Graphics Reference
In-Depth Information
We cannot determine surface orientations uniquely from the image irradi-
ance equation, even with supplementary boundary information. The problem
is ill-posed and regularization is used [162, 81, 21].
7.5.2 Method Based on Regularization
To find a smoothing spline (
f
∗
(
x, y
)
,g
∗
(
x, y
)), Ikeuchi and Horn used regu-
larization [81] that minimizes the error
E
(
f, g
)=
D
((
f
x
(
x, y
)+
f
y
(
x, y
)+
g
x
(
x, y
)+
g
y
(
x, y
))
+
λ
(
R
(
f
(
x, y
)
,g
(
x, y
))
(7.36)
−
I
(
x, y
))
2
)
dxdy.
The first term, the squared gradient of the surface orientations, in the inte-
grand is the departure from smoothness and the second term is the error in the
image irradiance equation.
λ
is the penalty parameter. When the brightness
measurements are accurate,
λ
is chosen large.
Three critical issues in regularization method are as follows:
(1) The existence of the solution.
(2) The uniqueness of the solution.
(3) The well-conditioning of the problem.
Of these three issues, existence of smoothing splines is ensured but the unique-
ness and well-conditioning cannot be guaranteed. Smoothing splines without
boundary conditions, in general, are not unique.
Theorem 1
:
Without any boundary conditions, the smoothing splines are in general not
unique, and the problem of computing a smoothing spline is ill-conditioned.
Ikeuchi and Horn [81] mentioned a number of boundary conditions, e.g.,
occluding boundaries, self-shadow boundaries, specular points, and singular
points.
7.5.3 Discrete Smoothing Splines
Any image domain
D
can be embedded into a rectangular region
D
where all
four sides can always be thought to intersect
∂D
through proper shrinking of
D
.
Suppose we discretize
D
with mesh size
h
. Further assume the region
D
is
divided into
m
+ 2 rows and that the
i
−
th
row contains
n
i
+ 2 grid points,
for
i
=0
,
1
,
···
,m
= 1. The total number of interior grid points in
D
i
is
m
N
=
n
i
.
i
=1
Let
n
= max
i
{
n
i
}
.
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