Biomedical Engineering Reference
In-Depth Information
5.4.2 The Wavelet-Based SFS
A wavelet-based method was developed in [31]. Instead of using the constraints
in Zheng-Chellappa's method (see Section 5.3.2.1), the authors introduced a new
constraint (5.20). It is said that “the new constraint not only enforces integrability
but also introduces a smoothness constraint in an implicit manner.” Now the
energy function is defined as
[(
E
(
x
,
y
)
−
R
(
p
,
q
))
2
+
(
p
x
+
p
y
+
q
x
+
q
y
)
W
=
(5.70)
+
(
z
x
−
p
)
2
+
(
z
y
−
q
)
2
+
(
z
xx
−
p
)
2
+
(
z
yy
−
q
)
2
]
dxdy
.
The objective function is first replaced by its approximation in scaling space
V
0
of Daubechies wavelets. Then the variational problem is solved by an iterative
algorithm. We now describe this method.
We assume that the given image size is
M
×
M
. The surface
Z
(
x
,
y
), its partial
derivatives
∂
Z
∂
x
=
p
(
x
,
y
), and
∂
Z
∂
y
=
q
(
x
,
y
) have projection to
V
0
,
the scaling
space at level 0:
M
−
1
M
−
1
Z
(
x
,
y
)
=
Z
k
,
l
φ
0
,
k
,
l
(
x
,
y
)
,
k
=
0
l
=
0
M
−
1
M
−
1
p
(
x
,
y
)
=
p
k
,
l
φ
0
,
k
,
l
(
x
,
y
)
,
(5.71)
k
=
0
l
=
0
M
−
1
M
−
1
q
(
x
,
y
)
=
p
k
,
l
φ
0
,
k
,
l
(
x
,
y
)
.
k
=
0
l
=
0
Denoting
2
∂
x
2
φ
0
,
k
,
l
(
x
,
y
)
,
0
,
k
,
l
(
x
,
y
)
=
∂
0
,
k
,
l
(
x
,
y
)
=
∂
(
x
)
(
xx
)
∂
x
φ
0
,
k
,
l
(
x
,
y
)
,
φ
φ
2
∂
y
2
φ
0
,
k
,
l
(
x
,
y
)
,
0
,
k
,
l
(
x
,
y
)
=
∂
0
,
k
,
l
(
x
,
y
)
=
∂
(
y
)
(
yy
)
φ
∂
y
φ
0
,
k
,
l
(
x
,
y
)
,
φ
substitute (5.71) in each term of (5.70) to get
2
M
−
1
M
−
1
W
=
E
(
x
,
y
)
−
R
(
p
k
,
l
φ
0
,
k
,
l
(
x
,
y
)
,
q
k
,
l
φ
0
,
k
,
l
(
x
,
y
))
dxdy
(5.72)
k
,
l
=
0
k
,
l
=
0
⎛
M
−
1
2
M
−
1
2
⎝
(
x
)
0
,
k
,
l
(
x
,
y
)
(
y
)
0
,
k
,
l
(
x
,
y
)
+
p
k
,
l
φ
+
p
k
,
l
φ
k
,
l
=
0
k
,
l
=
0
