Beta-Splines: A Flexible Model - Bezier and Splines in Image Processing and Machine Vision - page 128

6

Beta-Splines: A Flexible Model

6.1 Introduction

In general, a spline with greater flexibility is always desired because it en-
hances the strength for modeling a set of data points. A beta spline is a spline
with such an ability. For its flexibility, it can be used in image processing as
well as in vision problems in many different ways. Beta spline was developed
by Barsky [15] and our discussion in this chapter is based on his thesis.

6.2 Beta-Spline Curve

A β -spline curve is a piecewise parametric cubic beta curve that is the weighted
average of its control vertices. For every point of the curve, the weight w is
different and depends on two different shape parameters β 1, β 2 , and position
parameter, t itself. Hence, we can represent the i th piece of a beta curve as

n =1

P i ( t )= w n ( β 1 ,β 2 ,t ) V i + n , 0 ≤ t< 1 . (6.1)

n = − 2

The weight, w , is a basis function of β 1, β 2 and can be computed for some
values of the parameters β 1, β 2, and t . V i + n are the control points. Weight,
w , is given by

m =3

c mn ( β 1 ,β 2) t m w n ( β 1 ,β 2 ,t )= for n = −

2 ,

−

1 , 0 , 1 .

(6.2)

m =0

Consider two beta curve segments, say P i ( t )and P i +1 ( t ). Then from the
position, first order and second order continuity we can write,

P i +1 (0) = P i (1)

(6.3)