Graphics Reference
In-Depth Information
3.4 Approximation Technique
It is seen in the previous section that to approximate a raster scanned gray
tone image row wise (or column wise), the order of the Bernstein polynomial
varies from row to row (or column to column), and for an image with one unit
error in approximation (
= 1) this order becomes close to the maximum value
present in each row (or column). The large order of the polynomial, in turn,
makes the approximation time as well as the reconstruction time relatively
high. Again, the variation in order of the polynomial from row to row (or
column to column) makes the coding scheme complicated.
An attempt is made in this chapter to develop an approximation scheme
that keeps the order of the polynomial equal to two. Since the order is chosen
two, the amount of error
, as expected, will be significantly high. In order
to circumvent this, a modification of the conventional approximation scheme
based on Bezier-Bernstein polynomial is proposed. This leads to the formu-
lation of a new scheme by which it is also possible to obtain any degree of
accuracy in approximation.
Given
n
points, the approximation algorithm requires
n
-2 unique quadratic
B-B spline functions for their representation. Unlike the method described in
section 3.3, the scheme, proposed here, decomposes a row (column) either into
a single gray segment or into a number of segments so as to enable them to
be approximated properly. An error bound has been defined that guides the
process of segmentation.
3.4.1 Bezier-Bernstein (B-B) Polynomial
Equation (3.1), which represents a
p
-th degree Bernstein polynomial for ap-
proximating a function
f
(
t
),0
≤
t
≤
1 can be written as
B
ip
(
t
)=
φ
op
(
t
)
f
(0) +
φ
1
p
(
t
)
f
(
1
p
)+
φ
2
p
(
t
)
f
(
2
p
)+
·
+
φ
pp
(
t
)
f
(1)
.
B
ip
(
t
)
1) along with
some fixed points of the function
f
(
t
)in[0
,
1] for its approximation. With
the choice of some arbitrary points for
f
(
p
), one can determine
B
ip
(
t
)foreach
value of t.
Let
v
i
represent a point in a multi-dimensional space and that
v
i
=
f
(
p
).
Thus
B
ip
(
t
) becomes,
is seen to consider a set of weights
φ
ip
(
t
)(0
≤
t
≤
p
B
ip
(
t
)=
φ
ip
(
t
)
v
i
.
(3.3)
i
=0
Equation (3.3) can be viewed as a vector valued Bernstein polynomial and it
approximates a polygon with vertices
v
i
and t in [0
,
1].
B
ip
(
t
)isthusseen
to generate a space curve. Equation (3.3) is known as
p
-th degree Bezier-
Bernstein (B-B) polynomial. For
p
= 2, the quadratic B-B polynomial (drop-
ping the index i in
B
ip
)is
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