Image Processing Reference
In-Depth Information
parameter and it only addresses a particular point in the locus of the ellipse (just as it was
used to trace the circle in Equation 5.32). However, one parameter is redundant since it can
be computed by considering the orthogonality (independence) of the axes of the ellipse
(the product
a
x
b
x
+
a
y
b
y
= 0 which is one of the known properties of an ellipse). Thus, an
ellipse is defined by its centre (
a
0
,
b
0
) and three of the axis parameters (
a
x
,
b
x
,
a
y
,
b
y
). This
gives five parameters which is intuitively correct since an ellipse is defined by its centre
(two parameters), it size along both axes (two more parameters) and its rotation (one
parameter). In total this states that five parameters describe an ellipse, so our three axis
parameters must jointly describe size and rotation. In fact, the axis parameters can be
related to the orientation and the length along the axes by
a
a
y
x
2
2
2
2
tan(
) =
=
a
a
+
a
=
b
b
+
b
(5.36)
x
y
x
y
where (
a
,
b
) are the axes of the ellipse, as illustrated in Figure
5.13
.
y
a
b
a
y
b
y
a
x
b
x
x
Figure 5.13
Definition of ellipse axes
In a similar way to Equation 5.31, Equation 5.35 can be used to generate the mapping
function in the HT. In this case, the location of the centre of the ellipse is given by
a
0
=
x
-
a
x
cos(θ
) +
b
x
sin(θ
)
(5.37)
b
0
=
y
-
a
y
cos(θ ) +
b
y
sin(θ )
The location is dependent on three parameters, thus the mapping defines the trace of a
hypersurface in a 5D space. This space can be very large. For example, if there are 100
possible values for each of the five parameters, then the 5D accumulator space contains
10
10
values. This is 10 GB of storage, which is of course tiny nowadays (at least, when
someone else pays!). Accordingly there has been much interest in ellipse detection techniques
which use much less space and operate much faster than direct implementation of Equation
5.37.
