Image Processing Reference
In-Depth Information
formulas [41]:
∗
∗
Domain(D
o
)
=
[b
l
(a),b
l
(a)); or
a
l
<a<a
u
∗
l
(a),a
∗
u
(a)).
Domain(D
o
)
=
[a
b
l
<b<b
u
€
Corollary 5.1. R
ul
is the smallest rectangle in the (a,b)-space such that
Domain(D
o
) ⊆R
ul
.
€
In view of the above corollary, we immediately get an algorithm from itera-
tive refinement to test whether a given digital set can at all be the digitization
of an ellipse in canonical form or not. Mathematically it implies that we need
to check whether a Domain(D
o
) is an empty set or not.
Corollary 5.2. Domain(D
o
) = {} if and only if a
l
> a
u
or b
l
> b
u
or both.
That is, if the digital set D
o
is not a valid digitization of an ellipse, then there
exists a k such that a
l
> a
u
,b
l
> b
u
.
€
Corollary 5.3. Domain(D
o
) is axially convex and connected.
€
It may be noted that the above theorem provides the necessary algebraic
characterization of the domain Domain(D
o
). But it does not help in ana-
lytically finding the boundaries of the domain. However, it helps to numeri-
cally compute the domain (up to any desired accuracy), because we find that
for a given a, (a
l
< a < a
u
) there exists a single interval of b values, i.e.,
[b
∗
l
(a),b
∗
u
(a)) corresponding to the domain. Immediately, we also conclude
that (a,b
∗
l
(a)) and (a,b
∗
u
(a)) are points on the boundary of the domain. By
computing these points for various values of a, we can determine the domain
as we show in the following example. We can even numerically integrate some
quantity like the property estimator error over this domain to derive further
results. It may also be noted that Theorem 5.2 renders the reconstruction al-
gorithm redundant. It is presented for the sake of completeness in an attempt
to maintain coherence with other forms of analysis such as [157].
Example 5.5. For the ellipse in Example 5.1 (Fig. 5.1), we estimate
the domain of digitization using the above theorem. We have a
o
=
12.5, b
o
= 10.5 and D
o
X
= {12,12,12,11,11,10,10,9,8,6,3}, D
Y
=
{10,10,10,10,9,9,9,8,8,7,6,4,2}.The domain is plotted in Fig. 5.5.
5.4 Reconstruction of Hyperbola in Canonical Form
The analysis of a hyperbola in canonical form is similar to that of an
ellipse, so we present only an outline of the analysis. First, let us define the
