Digital Geometry in Image Processing

164 Digital Geometry in Image Processing

FIGURE 5.2: Digitization of a circle with r

o

= 9.7, where (9,3) defines r

l

and (9,4) defines r

u

(Example 5.2).

Reprinted from CVGIP: Graphical Models and Image Processing, 54(5)(1992), S. Chattopadhyay et al., Parameter Es-

timation and Reconstruction of Digital Conics in Normal Positions, 385–395, Copyright (1992), with permission

from Elsevier.

It is immediately evident that r

l

6 r

o

< r

u

.

Theorem 5.1. r

l

6 r < r

u

if and only if D(C(r)) = D

o

. In other words,

Domain(D

o

) = [r

l

, r

u

).

Proof: The proof follows from the definitions of r

l

and r

u

. 

Example 5.2. Let r

o

= 9.7. So, D

o

X

= D

o

Y

= {9, 9, 9, 9, 8, 8, 7, 6, 5, 3}.

Clearly ...