Graphics Reference
In-Depth Information
7.6.1 Convergence
I
ij
)
∂R
(
f,g
)
∂f
To discuss convergence, Lee assumed (
R
(
f, g
)
−
and (
R
(
f, g
)
−
I
ij
)
∂R
(
f,g
)
∂g
are Lipschitz functions for all (
i, j
). This means for (
f, g
)
,
(
f
,g
)
∈
S
,
I
ij
)
∂R
(
f
,g
)
∂f
I
ij
)
∂R
(
f,g
)
∂f
(
R
(
f
,g
)
|
(
R
(
f, g
)
−
−
−
|
ij
(
f
L
(1)
≤
−
f
)
2
+(
g
−
g
)
2
,
and
I
ij
)
∂R
(
f
,g
)
∂g
I
ij
)
∂R
(
f,g
)
∂g
(
R
(
f
,g
)
|
(
R
(
f, g
)
−
−
−
|
ij
(
f
L
(2)
f
)
2
+(
g
g
)
2
,
≤
−
−
where
L
ij
s are Lipschitz constants. Then for
x
,
x
S
N
,
∈
b
(
x
)
x
)
b
(
x
)
−
2
≤
ν
x
−
2
,
(7.42)
where
ν
=
max
ij
(
L
(1)
ij
)
2
+(
L
(2)
ij
{
)
2
}
.
Note that
ν
is also a Lipschitz constant and
2
is the
L
2
-norm.
Some of the interesting results in connection to
DSS
are as follows:
(1)
Theorem 3
:
If
x
∗
is a discrete smoothing spline, then for the range
.
[0
,
4
s
(
n, m
)
h
2
ν
π
2(
m
+1)
+
sin
2
π
2(
n
+1)
(
n, m
)=
sin
2
λ
∈
)
,
x
(
k
)
in Algorithm 2 converges to
x
∗
.
λ
is the penalty parameter in expression
(7.37).
h
is the mesh size of discretization,
m
is the number of rows in
D
i
,
n
is the maximum number of grid points in a single row, and
ν
is Lipschitz
constant determined by the function
R
(
f, g
)and
I
(
x, y
).
(2)
Theorem 4
:
For
[0
,
4
s
(
n, m
)
h
2
ν
λ
∈
)
,
Algorithm 2 converges to the unique regular discrete smoothing spline, which
is also the unique solution of equation (7.38). Algorithm 2 can be modified in
a number of ways to make it more ecient for regular and irregular regions.
Interested readers may have look at the article of David Lee [101]. Some of
the drawbacks of Algorithm 2 are:
•
It does not consider the integrability constraint, which plays an important
part in surface description.
•
Implementation of Algorithm 2 is not straightforward for irregular regions.
This is for the computation of
M
−
1
, the matrix
M
being equal to
diag
(
A, A
),
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