Image Processing Reference
In-Depth Information
υ ′
(
)
(
θ
)
υ ″
(
)
(x
0
, y
0
)
Figure 5.16
Definition of the first and second directional derivatives for a circle
y
x
()
()
= -
1
tan(
() =
(5.45)
)
Angles will be denoted by using the symbol ^. That is,
ˆ
(
) = tan
-1
(
(
))
(5.46)
Similarly, for the tangent of the second directional derivative we have that,
y
x
()
()
= tan(
ˆ
-1
(5.47)
() =
) and
(
) = tan
(
(
))
By observing the definition of
(
), we have that
y
x
()
()
=
y
() -
() -
y
0
0
() =
(5.48)
x
x
This equation defines a straight line passing through the points (
x
(θ ),
y
(θ )) and (
x
0
,
y
0
) and
it is perhaps the most important relation in parameter space decomposition. The definition
of the line is more evident by rearranging terms. That is,
y
(θ ) = φ′′ (θ )(
x
(θ ) -
x
0
) +
y
0
(5.49)
This equation is independent of the radius parameter. Thus, it can be used to gather
evidence of the location of the shape in a 2D accumulator. The HT mapping is defined by
the dual form given by
y
0
= φ′′ (θ )(
x
0
-
x
(θ )) +
y
(θ ) (5.50)
That is, given an image point (
x
(θ ),
y
(θ )) and the value of φ′′ (θ ) we can generate a line of
votes in the 2D accumulator (
x
0
,
y
0
). Once the centre of the circle is known, then a 1D
accumulator can be used to locate the radius. The key aspect of the parameter space
decomposition is the method used to obtain the value of φ′′
(θ
) from image data. We will
consider two alternative ways. First, we will show that φ′′
(θ
) can be obtained by edge
