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Hence, as depth differences are evaluated along image rows only, the second term
becomes zero, and we obtain
u
(s)
e
p
u, v
(s)
.
(z)
(s)
sfs
=
(5.11)
u
(s)
i
u
=
The derivative of
e
z
for pixel
(u, v)
with respect to the surface gradients
p
and
q
is
then
⎧
⎨
u,v
=
2
(z)
(s)
sfs
−
(z)
(s)
if
u
(s)
i
≤
u
≤
u
(s)
and
v
=
v
(s)
∂e
z
∂p
shadow
e
(
|
u
(s)
−
u
(s)
i
|+
1
)
2
e
⎩
0
otherwise
(5.12)
u,v
=
∂e
z
∂q
0
.
This leads to an extended update rule for the surface gradient
p
(cf. (
3.24
)):
p
(n)
u,v
λ
I
R
¯
uv
∂R
∂p
η
∂e
z
∂p
p
(n
+
1
)
p
(n)
p
(n)
q
(n)
=¯
+
−
¯
uv
−
uv
,
.
(5.13)
uv
uv
uv
, q
(n)
The update rule for
q
in (
3.24
) remains unchanged. The iteration is initialised with
the result of the algorithm described in Sect.
5.2.1
. After each iteration step the
depth profile
z
uv
needs to be reconstructed by numerical integration based on the
current values of
p
(n)
uv
and
q
(n)
(cf. Sect.
3.2.3
) in order to determine the values of
uv
(z)
(s)
sfs
for the next iteration step.
A further condition is that if pixel
(u, v)
is outside the shadow, the angle between
surface normal
n
and vector
s
of incident light must be less than
π/
2, and the values
p
uv
of the pixels outside the shadow have to be limited accordingly during the iter-
ation process. This means that
p
uv
and
q
uv
must fulfil the condition
n
s
shadow
>
0,
which for incident light from the left- or right-hand side (zero component of
s
shadow
in the vertical image direction) becomes
p
uv
<
tan
μ
shadow
if
μ
shadow
∈[
·
0
,...,π/
2
[
and
p
uv
>
tan
μ
shadow
if
μ
shadow
∈]
.
It is important to note that the shadow-related error term can be incorporated
into any error function applied to a variational optimisation scheme, an overview
of which is given in Sect.
3.2
. In particular, the iterative update rule (
3.30
) based
on the integrability error term (
3.25
) is readily extended to take into account depth
differences indicated by shadow analysis in a manner analogous to (
5.13
). Three-
dimensional reconstruction results of lunar tectonic faults and a wrinkle ridge ob-
tained with this approach are presented in Sect.
8.3
.
π/
2
,...,π
]
5.2.3 Initialisation of the Shape from Shading Algorithm Based on
Shadow Analysis
In this section, the integration of shading and shadow features described in
Sect.
5.2.1
is modified in that two or more shadows are used to initialise the shape
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