Beta-Splines: A Flexible Model - Bezier and Splines in Image Processing and Machine Vision - page 137

m =3 n =1

c mn ( β 1 ,β 2)) u m

S i,j ( u, v )= ( ×

m =0 n = − 2

(6.44)

p =3 q =1

e pq ( β 1 ,β 2)) v p V i + n,j + q .

(

p =0 q = − 2

End conditions for a surface can be written exactly in the same way as for a
curve.

6.5 Possible Applications in Vision

Since β -spline has two more shape parameters, it provides more flexibility and
hence data can be approximated in a much better way. Normally, a β -spline
surface interpolates the corner points but not all the other control points.
Hence, a suitable interpolation technique can be envisaged and used to model
the disparity data in stereo vision. The β surface with minimum energy may
produce a continuous smooth surface with suitable discontinuities controlled
by shape parameters. A comparison between the Laplacian or biharmonic
operator yielded surface and the β surface, each based on disparity data, may
be useful to judge the merit of the β surface. It should be noted that both the
Laplacian and biharmonic operator yield a good surface where the disparity is
continuous but will provide a poor result when the disparity is discontinuous,
e.g., over the region where one object occludes the other.
Another potential application of β -spline may be in feature extraction in
pattern recognition. An object may be decomposed into many surface patches
and each of them can be approximated well by the β -spline. The approxima-
tion parameters, which are essentially the approximated control points along
with the values of two shape parameters, namely the β 1 and β 2 parameters,
for each surface patch may act as its feature vector.

6.6 Concluding Remarks

β -spline has been examined from the standpoint of computer graphics and
not from the viewpoint of other research areas. Very little work using β -spline
has been done in image processing and machine vision. It may, therefore, be
effective if the field is investigated thoroughly.