Graphics Reference
In-Depth Information
3.3 Shortcomings of Bernstein Polynomial and Error of
Approximation
Bernstein polynomial is a powerful tool to approximate a continuous function
within any degree of accuracy. It uses the global information while approxi-
mating a function and the order of the polynomial increases with accuracy in
approximation. The Bernstein polynomial of degree
p
from is
p
f
(
i
B
ip
(
t
)=
p
)
φ
ip
(
t
)
(3.1)
i
=0
for approximating a function
f
(
t
). Here
f
(
t
) is defined and finite on the closed
interval [0
,
1]. Also,
φ
ip
(
t
)=
p
i
t
i
(1
t
)
p−i
−
and
p
i
=
p
!
(
p
−
i
)!(
i
)!
with
i
=1
,
2
,
p
.
The order
p
of the Bernstein polynomial
B
ip
(
t
) satisfies the inequality
···
k
m
δ
2
<p
(3.2)
in order to have the error of approximation less than
, where
k
m
is the
maximum value of the approximating function
f
(
t
) in the interval [0
,
1].
δ
is
a positive number such that for points
t
1
,t
2
∈
(0
,
1)
2
,
|
f
(
t
1
)
−
f
(
t
2
)
|
<
whenever
<δ
.
Since a graylevel image in a raster scan can be approximated either row
wise or column wise, it appears from the inequality (3.2) that the order of the
approximating polynomial may be different for different rows (or columns)
depending on the value of
k
m
(assuming
and
δ
do not change appreciably). As
an illustration, let us consider the case of approximating, row wise, a 32 level
(0
,
1
,
|
t
1
−
t
2
|
32. If a row has its maximum value
k
m
= 31,
then for
= 1, (i.e., one unit error in gray value)
p>
31
×
31
×
31
29
···
31) image of size 32
×
≈
35
.
42, i.e.,
×
29
36. Note that the maximum value of
δ
=
2
31
, because
|
t
1
−
t
2
|
=1
/
31
−
30
/
31
(
t
1
,t
2
∈
(0
,
1). Therefore, for
k
m
= 31, one can choose
p
to be equal to 36.
On the other hand, if
k
m
= 2, then
m
1
.
06, i.e.,
p
=2.
k
m
= 2 means
some of the graylevel values in the row are same and is equal to 1. Since in
a gray image it is very likely to have the maximum value anywhere in each
row, the order may be as high as the maximum graylevel in the image. This
makes the method ineffective.
≈
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