Biomedical Engineering Reference
In-Depth Information
for
i
=
1to
w
idth
(
I
)do
for
j
=
1to
height
(
I
)do
dfz
←−
1
·{
(cos
tan
γ
+
sin
tan
)
/
τ
τ
γ
(
p
2
+
q
2
+
1)(tan
2
γ
+
1)
−
(
p
+
q
)(
p
cos
τ
tan
γ
+
q
sin
τ
tan
γ
+
1)
/
(
p
2
+
1)
3
(tan
2
γ
+
1)
}
Z
(
i
,
j
)
←
Z
0
(
i
,
j
)
+−
f
(
Z
0
(
i
,
j
))
/
dfz
p
←
Z
(
i
,
j
)
−
Z
(
i
,
j
−
1)
q
←
Z
(
i
,
j
)
−
Z
(
i
−
1
,
j
)
+
q
2
end do
end do
Normalize
(
Z
(
x
,
y
)
,
Z
max
,
Z
min
)
Output Z
(
x
,
y
)
The subfunction
Normalize
is a standard math function used in signal and
image processing.
We now demonstrate this method by using the following example.
Example 5.
Reconstruct the surface of a synthetic vase using Tsai-Shah's
method.
In order to compare with Pentland's method, here we consider reconstruc-
tion of the same surface as in Example 2—the surface of a synthetic vase.
Figure 5.3 shows the synthetic vase and the reconstruction results using Tsai-
Shah's algorithm from three different directions. The light is from above at
(x
=
0
,
y
=
0
,
z
=
1
). The input image is showed in Fig. 5.3(a). The surface,
shown in Fig. 5.3(b), (c), and (d), is the reconstructed surface from three
different directions. Tsai-Shah's algorithm works well and produces good re-
sults as expected for the vase. However, it is sensitive to noises as we will point
out in the next section. In general, the experiment shows that Tsai-Shah's al-
gorithm can reconstruct the object well on the surface where the reflectance
changes linearly with respect to the surface shape.
5.3.2 Optimization Approaches
As we pointed out earlier, the problem of recovering the shape from shading
can be based on solving the irradiance equation (5.2). The irradiance equa-
tion is a first-order PDE. Unfortunately, in general, this PDE is nonlinear and
only well posed under limited conditions. To make things worse, in practice,
