Image Processing Reference
In-Depth Information
The analysis for the parabola y
2
= 4a
o
x can be carried out in an analogous
manner. Here, x
i
= ⌈i
2
/4a
o
⌉,0 6 i 6 y
n
and y
i
= ⌊
(4a
o
i)⌋,0 6 i 6 n, where
the parabola is truncated by x = n. The bounds on a
o
can be formed as in
the case of a circle:
(y
i
/4i),max
i
(i
2
/4x
i
)) and
a
l
=
max(max
i
((y
i
+ 1)
2
/4i),min
i
(i
2
/4(x
i
a
u
=
min(min
i
−1))) and i > 1.
Clearly then, Theorem 5.2 follows.
Theorem 5.2. a
l
6 a < a
u
if and only if D(P(a)) = D
o
and Domain(D
o
) =
[a
l
,a
u
).
€
Example 5.3. (Fig. 5.3) Let a
o
= 4.3 and n = 10. So, D
o
X
=
(0,1,1,1,1,2,3,3,4,5,6,8,9,10) and D
Y
= (0,4,5,7,8,9,10,10,11,12,13).
We get a
l
= 4.225000 and a
u
= 4.321429. The grid point defining a
l
is (10,13)
and that defining a
u
is (7,11).
FIGURE 5.3: Digitization of a parabola with a
o
= 4.3 (Example 5.3).
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.
