fis the frequency of the sinusoidal plane wave at an angle with the x-axis,and x and y are the standard deviations of the Gaussian envelope along the x andy axes, respectively.For extracting the response of the ridge at various orientations of the Gabor filter,the parameters (f, x, y, ) are set to the following values:

(i) The frequency, f, corresponds to the inter-ridge distance in fingerprint images.For the 300 300 (500 dpi) images obtained using the Veridicom sensor and resizedto 240 240 (see section 2.6.5), the average inter-ridge spacing is about 8 pixels.Hence, f= 18 = 0.125.(ii) The selection of the standard deviation values, x and y, involves a trade-off.Larger values are more robust to noise, but will not capture ridge information at afine level. Smaller values, on the other hand, are less robust to noise in the image,but capture ridge information very well. Based on empirical data [33], both thesevalues were set to 4, i.e., x = y = = 4.(iii) Eight different orientations are examined. These correspond to values of 0o,

22.5

o

, 45

o

, 67.5

o

, 90

o

,112.5

o

, 135

o

, 157.5o

(Figure 2.6).These parameters are fixed during the matching process, allowing for pre-storing26the Gabor filter representations in a lookup table referred to as the Gabor filter bank.This filter bank precalculates the Fourier representation of the Gabor filter for allorientations of interest. This formulation substantially improves the matching timein a one-to-many matching scheme.

(a) 0o b) 22.5o (c) 45o (d) 67.5o (e) 90o

(f) 112.5o (g) 135o (h) 157.5o

Figure 2.6: Gabor filters in spatial domain with eight different orientations used for

feature extraction. f= 0.125, x = y = = 4.