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
) 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
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