Image Processing Reference
In-Depth Information
Example 5.9. Let f : y −mx−c. Its PDSV = (-,+,-,-), i.e., a variant of
PDSV
1
in Table 5.2. Also we consider the segment from x = 0 to x = n. We
can develop a similar iterative refinement scheme using
m
l
= −1/n,m
u
= (n + 1)/n,c
l
= y
0
,c
u
= y
0
+ 1
as initial choices of the bounds of the parameters.
It is easy to verify that f satisfies all assumptions. Consequently, the anal-
ysis presented to introduce I R follows as a special case now.
5.6 Summary
In this chapter we addressed the reconstruction and the domain construc-
tion problems for digital conics in canonical form and a class of digitized
planar curves having one or two parameters. In Chapter 3, a new framework
of analysis has been introduced to determine the domain of a given DSLS.
Through this analysis, a general methodology for reconstruction, namely the
iterative refinement (I R)technique, has also been developed.
The I R technique is first applied to digital conics in canonical form. A
detailed analysis of such digital ellipses has been carried out to obtain the
smallest rectangle in the parametric space that encloses the domain. For the
sake of brevity, relevant results are only presented for digital hyperbola. Since
the method is iterative, heuristics have also been suggested to enhance the
convergence of the I R algorithm.
Next, the I R technique is developed as a unified methodology to solve the
reconstruction problem for a class of digital curves. It is shown that the domain
of the given digitization can be exactly formulated for separable, monotone
curves with one unknown parameter. In case of two parameters, the domain
can be numerically computed if the curve satisfies another additional property.
Exercises
1. Prove Lemma 5.1
2. Prove Theorem 5.8.
3. Prove Theorem 5.10.
4. Prove Theorem 5.15.
