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180 CHAPTER 6 Variational Schemes for Bounded Domains
(upper left). Note that here the scheme is applied parametrically; that is, the control
points are points in two dimensions, and subdivision is applied to both coordinates
to obtain the next denser shape.
6.3 Minimization of the Variational Scheme
The previous section constructed a relation of the form E
k
S
k−1
== U
k−1
E
k−1
that
models the finite difference relation
d
k
[x] s
k−1
[x] == 2d
k−1
[x
2
] arising from the dif-
ferential method. However, we offered no formal proof that subdivision schemes
constructed using the differential method converged to solutions to the differen-
tial equation. Instead, we simply analyzed the convergence and smoothness of
the resulting schemes. In this section, we show that subdivision matrices satisfying
equation 6.17 define a subdivision scheme whose limit functions
p
∞
[x] are minimiz-
ers of
E[p] in the following sense: over the space of all functions (with well-defined
energy) interpolating a fixed set of values on
∩Z, the limit function p
∞
[x] produced
by this subdivision scheme has minimal energy.
Our approach in proving this minimization involves three steps. First, we con-
sider the problem of enforcing interpolation for natural cubic splines using interpo-
lation matrices. Next, we show that these interpolation matrices can also be used
to compute the exact value of the variational functional
E[p
∞
]. Finally, we set up a
multiresolution framework based on the inner product associated with
E and prove
that
p
∞
[x] is a minimizer of E in this framework.
6.3.1 Interpolation with Natural Cubic Splines
Due to the linearity of the subdivision process, any limit function p
∞
[x] can be
written as the product of the row vector of basis functions
N
0
[x] associated with
the subdivision scheme and a column vector
p
0
; that is, p
∞
[x] = N
0
[x]p
0
. Remember
that
N
0
[x] represents the vector of basis functions associated with the variational
subdivision scheme;
N
0
[x] is the finite element basis used in constructing the inner
product matrix
E
0
. Due to the convergence of the subdivision scheme for natural
cubic splines, the continuous basis
N
0
[x] can be evaluated on the grid Z (restricted
to
) to form the interpolation matrix N
0
.
In the stationary case where
= [0, ∞], all of the interpolation matrices N
k
agree
with a single interpolation matrix
N that satisfies the recurrence of equation 6.2
(i.e.,
U
T
NS == N). This matrix N is tridiagonal because the natural cubic spline basis
functions are supported over four consecutive intervals. If the non-zero entries