Graphics Reference
In-Depth Information
where (
T
)=(
t
p
,t
p−
1
,
···
1),
⎛
⎞
0
p
(
1
p−
1
p−
1
(
···
p
p−p
p−p
(
1)
p
1)
p−
1
1)
0
−
−
−
0
p
p−
1
(
1)
p−
1
1
p−
1
p−
2
(
⎝
⎠
1)
p−
2
−
−
···
0
.
.
.
.
(
C
)=
(1.22)
0
1
(
p−
0
(
1)
1
1)
0
−
−
···
0
0
0
(
1)
0
−
0
···
0
and (
V
)
T
is (
V
0
,V
1
,V
2
,
···
V
p
).
1.4.2 Bezier-Bernstein Surfaces
ABezier-Bernstein surface is a tensor product surface and is represented by
a two-dimensional Bezier-Bernstein (B-B) polynomial. If we designate the
surface patch by
S
(
u, v
), then
p
q
S
(
u, v
)=
φ
ip
(
u
)
φ
jq
(
v
)
V
ij
,
(1.23)
i
=0
j
=0
where 0
1.
V
ij
is the (i,j)th control point.
φ
ip
is the
i
th basis Bernstein basis function of order p and
φ
jq
is the Bernstein basis of
order q. When
p
≤
u
≤
1 and 0
≤
v
≤
=
q
,theBezier-Bernstein surface is defined on a rectangular
support. This support becomes a square for
p
=
q
.Thus,for
p
=3and
q
=3,
we get a bicubic surface on a square support.
All the properties mentioned for 1-d B-B curves also hold for 2-d B-B
surfaces. Once again, for selection of control points for two pieces of a surface,
it is possible to draw a single piece of a spline surface.
1.4.3 Curve and Surface Design
One dimensional Bezier-Bernstein splines are used to design curves. To draw
a curve with a definite shape, a designer inputs a set of ordered control points,
which when joined in succession, produces the polygonal shape corresponding
to the shape of the object that the designer wants to draw. The designer refines
the shape, changing a few control points, through adequate interaction. Figure
1.2 shows two important cubic curves.
A2-dBezier spline is used to design a surface. The control points in this
case define a control polygonal surface, which upon interactive refinement pro-
duces a desired surface. However, a quadratic spline provides some advantage
from the computational point of view. For actual drawing, interested readers
can consult topics on computer graphics.
We now discuss the problem of data approximation in relation to binary
image approximation and reconstruction.
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