Master Thesis - Department of Computer Science

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