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We can obtain M eigenvector and eigenvalue of covariance matrix C. Then, we
sort out R
RM
largest eigenvectors by the corresponding eigenvalues, denoted
as:
U
u
u
…u
R
N
R
.
u
is the eigenface, these eigenfaces are orthogonal to each
other. The image of a user can be transformed to the R-dimensional face space by
linear mapping:
Ω
Γ
(4)
3.3
Sine Transformation
Before going into details about our transformation, we introduce some notations
which are often used in this section.
─
X = {x
1
, x
2
, … , x
n
}: a biometric feature vector extracted from the face image of a
user.
─
Y = {y
1
, y
2
, … , y
n
}: an intermediate vector after applied sine transformation on X.
─
X
'
= {x
'
1
, x
'
2
, … , x
'
n
}: a final transformed vector which is used for fuzzy vault
encoding and decoding.
When the feature extraction step completes, the feature vector X of all users will be
transformed into an intermediate vector Y by the function (5):
sin (x
i
+ y
i
) = c (c is chosen randomly)
(5)
As we all know, sine function has period of 2
Π
rad. So, with each value of x
i
, we
will find exactly one value y
i
. But, given a value of y
i
, you cannot derive an exactly x
i
,
because there are many value of x
i
corresponding with that y
i
. In another words, the
sine function is a non-invertible transformation. You can also choose another periodic
function has the same characteristic with sine function (such as cosine function). In
this paper, we use the sine function to present our works.
The value of
is obtained by finding the minimum
0
such as:
sin
(6)
It means that the value of
lies between
0; 2Π
. The sine transformation is simp-
ly illustrated in the Fig. 2. For example, assume that c = 1, x
i
=
Π/3
, so we have y
i
=
2
Π/3
.
However, the fuzzy vault requires that the points in vault are disordered. So we
cannot use this
as x-coordinate values, because if so, it will eliminate the ordered
property of feature vector Y. This reason causes the increase of FAR (False Accept
Rate). To eliminate this risk, we apply a minor transformation to the vector Y for
preserving the order of the elements in fuzzy vault set. The result is the final trans-
formed vector
generated by the following rules:
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