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entire processing, through checking time cost of each procedure of finger texture
QEPD.
The rest of paper is organized as follows: section 2 introduces basic issues
of quaternion, concept of wavelet decomposition and parallel fusion. Then we
propose our scheme for finger texture verification in the section 3. After that,
section 4 shows that experimental results by the approach. Finally, we perform
a conclusion.
2 Related Work
2.1 Quaternion
The quaternion is a non-commutative extension of complex numbers, which
first described by the Irish mathematician Sir William Rowan Hamilton in 1843
and applied to mechanics in three-dimensional space [5,9]. Quaternion has one
real part and three imaginary parts, not only one like complex number. Let
P
=
,where
a, b, c, d
arerealnumbers,and
i, j, k
are imaginary identities, respectively. According to the definition, quaternion
has basic properties of
{a
+
bi
+
cj
+
dk|a, b, c, d ∈ R}
i
2
=
j
2
=
k
2
=
ijk
=
−
1
,
ij
=
k, ji
=
−k, jk
=
−i,
(1)
kj
=
−i, ki
=
j, ik
=
−j
Also, quaternion has a lot of similarities as complex number, e.g.
Conjugate
P
=
a − bi − cj − dk
(2)
Modulus
=
PP
=
a
2
+
b
2
+
c
2
+
d
2
|P |
≥
0
(3)
Q
=
is another quaternion, and we discuss quater-
nion arithmetic operations in the following:
Equality
{t
+
xi
+
yj
+
zk|t, x, y, z ∈ R}
P
=
Q ⇔ a
=
t, b
=
x, c
=
y, d
=
z
(4)
Addition
P
+
Q
=(
a
+
t
)+(
b
+
x
)
i
+(
c
+
y
)
j
+(
d
+
z
)
k
(5)
Multiplication
P × Q
=(
at − bx − cy − dz
)+(
bt
+
ax
+
cz − dy
)
i
+(
ct
+
ay
+
dx − bz
)
j
+(
dt
+
az
+
by − cx
)
k
(6)
Inner Product
<P,Q>
=
at
+
bx
+
cy
+
dz
(7)
And Quaternion Euclidean Product (QEP) is defined as:
QEP
=
PQ
(8)
2
Consider that when
P
=
Q
,QEPisequalto
PP
=
|P |
, i.e. the square of
modulus.
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