Image Processing Reference
In-Depth Information
5.5.3 Two-Parameter Class
As in the last section, we shall consider one particular MM to highlight
the theme of our analysis. The MM we are considering is given in Table 5.4.
So, x
i
= ⌊f
x
(i,a
o
,b
o
)⌋ and y
i
= ⌊f
y
(i,a
o
,b
o
)⌋. We present the main result in
the following theorem [40].
Theorem 5.14. Let a
k+1
l
, b
k+1
u
, a
k+1
u
, and b
k+1
l
, be defined by the following
iterative algorithm for k > 0.
a
k+1
l
(f
a
(x
i
,i,b
u
))
=
max
i
b
k+1
u
(f
b
(i,y
i
+ 1,a
l
))
=
min
i
a
k+1
u
(f
a
(x
i
+ 1,i,b
l
))
=
min
i
b
k+1
l
(f
b
(i,y
i
,a
u
))
=
max
i
With a proper choice of a
l
, b
u
, a
u
, and b
l
that satisfies
(a) a
l
< a
o
< a
u
,b
l
< b
o
< b
u
and
(b) a
l
> a
l
,b
u
6 b
u
,a
u
6 a
u
,b
l
> b
l
,
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
(B) b
l
< b
o
< b
u
and a
l
< a
o
< a
u
.
Proof: Proof of the theorem is similar to the proof of Theorem 5.3. This uses
the definition of the floor function and the fact that f
b
is a decreasing function
in a.
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Now, if we consider the curves f(x,y,a
l
,b
u
) = 0 and f(x,y,a
u
,b
l
) = 0, we
can find a few interesting properties which are summarized as the following
theorems and lemmas.
The following Theorem is required to prove the tightness of the bounds of
a
o
and b
o
.
Theorem 5.15. The curves f(x,y,a
l
,b
u
) = 0 and f(x,y,a
u
,b
l
) = 0 intersect
[40].
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The proof is by contradiction and is left as an exercise.
Corollary 5.4. The open region between f
lu
and f
ul
(i.e., excluding the arcs
of the curves) do not contain any grid point.
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