Shape representation using fourier transforms
View Patent ↗A method of approximating the inner or outer boundary of an iris comprises generating an approximate boundary representation ( 20 ) comprising a least squares approximation by a Fourier Series of a function of the angle (θ) about a fixed point (A) of the distance of measured points ( 10 ) on the boundary from the fixed point (A). More broadly, the method may be used to approximate the shape of any two-dimensional curve or figure.
1. A method of approximating an iris boundary, comprising the steps of:
acquiring an image of an eye, including an iris boundary;
noting a plurality of unequally spaced boundary points on the boundary;
selecting a fixed reference point;
generating an approximate boundary representation comprising a least squares approximation by a Fourier Series as a function of angle θ i about said fixed point of the distance of said boundary points in the image from said fixed point; and
the spaced boundary points being not all equally spaced in θ i , and in which said least-squares approximation minimizes the sum of squares of errors in the distances taken over the boundary points, the approximate boundary representation being generated by numerically solving the system of linear equations
PV=C
or a mathematical equivalent thereof, for the unknown matrix V, where:
V
=
[
a
0
a
1
⋮
a
N
b
1
⋮
b
N
]
C
=
[
C
0
C
1
⋮
C
N
C
N
+
1
⋮
C
2
N
]
with
C
0
=
∑
i
=
0
M
r
i
,
C
k
=
∑
i
=
0
M
r
i
cos
k
θ
i
and
C
N
+
k
=
∑
i
=
0
M
r
i
sin
k
θ
i
for
k
=
1
…
N
,
and
where
P
is
a
2
n
+
1
by
2
N
=
1
matrix
given
by
:
P
k
,
n
=
∑
i
=
1
M
cos
n
θ
i
cos
k
θ
i
for
k
=
0
…
N
and
n
=
0
…
N
(
Upper
Left
Matrix
)
P
k
,
N
+
n
=
∑
i
=
1
M
sin
n
θ
i
cos
k
θ
i
for
k
=
0
…
N
and
n
=
1
…
N
(
Upper
Right
Matrix
)
P
N
+
k
,
n
=
∑
i
=
1
M
cos
n
θ
i
sin
k
θ
i
for
k
=
1
…
N
and
n
=
0
…
N
(
Lower
Left
Matrix
)
P
N
+
k
,
N
+
n
=
∑
i
=
1
M
sin
n
θ
i
sin
k
θ
i
for
k
=
1
…
N
and
n
=
1
…
N
(
Lower
Right
Matrix
)
.
2. A method as claimed in claim 1 in which the boundary is the pupil/iris boundary.
3. A method as claimed in claim 1 in which the boundary is the outer iris boundary.
4. A method as claimed in claim 1 in which the fixed reference point is in the approximate center of the iris boundary.
5. A method as claimed in claim 1 in which, following the generating step, any boundary points which lie more than a selected distance from the boundary representation are excluded, and the generating step is then repeated.