Image Processing Reference
In-Depth Information
4
1
3
2
0.8
1
0.6
0
0.4
−1
0.2
−2
−3
0
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0.4
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
0.85
0.9
Fig. 10.32. The same as in Fig. 10.31 but the graphs on the
left
represent angle estimations
on the line joining point 5 to the center (green) and on the line joining point 6 to the center
(blue) marked in Fig. 10.15. The graphs on the right represent the corresponding
|I
20
|
(solid)
and
I
11
(dashed), respectively. The measurements on the noisy line are at the bottom.
through all discrete
φ
values of the grid
A
established in step 3. For each such
φ
k
, one
b
is obtained, which is rounded off towards the closest discrete
b
value
of the grid cells. The final
A
(
φ
m
,b
m
) obtained after the voting procedure is the
Hough transform for lines.
5. A long line causes a high peak in the (
φ, b
) plane, because every long line in
the (
x, y
)-plane generates a point in the (
φ, b
) plane. Accordingly, the position
of each such peak yields a specific (
φ, b
) parameter that represents an infinitely
long line in the (
x, y
)-plane.
The Hough transform procedure above can be simplified further by voting only for
one parameter cell (
φ, b
) per edge point (
x, y
) in step 4, the most likely direction
φ
k
≈
θ
(
x, y
) and its corresponding
b
value closest to a grid node. The robustness can
be improved by smoothing the accumulator before finding the peaks. Accordingly,
the votes cast for the line (
φ
j
,b
j
) are
A
(
φ
j
,b
j
)=
l
δ
[
θ
(
x
j
+
x
l
,y
j
+
y
l
)
−
θ
(
x
j
,y
j
)]
(10.95)
where (
x
l
,y
l
) are the edge points along the line given
φ
j
,b
j
having the direction
θ
(
x
j
,y
j
). The sum represents a filtering that counts the occurrence of the edges
encountered along the line. The occurrence is w.r.t. edges
and
edge directions, i.e.,
if there is a directional conflict the value of the cast vote is zero.
Nonuniqueness of Line Directions
In step 1 we have a model of a line for every fixed pair of parameters of (
ϕ, b
).
However, the model parameters and the lines do not uniquely correspond to each
other because by substituting (
φ
+
π,
−
b
) in Eq. (10.91), one sees that this parameter