Image Processing Reference
In-Depth Information
digitization of the hyperbola H
o
: x
2
/a
2
− y
2
/b
2
= 1. As in the case of an
ellipse, we treat the OBQ image of H
o
as the union of the following two sets
of digital points, namely D
X
and D
Y
.
D
o
X
= {(x
i
,i) : 0 6 i 6 y
n
and x
i
= ⌈a
o
(1 + i
2
/b
o
)⌉} and
D
Y
= {(i,y
i
) : ⌈a
o
⌉
(i
2
/a
o
6 i 6 n and y
i
= ⌊b
o
−1)⌋}.
The iterative refinement equations for computing tight bounds of a
o
and
b
o
are given in the following theorem.
Theorem 5.8. Let the upper and lower bounds of a
o
and b
o
are defined by
the following iterative algorithm where k > 0:
a
l
= x
0
−1;a
k+1
l
(1 + i
2
/(b
l
)
2
)),
= max
i
((x
i
−1)/
((r/(r−1)
2
−1) where r = x
0
,
b
l
= y
r
/
b
k+1
l
(i
2
/(a
l
)
2
− 1)),
=
max
i
(y
i
/
a
u
= x
0
;a
k+1
(1 + i
2
/(b
l
)
2
));
= min
i
(x
i
/
u
b
u
=
(((r + 1)/r)
2
−1) where r = x
0
,
(y
r+1
+ 1)/
b
k+1
u
(i
2
/(a
u
)
2
−1)).
=
min
i
((y
i
+ 1)/
Then there exist b
l
, b
u
, a
l
, and a
u
such that
lim
k→∞
b
l
= b
l
, lim
k→∞
b
u
= b
u
, lim
k→∞
a
l
= a
l
, lim
k→∞
a
u
= a
u
, and
b
l
< b
o
< b
u
and a
l
< a
o
< a
u
.
€
The subsequent theorem establishes that the rectangle with diagonally
opposite vertices (a
l
,b
l
) and (a
u
,b
u
) in the a-b space properly contains the
domain of D
o
.
Theorem 5.9. If D = D(H(a,b)) = D
o
, then both the following conditions
hold: (A)a
l
< a < a
u
and (B)b
l
< b < b
u
.
€
Finally, we state the domain theorem, which enables us to compute the
domain numerically.
Theorem 5.10. (The Domain Theorem) The domain Domain(D
o
) of all
possible values of (a,b) parameters of a hyperbola in canonical form to give
reconstruction is given by the following formulas:
∗
l
(a),b
∗
l
(a)); or
(A) Domain(D
o
)
=
[b
a
l
<a<a
u
∗
l
(a),a
∗
u
(a));
(A) Domain(D
o
)
=
[a
b
l
<b<b
u
