B-Splines and Its Applications - Bezier and Splines in Image Processing and Machine Vision - page 105

5
B-Splines and Its Applications
5.1 Introduction
Though Bezier-Bernstein (B-B) splines are very similar to B-splines in design-
ing a curve or a surface, the latter provides more flexibility during interactions.
Consequently, B-splines are more effective and more ecient, and hence are
more widely used. Since B-B splines use the Bernstein basis, we cannot deny
its influence over the design of B-B curves and surfaces. Any point on a B-B
curve is the weighted average of all the control points, of course, excepting
the end control points. Therefore, the effect of a change in one control point
is transmitted over the entire curve. Thus, any change in one control point
globally affects the curve. We cannot make a local change within a curve, even
when we are badly in need of one. The other limitation of the B-B spline is the
degree of the polynomial. For a cubic B-B spline, the number of control points
is always four while for an m th degree curve, the number of control points is
m+1, or in other words, the degree of the spline function is always one less
than the number of control points. Hence, the degree of the B-B spline curve
is restricted by the number of control points. The lack of local control and the
hard relation of degree of the polynomial function with the number of control
points are the major drawbacks of B-B splines.
To design curves and surfaces in a more versatile way, Schoenberg [146]
formulated the B-spline theory. He introduced a unique non-global basis func-
tion associated with each control point. This basis is called the B-spline basis.
Here, each control point is capable of controlling the curve over a range of
parameter values. Within this range of parameter values, the associated basis
function is non-zero and is zero beyond the parameter values. As a result,
B-spline basis functions are found to introduce better interactive flexibility in
curve and surface design. One of the great advantages of B-spline basis is that
one can change the order of the basis function without changing the number
of the control points in the control graph of an object.
In a special situation, B-spline contains the Bernstein basis.