Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
Master Thesis - Department of Computer Science
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Q = �<br />
(q 1 x, q 1 y, β 1 ), ....., (q Q x , q Q y , β Q ) �<br />
. (A.35)<br />
where |P| = P , |Q| = Q, and (p i x , pi y , αi ) are the three features associated with the<br />
i th minutiae in set P. The basic assumption made here is that the second point set<br />
Q is a rotated, scaled, and translated version <strong>of</strong> the first set P, where points may be<br />
shifted by random noise, some points may be added and some points deleted. The<br />
task <strong>of</strong> fingerprint registration is to recover this unknown transformation. Since we<br />
do not know whether the two fingerprints are the same or not (i.e., images <strong>of</strong> the same<br />
finger), we attempt to find the ’best’ transformation in the sense that when applying<br />
the transformation to the minutiae points <strong>of</strong> the set P, as many <strong>of</strong> these points as<br />
possible overlap with the minutiae points from the set Q. Two overlapping points are<br />
considered as a match only if they have the same direction. There may be minutiae<br />
points in either set that do not match with any point in the other set.<br />
The matching score is computed for each transformation after discretizing the set<br />
<strong>of</strong> all allowed transformations. Consider a transformation Fs,θ,∆x,∆y : R 2 → R 2 , given<br />
by,<br />
⎛<br />
⎜<br />
Fs,θ,∆x,∆y ⎝ x<br />
⎞ ⎛<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎠ = s ⎝<br />
y<br />
cosθ<br />
−sinθ<br />
sinθ ⎟ ⎜<br />
⎠ ⎝<br />
cosθ<br />
x<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎠ + ⎝<br />
y<br />
∆x<br />
⎞<br />
⎟<br />
⎠ ,<br />
∆y<br />
(A.36)<br />
where s, θ, and (∆x, ∆y) are the scale, rotation, and shift parameters, respectively.<br />
The space <strong>of</strong> transformations consists <strong>of</strong> quadruples (s, θ, ∆x, ∆y), where each pa-<br />
rameter is discretized into a finite set <strong>of</strong> values:<br />
s ∈ s1, .., sK , θ ∈ θ1, ..., θL,<br />
∆x ∈ ∆x1, ..., ∆xM and ∆y ∈ ∆y1, ..., ∆yN<br />
Matching scores for the transformations are collected in the accumulator array A,<br />
where the entry A(k, l, m, n) counts the evidence for the transformation Fsk,θl,∆xm,∆yn.<br />
The array A is filled as follows: For each pair (p, q), where p = (pi x , piy ) is a point in<br />
the set P and q = (q j x, q j y) is a point in the set Q, find all possible transformations that<br />
map p to q and increment the evidence for these evidence for these transformations in<br />
array A. For every pair <strong>of</strong> values, (sk, θl), there is exactly one shift vector (∆x, ∆y) t<br />
such that Fsk,θl,∆x,∆y(p) = q, and it can be obtained as,<br />
⎛<br />
⎜<br />
⎝ ∆x<br />
⎞ ⎛<br />
⎟ ⎜<br />
⎠ = q − sk ⎝<br />
∆y<br />
cosθl<br />
⎞<br />
−sinθl<br />
sinθl ⎟<br />
⎠ p.<br />
cosθl<br />
(A.37)<br />
140