Image Processing Reference
In-Depth Information
Chapter 5
Parametric Curve Estimation and
Reconstruction
5.1
Digital Conics in Canonical Form ............................... 160
5.2
Circles and Parabolas in Canonical Form ....................... 163
5.3
Estimation of Major and Minor Axes of an Ellipse in Canonical
Form ............................................................. 166
5.3.1
Reconstruction of Ellipse ................................ 166
5.3.2
The Reconstruction Algorithm .......................... 170
5.3.3
The Domain Theorem ................................... 171
5.4
Reconstruction of Hyperbola in Canonical Form ................ 172
5.5
A Restricted Class of Digitized Planar Curves .................. 175
5.5.1
Characterizing Properties of the Class .................. 176
5.5.2
One-Parameter Class .................................... 181
5.5.3
Two-Parameter Class .................................... 183
5.6
Summary ......................................................... 186
Exercises ......................................................... 186
Reconstruction of the original continuous curve from a given set of digital
points representing its digitization is an important problem in digital image
analysis. It is understood that digitization is a lossy transformation. There-
fore, reconstruction of the original curve from its digitization is, in general,
impossible. This loss of information opens up a number of questions, the most
important of them being: Is it possible to obtain the given digital image by
digitizing a particular kind of curve? In Chapter 3, we considered this prob-
lem for straight lines. In this chapter we investigate the problem under a more
general context in 2-D.
To pose the question formally, consider the class of planar curves alge-
braically by an equation f(x,y,P) = 0 where x and y are spatial variables,
P = (p
1
,p
2
,...,p
k
) is a set of control parameters and f is a function relating
x, y, and P. For example, f(x,y,m,c) may be y −mx−c where P = (m,c)
and k = 2. Let D
o
be the given set of digital points. Then the earlier question
can be restated as: Does there exist a vector P
1
∈ R
k
(R denotes the set of
real numbers) so that the digitization of f(x,y,P
1
) = 0 yields the given set
D
o
?
This problem is known as a reconstruction problem of a digitized image
159
