Image Processing Reference
In-Depth Information
x
1
=
a
x
cos(θ
1
)
x
2
=
a
x
cos(θ
2
)
x
(θ
)=
a
x
cos(θ
)
(5.61)
y
1
=
b
x
sin(θ
1
)
y
2
=
b
x
sin(θ
2
)
y
(θ
)=
b
x
sin(θ
)
The point (
x
(θ
),
y
(θ
)) is given by the intersection of the line in Equation 5.60 with the
ellipse. That is,
y
() -
() -
y
a
b
y
x
0
0
x
y
m
m
=
(5.62)
x
x
By substitution of the values of (
x
m
,
y
m
) defined as the average of the co-ordinates of the
points (
x
1
,
y
1
) and (
x
2
,
y
2
) in Equation 5.56, we have that
b
sin(
) +
b
sin(
)
a
b
y
1
y
2
x
y
tan(
) =
(5.63)
a
cos (
) +
a
cos (
)
x
1
x
2
1
2
Thus,
(5.64)
tan(
) = tan (
(
+
))
1
2
From this equation it is evident that the relationship in Equation 5.54 is also valid for
ellipses. Based on this result, the tangent angle of the second directional derivative can be
defined as
b
a
y
x
(
) =
tan(
)
(5.65)
By substitution in Equation 5.62 we have that
y
x
m
m
θ () =
(5.66)
This equation is valid when the ellipse is not translated. If the ellipse is translated then the
tangent of the angle can be written in terms of the points (
x
m
,
y
m
) and (
x
T
,
y
T
) as
y
-
-
y
T
m
θ () =
(5.67)
x
x
T
m
By considering that the point (
x
T
,
y
T
) is the intersection point of the tangent lines at (
x
1
,
y
1
)
and (
x
2
,
y
2
) we obtain
AC BD
ABC
+ 2
2 +
θ () =
(5.68)
A
=
y
1
-
y
2
B
=
x
1
-
x
2
where
(5.69)
C
=
φ
1
+
2
D
=
1
·
φ
2
and
φ
2
are the slope of the tangent to the points. Finally, by considering Equation 5.60,
the HT mapping for the centre parameter is defined as
1
,
AC BD
ABC
xx
+ 2
2 +
yy
=
+
( -
)
(5.70)
0
m
0
m
This equation can be used to gather evidence that is independent of rotation or scale. Once
the location is known, a 3D parameter space is needed to obtain the remaining parameters.
However, these parameters can also be computed independently using two 2D parameter
