Image Processing Reference
In-Depth Information
The domain of D
o
can be computed using the following theorem.
Lemma 5.10. The following hold [40]:
∗
∗
(A) a
l
< a < a
u
if and only if b
1
(a) < b
u
(a)
∗
∗
where b
l
(a)
=
max
i
(f
b
(i,y
i
,a)) and b
u
(a) = min
i
(f
b
(i,y
i
+ 1,a))
∗
∗
(B) b
l
< b < b
u
if and only if a
l
(b) < a
u
(b)
∗
∗
where a
l
(b)
=
max
i
(f
a
(x
i
,i,b)) and a
u
(b) = min
i
(f
a
(x
i
+ 1,i,b)).
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Using the above lemma, we can compute the domain that is stated in the
following theorem [40].
Theorem 5.18.
∗
l
(a),b
∗
u
(a))
Domain(D
o
)
= ∪
a
l
<a<a
u
[b
∗
l
(b),a
∗
u
(b)).
= ∪
b
l
<b<b
u
[a
€
We complete this section by considering the following two examples.
Example 5.8. Let f be an ellipse with center at origin and axes parallel to
the coordinate axes. So, f : x
2
/a
2
+ y
2
/b
2
− 1 = 0 and the parameters to
estimate are a and b. Clearly, f is continuous and its MM is the same as
given in Table 5.2. f is also separable where the separated functions are:
(1−y
2
/b
2
),y = f
y
= b
(1−x
2
/a
2
)
x = f
x
= a
(1−y
2
/b
2
),b = f
b
= y/
(1−x
2
/a
2
).
a = f
a
= x/
∂f
y
∂x
(1 − x
2
/a
2
)). So g(x,a
1
,b
1
) <
g(x,a
2
,b
2
) for a
1
< a
2
, and b
1
> b
2
and f satisfies Assumption 3 as well.
= −bx/(a
2
Further, g(x,a,b) =
We have dealt thoroughly with this curve in the previous section. It is seen
that the initial choice of the lower and upper bounds of a and b for Theorem
5.14 can be given by the following set of equations:
a
l
= x
0
,a
u
= x
0
+ 1,b
l
= y
0
,b
l
= y
0
+ 1.
We have presented an analysis of the class of curves whose nature of mono-
tonicity is depicted by the particular MM as given in Table 5.2. However, along
a similar line the other four MMs may be treated to achieve similar results.
In the next example we consider a straight line because it has a different MM
than the earlier one.
