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complex feature vector form that aims at obtaining a higher recognition effect

and a faster speed than serial fusion. According to the advantage above, a wavelet

decomposition feature based quaternion fusion with QEPD [1,2] for palmprint

or finger texture authentication was proposed. Also, another one with CSID [11]

migrated from face verification is implemented for palmprint or finger in this

paper. We propose a reasonable interpretation for the mathematical significance

of 4-feature parallel fusion by introducing quaternion prior to a discrimination by

such two distances respectively. Then we conduct a comparison on experimental

aspects for providing a conclusion which algorithm is better. The rest of paper

is organized as follows: section 2 introduces QEPD and CSID in turn. The next

section we propose our experimental comparison between such two distances for

handmetrics verification in details. After that, section 4 provides a conclusion.

2QEPDandCSID

2.1 Quaternion Euclidean Product Distance (QEPD)

We here make QEPD available as a discriminant distance. The relationship be-

tween QEP and quaternion modulus was discussed in [1,2]. Thus for an ar-

bitrary pixel corresponding to 4 separable wavelets decomposition sub-image,

consider two quaternions, 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

2
, where is a particular case of Quater-

nion Euclidean product. We can estimate the difference between the template

quaternion matrix and the tester one by QEPD
D
(
PP,|PQ|

=
Q
such that
PP
=

|P |

) as a discriminant

distance. In which

is an entry of the matrix with the modulus of QEP,

and the operator
D
is certain kind of distance, e.g.
L
1
norm,
L
∞
norm distance,

Euclidean distance etc. Such two matrices have the same size as the subimage

above. The reason use the modulus is that the multiplication of two different

quaternions is a quaternion so that it is impossible to compare directly with
PP
.

Because
PQ
is not a scalar as follows.

|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

(1)

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 (1) and that of template's modulus square
|P |

2
.

2.2 Cauchy-Schwartz Inequality Distance (CSID)

(1) Autocorrelation for signal processing.
Autocorrelation is used fre-

quently in signal processing for analyzing functions or series of values, such

as time domain signals [6]. We here discuss the case of discrete signal sequence,

defining

R
xx
(
j
)=

n

x
n
x
n−j

(2)

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