Graphics Reference
In-Depth Information
1.3.1 Determination of the Order of the Polynomial
To judiciously fit a Bernstein curve over a set of data points, we need to know
the order of the polynomial. Once the order is known, one can fit a curve
over the data points using any standard method. We shall present here a
classical approach to determine the order of the polynomial to approximate a
one dimensional function. Extension to two or higher dimensions is not very
dicult. We shall later consider a relatively simple approach to determine
the order of a Bezier-Bernstein polynomial for approximating image intensity
(pixels) values.
Let
f
(
t
) be defined and finite on the closed interval [0
,
1]. The Bernstein
polynomial [113] of degree p for the function
f
(
t
)is
p
k
f
(
k/p
)
t
k
(1
p
t
)
p−k
.
B
kp
(
t
)=
−
(1.5)
k
=0
Since
f
(
t
) is continuous on [0
,
1], it is uniformly continuous, i.e., for every
>
0
there will exist a
δ>
0 such that
|
f
(
t
1
)
−
f
(
t
2
)
|
<
whenever
|
t
1
−
t
2
|
<δ
.
Let us select an arbitrary t on [0
,
1]. Then
f
(
t
)
p
k
t
k
(1
p
t
)
p−k
f
(
t
)=
−
k
=0
since
p
k
t
k
(1
p
t
)
p−k
=1
.
−
k
=0
Hence,
f
(
t
))
p
k
t
k
(1
p
t
)
p−k
|
B
kp
(
t
)
−
f
(
t
)
|
=
|
(
f
(
k/p
)
−
−
|
k
=0
p
k
t
k
(1
(1.6)
k
=0
|
p
t
)
p−k
.
≤
f
(
k/p
)
−
f
(
t
)
|
−
Now we divide the set of integers 0
,
1
,
2
,
···
into two sets
A
and
B
according
to the following rule: an integer
k
<δ
, k is in
B
otherwise.
Therefore, the sum on the right of the equation (1.6) can be broken into two
different sums, one for each of the two sets
A
and
B
.
Ifkisin
A
, we have according to the definition of
δ
∈
A
if
|
k/p
−
t
|
|
f
(
k/p
)
−
f
(
t
)
|
<.
Therefore,
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