Image Processing Reference
In-Depth Information
backmapping
(Gerig, 1986) can be used to determine exactly which edge points contributed
to a particular peak. Backmapping is an
inverse
mapping from the accumulator space to the
edge data and can allow for shape analysis of the image by removal of the edge points
which contributed to particular peaks, and then by re-accumulation using the HT. Note that
the computational cost of the HT depends on the number of edge points (
n
e
) and the length
of the lines formed in the parameter space (
l
), giving a computational cost of
O
(
n
e
l
). This
is considerably less than that for template matching, given earlier as
O
(
n
2
m
2
).
One way to avoid the problems of the
Cartesian parameterisation in the HT is to base the
mapping function on an alternative parameterisation. One of the most proven techniques is
called the foot-of-normal parameterisation. This parameterises a line by considering a
point (
x
,
y
) as a function of an angle normal to the line, passing through the origin of the
image. This gives a form of the HT for lines known as the
polar HT for lines
(Duda, 1972)
.
The point where this line intersects the line in the image is given by
ρ =
x
cos(θ ) +
y
sin(θ ) (5.28)
where θ
is the angle of the line normal to the line in an image and ρ is the length between
the origin and the point where the lines intersect, as illustrated in Figure
5.8
.
x
y
c
Figure 5.8
Polar consideration of a line
By recalling that two lines are perpendicular if the product of their slopes is -1, and by
considering the geometry of the arrangement in Figure
5.8
, we obtain
1
tan( )
c
=
sin(
m
= -
(5.29)
By substitution in Equation 5.24 we obtain the polar form, Equation 5.28. This provides a
different mapping function: votes are now cast in a sinusoidal manner, in a 2D accumulator
