Image Processing Reference
In-Depth Information
direction information. Second, how it can be obtained from the information of a pair of
points.
In order to obtain φ′′ (θ ), we can use the definitions in Equation 5.46 and Equation 5.47.
According to these equations, the tangents φ′′
(θ
) and φ′
(θ
) are perpendicular. Thus,
1
()
() = -
(5.51)
Thus, the HT mapping in Equation 5.50 can be written in terms of gradient direction
(
)
as
x
() -
()
x
0
yy
= ( ) +
(5.52)
0
This equation has a simple geometric interpretation illustrated in Figure
5.17
(a). We can
see that the line of votes passes through the points (
x
(θ
),
y
(θ
)) and (
x
0
,
y
0
). The slope of the
line is perpendicular to the direction of gradient direction.
(x(
θ
1
), y(
θ
1
))
(x(
θ
), y(
θ
))
(x(
θ
), y(
θ
))
(x
m
, y
m
)
ˆ
θ
()
(x
0
, y
0
)
(x(
θ
2
), y(
θ
2
))
ˆ
θ
()
ˆ
ˆ
θ
()
θ
()
(a) Relationship between angles
(b) Two point angle definition
Figure 5.17
Geometry of the angle of the first and second directional derivatives
An alternative decomposition can be obtained by considering the geometry shown in
Figure
5.17
(b). In the figure we can see that if we take a pair of points (
x
1
,
y
1
) and (
x
2
,
y
2
),
where
x
i
=
x
(θ
i
) then the line that passes through the points has the same slope as the line
at a point (
x
(θ
),
y
(θ
)). Accordingly,
yy
xx
-
-
2
1
() =
(5.53)
2
1
1
2
where
= (
+
)
(5.54)
1
2
Based on Equation 5.53 we have that
xx
yy
-
-
2
1
() = -
(5.55)
2
1
The problem with using a pair of points is that by Equation 5.53 we cannot determine the
location of the point (
x
(θ ),
y
(θ )). Fortunately, the voting line also passes through the
midpoint of the line between the two selected points. Let us define this point as
