Image Processing Reference
In-Depth Information
D
o
with respect to a given function f. A solution to this problem computes
at least one P such that digitization of f(x,y,P) = 0 produces D
o
.
As a generalization of the reconstruction problem, we may want to find
out the set of all P ∈ R
k
so that the digitization of f(x,y,P) = 0 produces
D
o
. The solution to this problem, known as a domain construction problem,
is a region S in k-dimensional space such that
∀P ∈S,D(f(x,y,P)) = D
o
where D represents the digitization function.
We may view the digitization procedure D as a transformation of a curve
f(x,y,P
o
) = 0 to the corresponding digitized data D
o
. Then the domain
construction problem is one of finding an inverse mapping from digitized data
to the specification of the curve. Formally, if D(f(x,y,P
o
)) = D
o
, then the
domain of digitization is defined as
Domain(D
o
,f) = {P|P ∈ R
k
and D(f(x,y,P)) = D
o
}.
Note that due to the intrinsic nature of this one-to-many inverse mappings,
we cannot compute the exact original of a digital image and have to be content
with specifying one or all of its possible original(s). It is in this view we say
that a solution to the domain construction problem involves the estimation of
the actual parameters (though the term estimation occurs in statistics with
a different connotation).
The problems of reconstruction and domain construction for a straight
line segment have been discussed in detail in Chapter 3. Conics are next to
straight lines in order of complexity. In this chapter, we discuss algorithms to
solve the reconstruction and domain construction problem of canonical digital
conics. We go further and delve into designing algorithms to solve the domain
construction problem for a class of digitized curves including digital circles.
5.1 Digital Conics in Canonical Form
A general conic is represented by the equation ax
2
+by
2
+cxy+dx+fy+g =
0. Therefore, six parameters are needed to characterize such conics. As the
number of parameters is large, we restrict our attention to an important subset
of general conics, which are in canonical form.
Definition 5.1. A conic is said to be in canonical form if its center lies on
the origin of the coordinate system and if its axes are parallel to the coordinate
axes.
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In canonical form, circles and parabolas are represented by one unknown
parameter, while ellipses and hyperbolas are characterized by two parameters.
