Image Processing Reference
In-Depth Information
where
y
x
()
()
=
(5.93)
In order to define
φ
i
we can consider the tangent angle at the point ω
i
. By considering the
derivative of Equation 5.72 we have that
=
[- sin (
) cos(
)]
y
x
(
)
(5.94)
i
[cos (
) sin(
)]
(
)
Thus,
i
= tan( - )
(5.95)
where
y
x
()
()
=
(5.96)
By considering Equation 5.92 and Equation 5.95 we define
ˆ
ˆ
= + (5.97)
The important point in this definition is that the value of
k
is invariant to rotation. Thus, if
we use this value in combination with the tangent at a point we can have an invariant
characterisation. In order to see that
k
is invariant, we solve it for Equation 5.97. That is,
k
k
i
i
ˆ
ˆ
=
-
(5.98)
i
i
Thus,
k
= ξ
- ρ
- (φ
- ρ
)
(5.99)
That is,
(5.100)
That is, independent of rotation. The definition of
k
has a simple geometric interpretation
illustrated in Figure
5.23
(b).
In order to obtain an invariant GHT, it is necessary to know for each point ω
i
, the
corresponding point υ
k
= ξ
- φ
(θ
) and then compute the value of
φ
i
. Then evidence can be gathered
by the line in Equation 5.91. That is,
y
=
(
x
-
) +
(5.101)
0
i
0
xi
yi
In order to compute
φ
i
we can obtain
k
and then use Equation 5.100. In the standard
tabular form the value of
k
can be precomputed and stored as function of the angle β .
Code
5.12
illustrates the implementation to obtain the invariant R-table. This code is
based on Code
5.10
. The value of α is set to π /4 and each element of the table stores a
single value computed according to Equation 5.98. The more cumbersome part of the code
is to search for the point ω
j
. We search in two directions from ω
i
and we stop once an edge
point has been located. This search is performed by tracing a line. The trace is dependent
on the slope. When the slope is between -1 and +1 we then determine a value of
y
for each
value of
x
, otherwise we determine a value of
x
for each value of
y
.
Code
5.13
illustrates the evidence gathering process according to Equation 5.101. This