Biomedical Engineering Reference
In-Depth Information
each value of the depth map can be iteratively calculated. In fact, (5.44) can be
read as
Z
(
x
,
y
)
−
Z
n
−
1
(
x
,
y
)
df
(
Z
n
−
1
(
x
,
y
))
dZ
(
x
,
y
)
0
=
f
(
Z
(
x
,
y
)
≈
f
(
Z
n
−
1
(
x
,
y
))
+
.
(5.45)
Rearranging Eq. (5.45), we obtain
Z
0
(
x
,
y
)
=
initial value
(5.46)
−
f
(
Z
n
−
1
(
x
,
y
))
d
Z
n
(
x
,
y
)
=
Z
n
−
1
(
x
,
y
)
+
n
=
1
,
2
,...,
dZ
(
x
,
y
)
f
(
Z
n
−
1
(
x
,
y
))
,
where
cos
τ
tan
γ
+
sin
τ
tan
γ
p
2
df
(
Z
n
−
1
(
x
,
y
))
dZ
(
x
,
y
)
=−
1
+
1
tan
2
+
q
2
γ
+
1
(
p
+
q
)(
p
cos
τ
tan
γ
+
q
sin
τ
tan
γ
+
1)
(
p
2
−
+
1)
3
tan
2
.
(5.47)
+
q
2
γ
+
1
By iteratively using formula (5.46), we obtain the approximation of the depth
map
Z
(
x
,
y
). Readers may have noticed that the iterative formula is Newton's
formula.
This method has a similar disadvantage as the algorithm based on linear
approach. However, it is faster since it does not need to compute the FFT and
IFFT.
The algorithm can be described by the following procedure:
Step 1.
Input the original parameters of the reflectance map,
Step 2.
Set the initial guess of
Z
0
(
x
,
y
)
=
0,
Step 3.
Refine the depth map
Z
k
(
x
,
y
) using Eq. (5.46).
The way to realize Pentland's algorithm can be described by the following
pseudocode.
Algorithm 2:
Tsai-Shah's linearization method
Input Z
min
(mindepthvalue),
Z
max
(maxdepthvalue), (
x
,
y
,
z
)(direction of the
light source),
I
(
x
,
y
)(inputimage)
z
0
←
0;
p
0
←
0;
p
←
q
←
p
0
←
q
0
←
q
0
;
x
2
D
←
+
y
2
+
z
2
,
sx
←
x
/
D
,
sy
←
y
/
D
,
sz
←
z
/
D
.
sin
γ
←
sin(arccos (
lz
))
,
sin
τ
←
sin(arctan(
sy
/
sx
))
,
cos
τ
←
cos(arctan (
sy
/
sx
))
.
