162 CHAPTER 6 Variational Schemes for Bounded Domains
scheme has a subdivision mask of the form
2s[x]
1+x
. Given the interpolation mask n[x]
for this derivative scheme, we observe that the derivative mask for the original
scheme (on
Z) has the form (1 − x)n[x]. The derivative mask on finer grids
1
2
k
Z has
the form
2
k
(1 − x)n[x ].
For example, the subdivision mask for the four-point scheme has the form
s[x ] =−
1
16
x
−3
+
9
16
x
−1
+ 1 +
9
16
x −
1
16
x
3
. Because this scheme is interpolatory, the
interpolation mask for this scheme is simply
1. However, due to the fact that the
four-point scheme is not piecewise polynomial, computing the exact value of first
derivatives of this scheme at integer grid points is nontrivial. Dividing the subdi-
vision mask
s[x ] for the four-point scheme by
1
2
(1 + x) yields the subdivision mask
for the first derivative of the four-point scheme,
−
1
8
x
−3
+
1
8
x
−2
+ x
−1
+ 1 +
1
8
x −
1
8
x
2
.
After applying the recurrence of equation 6.4, the interpolation mask for this deriva-
tive scheme has the form
−
1
12
x
−2
+
7
12
x
−1
+
7
12
−
1
12
x. Finally, multiplying this mask
by the difference mask
(1 − x)
yields the exact derivative mask for the four-point
scheme of the form
−
1
12
x
−2
+
2
3
x
−1
−
2
3
x +
1
12
x
2
. (Remember that this mask should
be multiplied by a factor of
2
k
to compute the derivative on
1
2
k
Z.)
6.1.2 Exact Inner Products
Having derived a method for computing the exact value and derivatives of the
scaling functions
N [x], we next derive a similar scheme that computes the exact
inner product of two functions defined using subdivision. To begin, we consider
a simple inner product
p, q of the form
p[x] q[x] dx, where is some bounded
domain. Given a convergent, stationary scheme with a subdivision matrix
S, our
task is to compute the exact values of this inner product for functions of the form
p[x] = N [x]p and q[x] = N [x]q, where the vector N [x] satisfies the matrix refinement
relation of equation 6.1. In particular, we are interested in a method that does not
rely on any type of underlying piecewise polynomial definition for
N [x]. The key
to this method is to compute an inner product matrix
E of the form
E =
N [x]
T
N [x] dx. (6.5)
Note that N [x]
T
N [x] is a matrix whose ijth entry is the product of the ith and jth
functions in the vector
N [x]. Therefore, the ijth element of E is the inner product of
the
ith and jth functions in the vector N [x]. The usefulness of this inner product
matrix
E lies in the fact that it can be used to compute the exact inner product of
two functions
p[x] and q[x] defined as N [x]p and N [x]q, respectively. In particular,