Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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(a) (b)<br />
Figure A.6: Frequency images (after filtering) <strong>of</strong> two input fingerprints shown in<br />
Fig. A.2.<br />
where φ is the orientation <strong>of</strong> the Gabor filter, f is the frequency <strong>of</strong> a sinusoidal plane<br />
wave, and δx and δy are the space constants <strong>of</strong> the Gaussian envelope along x and y<br />
axes, respectively. The modulation transfer function (MTF) <strong>of</strong> the Gabor filter can<br />
be represented as,<br />
H(u, v : φ, f) =<br />
�<br />
2πδxδyexp − 1<br />
�<br />
(uφ − u0)<br />
2<br />
2<br />
δ2 +<br />
u<br />
(vφ − v0) 2<br />
δ2 +<br />
��<br />
v<br />
�<br />
2πδxδyexp −<br />
(A.22)<br />
1<br />
�<br />
(uφ + u0)<br />
2<br />
2<br />
δ2 +<br />
u<br />
(vφ + v0) 2<br />
δ2 ��<br />
,<br />
v<br />
(A.23)<br />
uφ = u cos φ + v sin φ, (A.24)<br />
vφ = −u sin φ + v cos φ, (A.25)<br />
u0 =<br />
2π cos φ<br />
,<br />
f<br />
(A.26)<br />
v0 =<br />
2π sin φ<br />
.<br />
f<br />
(A.27)<br />
where δu = 1/2πδx and δv = 1/2πδy. To apply Gabor filters to an image, three<br />
parameters must be specified:<br />
• The frequency, f,<br />
• The filter orientation φ, and<br />
• The standard deviations <strong>of</strong> the Gaussian envelope, δx and δy.<br />
In this case, the frequency <strong>of</strong> the filter is given by local ridge frequency and the<br />
orientation by local ridge orientation. The selection <strong>of</strong> the values <strong>of</strong> δx and δy involves<br />
130