Graphics Reference
In-Depth Information
7
Discrete Splines and Vision
7.1 Introduction
This chapter presents a theoretical background of discrete splines: how it can
be used in the area of subdivision so that refinement can be done for better
representation and better visualization, and how to examine the feasibility of
discrete smoothing splines to detect shapes of opaque physical objects from
their shading. For this, we first look at the theory of discrete splines as de-
veloped by Cohen, Lyche, and Risenfeld [41] and use it for understanding the
underlying structure of subdivision algorithms.
Next, we try to view knots of smoothing discrete splines as the discrete
grid points defined in the greylevel image plane, and examine the feasibility
of using such a spline to detect shapes of objects with the help of a reflectance
map [77], defined in terms of image brightness values and surface gradients.
The feasibility of using smoothing splines in the shape from shading problem
has been discussed by David Lee [101].
7.2 Discrete Splines
Discrete splines were introduced by Mangasarian and Schumaker [118] as solu-
tions to certain minimization problems involving differences instead of deriva-
tives. Lyche [111, 112] studied approximation properties of discrete splines.
Schumaker [148] provided discrete B-splines on a uniform partition, while de-
Boor [55] provided the same on a nonuniform partition.
We have already discussed B-splines
B
i,k
of order
k
in a previous chapter.
We now consider a piecewise polynomial
f
(
x
) in terms of
B
i,k
, so that
n
f
(
x
)=
B
i,k
(
x
)
P
i
.
(7.1)
i
=1
The knots
τ
=
{
τ
1
,τ
2
,
···
τ
n
+
k
}
can be made uniform as well as multiple.
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