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as autocorrelation for
x
n
. This term has many properties, e.g.symmetry,
R
xx
(
−τ
)
−τ
)=
R
xx
(
τ
)when
x
(
n
)isa
complex one. In addition, a wavelet feature based quaternion representation for
parallel fusion is viewed as a real signal sequence.
=
R
xx
(
τ
)when
x
(
n
) is a real sequence, and
R
xx
(
(2) Cauchy-Schwartz inequality for autocorrelation.
According to the sec-
tion above, another significant property of autocorrelation is Cauchy-Schwartz
inequality [6]. Given two arbitrary vectors
a, b
their inner product space is depict
as
|a, b|
2
≤a, a·b, b
and the inequality is expressed as
|a, b| ≤ a·b
,
where
·
is inner product, and
·
denotes norm of the vectors. Such consequence
is utilized for property of autocorrelation as follows
|R
xx
(
τ
)
|≤R
xx
(0)
(3)
The equation (2) is used for construction of CSID distance, by which discrimi-
nate genuine and impostor.
(3) Cauchy-Schwartz inequality distance.
Consider two quaternions, which
the former is
P
=
a
+
bi
+
cj
+
dk
and the latter from the tester, in which
Q
=
t
+
xi
+
yj
+
zk
. If template and tester one belong to the same person,
P
will be more similar with
Q
than that from different persons, i.e.
,
modulus of
P − Q
, is smaller than that from different persons. Thus we obtain
|P − Q|
Q→P
|P − Q|
lim
=0
(4)
Thus we view
P
and
Q
as real signal sequences in sense of discrete-time signals, i.e.
transform quaternion
P
and
Q
into forms of
in
which
x
(
p
)and
x
(
q
) are discrete sequences at time
p
and
q
respectively. According
to the autocorrelation discussed above, the equation (3) can be rewritten as
R
xx
(
τ
)=
n
{x
(
p
)
|a, b, c, d}
and
{x
(
q
)
|t, x, y, z}
x
p
x
q
(5)
Where
p
=
q − τ
,time
q
can be viewed as a time delay to
p
.Nowevolvethe
equation (3)
R
xx
(0)
−|R
xx
(
τ
)
|≥
0
(6)
Replace (5) with (6), we obtain
≥
D
pixel
=
n
x
2
p
−
x
p
x
q
0
(7)
n
It is easily found that the equation above is larger than 0. To this end, it is
accommodate to set this as the distance of the pixel for 4 sub-images with a
reasonable physical significance. For all pixels of such sub-images, we define
Cauchy-Schwartz inequality distance
k
k
x
2
D
CSID
=
D
pixel
=
p
−
x
p
x
q
≥
0
(8)
n
n
i
=1
i
=1
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