Image Processing Reference
In-Depth Information
′
′
′
Let us define p,p
,q,q
such that a
l
= f
a
(x
p
′
,p
,b
u
), b
l
= f
b
(p,y
p
,a
u
),
b
u
= f
b
(q′,y
q′
+ 1,a
l
), and a
u
= f
a
(x
q
+ 1,q,b
l
).
′
Corollary 5.5. D(f
lu
) or D(f
ul
) differs from D
o
only at point(s) like (q
,y
q
′
+
1) or (x
q
+ 1,q), which define b
u
and a
u
.
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From the above corollary, we can say that f
lu
or f
ul
are very close approx-
imations of the original curve represented by the function f. If we assume one
more property about the curve, then we can prove
(i) that the Domain of D
o
in the a−b plane is contained in the rectangle
R
ul
whose diagonally opposite vertices are (a
l
,b
u
) and (a
u
,b
l
); and
(ii) that R
ul
is the smallest rectangle containing the domain. This assump-
tion is stated below.
Assumption 3. Let,
∂f
y
∂x
= g(x,a,b). Then for a
1
< a
2
and b
1
> b
2
either
∀x, g(x,a
1
,b
1
) < g(x,a
2
,b
2
), or
∀x, g(x,a
1
,b
1
) > g(x,a
2
,b
2
).
In terms of the limiting and original curves, the above assumption helps
us in proving the following properties.
Lemma 5.8. If f satisfies Assumption 3 then f
1
: f(x,y,a
1
,b
1
) = 0 and
f
2
: f(x,y,a
2
,b
2
) = 0 cannot intersect more than once for a
1
< a
2
and
b
1
> b
2
[40].
Proof: If f
1
and f
2
do not intersect at all, then the lemma trivially holds.
So we assume that f
1
and f
2
intersect at least twice. But Assumption 3 is
violated.
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In the sequel, we shall use only the first part of Assumption 3. Symmetric
results may be obtained using the other part.
Similar to the case of the ellipse, the following lemma relates the grid
points that are hit by a
u
and b
l
at the end of iterative refinement.
Lemma 5.9. If p and q are defined such that a
u
= f
a
(x
q
+ 1,p,b
l
) and
b
l
= f
b
(p,y
p
,a
u
), then q < y
p
and p < x
q
+ 1 [40].
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So we can say that the domain of D
o
is properly contained in the rectangle
R
ul
, which is defined by the diagonally opposite points (a
l
,b
u
) and (a
u
,b
l
) in
the parametric space. This is formally stated in the next theorem.
Theorem 5.16. D(f(x,y,a,b)) = D
o
implies that a
l
< a < a
u
and b
l
< b <
b
u
.
Proof: This proof is similar to Theorem 5.4.
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The next theorem claims that R
ul
is the smallest rectangle enclosing the
domain of D
o
.
Theorem 5.17. If we select some a so that a
l
< a < a
u
, then there exists
some b for which D(f(x,y,a,b)) = D
o
[40].
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