Image Processing Reference
In-Depth Information
5.3 Estimation of Major and Minor Axes of an Ellipse
in Canonical Form
In contrast to the one-parameter characterization of circles and parabolas
we need two parameters, namely a and b, to specify an ellipse or hyperbola in
canonical form. A straightforward analysis of the discrete point inequalities
cannot determine the possible domain of a and b values in this case. However,
we show that the iterative refinement scheme introduced in Chapter 3 (refer
to Section 3.2) can be appropriately modified to first find a tight rectangular
bound for the domain and then to compute the domain itself. In the ensuing
discussions, we first present the iterative refinement scheme for the bounds,
then prove their various properties, discuss the reconstruction algorithm, and
finally present the domain theorem [41].
Let us highlight a general property of two intersecting ellipses E
1
and E
2
.
Lemma 5.2. Let E
1
= E(a
1
,b
1
) and E
2
= E(a
2
,b
2
) be two ellipses such that
a
1
> a
2
and b
1
< b
2
. If E
1
∩E
2
= (x
′
,y
′
), then
′
′
(A)
At every x,x > x
(6 x
) E
1
lies above (below) E
2
and
′
′
(B)
At every y,y > y
(6 y
) E
1
lies to the left (right) of E
2
where ∩ denotes the point of intersection of two continuous ellipses in the first
quadrant.
Proof: The proof is immediate from the equation of the ellipse.
€
5.3.1 Reconstruction of Ellipse
We present the main result in the next theorem [41].
Theorem 5.3. Suppose that the upper and lower bounds of a
o
and b
o
are
defined by the following iterative algorithm where k > 0:
b
l
= ⌊b
o
⌋ = y
0
,b
k+1
(1−i
2
/(a
u
)
2
)),
= max
i
(y
i
/
l
b
u
= ⌊b
o
⌋+ 1 = y
0
+ 1,b
k+1
(1−i
2
/(a
l
)
2
)),
= min
i
((y
i
+ 1)/
u
(1−i
2
/(b
u
)
2
)),
a
l
= ⌊a
o
⌋ = x
0
,a
k+1
= max
i
(x
i
/
l
a
u
= ⌊a
o
⌋+ 1 = x
0
+ 1,a
k+1
(1−i
2
/(b
l
)
2
)),
= min
i
((x
i
+ 1)/
u
then there exist b
l
, b
u
, a
l
and a
u
such that
k→∞
b
l
= b
l
, lim
k→∞
b
u
= b
u
, lim
k→∞
a
l
= a
l
, lim
k→∞
a
u
= a
u
, and
(A)
lim
