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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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