Image Processing Reference
In-Depth Information
factors, the results provided by the GHT can become less reliable. A solution is to use an
analytic form instead of a table (Aguado, 1998). This avoids discretisation errors and
makes the technique more reliable. This also allows the extension to affine or other
transformations. However, this technique requires solving for the point υ (θ ) in an analytic
way, increasing the computational load. A solution is to reduce the number of points by
considering characteristic points defined as points of high curvature. However, this still
requires the use of a four-dimensional accumulator. An alternative to reduce this computational
load is to include the concept of invariance in the GHT mapping.
5.5.4
Invariant GHT
The problem with the GHT (and other extensions of the HT) is that they are very general.
That is, the HT gathers evidence for a single point in the image. However, a point on its
own provides little information. Thus, it is necessary to consider a large parameter space
to cover all the potential shapes defined by a given image point. The GHT improves
evidence gathering by considering a point and its gradient direction. However, since gradient
direction changes with rotation, then the evidence gathering is improved in terms of noise
handling, but little is done about computational complexity.
In order to reduce computational complexity of the GHT, we can consider replacing the
gradient direction by another feature. That is, by a feature that is not affected by
rotation
.
Let us explain this idea in more detail. The main aim of the constraint in Equation (5.77),
is to include gradient direction to reduce the number of votes in the accumulator by
identifying a point υ
(θ
). Once this point is known, then we obtain the displacement vector
γ (λ
, ρ
). However, for each value of rotation, we have a different point in υ
(θ
). Now let us
replace that constraint in Equation 5.76 by a constraint of the form
Q
(ω
i
) =
Q
(υ (θ )) (5.86)
The function
Q
is said to be invariant and it computes a feature at the point. This feature
can be, for example, the colour of the point, or any other property that does not change in
the model and in the image. By considering Equation 5.86, we have that Equation 5.77 is
redefined as
Q
(ω
i
) -
Q
(υ (θ )) = 0 (5.87)
That is, instead of searching for a point with the same gradient direction, we will search for
the point with the same invariant feature. The advantage is that this feature will not change
with rotation or scale, so we only require a 2D space to locate the shape. The definition of
Q
depends on the application and the type of transformation. The most general invariant
properties can be obtained by considering geometric definitions. In the case of rotation and
scale changes (i.e. similarity transformations) the fundamental invariant property is given
by the concept of angle. An angle is defined by three points and its value remains unchanged
when it is rotated and scaled. Thus, if we associate to each edge point ω
i
a set of other two
points {ω
j
, ω
T
} then we can compute a geometric feature that is invariant to similarity
transformations. That is,
XY
-
+
XY
ji
i j
Q
() =
(5.88)
i
XX
YY
i
j
i
j
where
X
k
=
ω
k
-
ω
T
,
Y
k
=
ω
k
-
T
. Equation 5.88 defines the tangent of the angle at the point
