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features of the same sample have not identical number, e.g. m>n . If assign 0
as b l n +1 ,b l n +2 , ···,b l m , this method will be lack of physical significance. Here our
algorithms enable to avoid this problem because each sub-image pixel has four
coecients for fusion at the same time.
3.3
Quaternion Euclidean Product Distance
As mentioned section above, we here make QEPD available as matching score.
The relationship between QEP and quaternion modulus has been discussed in the
section 2.1, thus consider two quaternions, for an arbitrary pixel corresponding
to 4 separable wavelets decomposition coecients sub-image, which the former
is from this pixel as the template, P = a + bi + cj + dk and the latter is from
the tester, in which Q = t + xi + yj + zk . Ideally, if P = Q such that PP =
2 ,
where is a particular case of QEP. We can estimate the difference between the
template quaternion matrix and the tester one by QEPD D ( PP,|PQ|
|P |
)asa
discriminant distance. Notice that the template quaternion matrix and tester
matrix refer to a matrix stores the value of PP corresponding to each pixel
of the subimage and a matrix with all of
|PQ|
corresponding to these pixels
respectively. Such two matrices have the same size as the subimage above. In
which
is the modulus of Quaternion Euclidean product, and the operator D
is certain kind of distance, e.g. L 1 norm, L norm distance, Euclidean distance
etc. In our scheme, Euclidean distance is chosen as the operator D . The reason
use the modulus of PQ is that the multiplication of two different quaternions
is a quaternion (equation (9)), combined (3) and (6), so that it is impossible to
compare directly with PP . From the equation, PQ is not a scalar.
|PQ|
PQ =( a − bi − cj − dk )( t + xi + yj + zk )
=( at + bx + cy + dz )+( ax − bt − cz + dy ) i
+( ay + bz − ct − dx ) j +( az − by + cx − dt ) k
(9)
Therefore, the similarity between template and tester QEPD D ( PP,|PQ|
)is
obtained by Euclidean distance between the matrix of the absolute value of the
equation (9) and that of template's modulus square
2 .
|P |
3.4
QEPD Matching
We make 2-D wavelets decomposition feature as parallel fusion discussed in the
subsection 3.2. After fusion, we employ QEPD for matching. Suppose A and B
as two finger texture feature quaternion vectors for each pixel of the subimage,
A =
where a, b, c, d are wavelets decomposition coecients
respectively. The same is true like B =
{a + bi + cj + dk}
{t + xi + yj + zk}
. According to QEPD
D ( PP,|PQ|
) , a matching score table (Table 1) is listed from the following Finger
texture ROI image (Fig.4):
From the table 1, QEPD matching scores are calculated by each two samples.
The most intra-class scores of this QEPD matching score estimated in our ex-
periment, usually span from 0 to 0.32, and majority of inter-class scores from              Search WWH ::

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