Image Processing Reference
In-Depth Information
Domain(D
o
) = {(a,b)|D(E(a,b)) = D
o
}.
€
Domains and digitizations of other conics are defined analogously.
5.2 Circles and Parabolas in Canonical Form
In this section, we study digitized images of circles and parabolas in canon-
ical form. We elucidate the method to construct the domain of digital circles
while a similar method for domain construction of digitized parabolas is easy
to get. Unless otherwise stated, circles and parabolas are always assumed to
be in canonical form.
The restrictions of a conic in canonical form allows us to characterize a
circle, say C(r), or a parabola, say P(a), by one parameter only, where r is
the radius of the circle and a is the semi-latus rectum of the parabola. Given
a digitization D
o
of a circle, we present a simple algorithm to determine the
domain Domain(D
o
) of the possible radius values of all continuous circles,
which can be quantized to D
o
.
Given D
o
, it is easy to separate D
o
X
and D
Y
.
Mathematically,
D
o
X
(r
o
−i
2
)⌋}, and
= {(x
i
,i) : 0 6 i 6
⌊r
o
⌋ and x
i
= ⌊
D
Y
(r
o
−i
2
)⌋}.
= {(i,y
i
) : 0 6 i 6
⌊r
o
⌋ and y
i
= ⌊
From D
o
X
,
(r
o
−i
2
)−1 < x
i
6
(r
o
−i
2
),0 6 i 6
⌊r
o
⌋.
(x
i
+ i
2
) 6 r
o
<
((x
i
+ 1)
2
+ i
2
).
Rearranging terms,
((y
i
+ 1)
2
+ i
2
),0 6 i 6 ⌊r
o
⌋.
Combining the above inequalities we get two bounds on r
o
, namely r
l
and
r
u
, as follows.
Similarly,
(y
i
+ i
2
) 6 r
o
<
(x
i
+ i
2
),
(y
i
+ i
2
))), and
r
l
=
max
i
(max(
((x
i
+ 1)
2
+ i
2
),
((y
i
+ 1)
2
+ i
2
))).
r
u
=
min
i
(min(
From the symmetry of the circle, D
o
X
= D
Y
in the list notation.
Hence,
(x
i
+ i
2
)) = max
i
(y
i
+ i
2
)), and
r
l
=
max
i
(
(
((x
i
+ 1)
2
+ i
2
)) = min
i
((y
i
+ 1)
2
+ i
2
)).
r
u
=
min
i
(
(
