Image Processing Reference
In-Depth Information
ˆ
i
=
( ) -
(5.76)
ˆ
where
φ
i
is the angle at the point ω
i
. Note that according to this equation, gradient
direction is independent of scale (in theory at least) and it changes in the same ratio as
rotation. We can constrain Equation 5.74 to consider only the points υ
(θ
) for which
i
- ( ) + = 0
ˆ
(5.77)
That is, a point spread function for a given edge point ω
i
is obtained by selecting a subset
of points in υ
(θ
) such that the edge direction at the image point rotated by ρ
equals the
gradient direction at the model point. For each point ω
i
and selected point in υ
(θ
) the point
spread function is defined by the HT mapping in Equation 5.74.
5.5.2
Polar definition
Equation 5.74 defines the mapping of the HT in Cartesian form. That is, it defines the votes
in the parameter space as a pair of co-ordinates (
x
,
y
). There is an alternative definition in
polar form. The polar implementation is more common than the Cartesian form Hecker
(1994) and Sonka (1994). The advantage of the polar form is that it is easy to implement
since changes in rotation and scale correspond to addition in the angle-magnitude
representation. However, ensuring that the polar vector has the correct direction incurs
more complexity.
Equation 5.74 can be written in a form that combines rotation and scale as
b
= ω (θ
) - γ (λ
, ρ
)
(5.78)
T
(λ
where γ
, ρ
) = [γ
x
(λ
, ρ
) γ
y
(λ
, ρ
)] and where the combined rotation and scale is
γ
x
(λ
, ρ
) = λ
(
x
(θ
) cos(ρ
) -
y
(θ
) sin(ρ
))
(5.79)
γ
y
(λ
, ρ
) = λ
(
x
(θ
) sin (ρ
) +
y
(θ
) cos(ρ
))
This combination of rotation and scale defines a vector, γ (λ
, ρ
), whose tangent angle and
magnitude are given by
λ ρ
(,
)
y
2
2
tan(
) =
)
=
r
(
λ ρ
, ) +
(
, )
(5.80)
x
y
λ ρ
(,
x
The main idea here is that if we know the values for
and
r
, then we can gather evidence
by considering Equation 5.78 in polar form. That is,
b
=
) -
re
α
(
(5.81)
Thus, we should focus on computing values for
and
r
. After some algebraic manipulation,
we have that
α
=
(
) +
r
=
(
)
(5.82)
where
y
x
()
()
-1
2
2
φ θ
(
) = tan
() =
x
() +
y
()
(5.83)
In this definition, we must include the constraint defined in Equation 5.77. That is, we
