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∂p
∂x
1
2
(p
u
+
1
,v
−
uv
=
p
u
−
1
,v
)
∂p
∂y
1
2
(p
u,v
+
1
−
uv
=
p
u,v
−
1
)
(3.22)
∂q
∂x
1
2
(q
u
+
1
,v
−
uv
=
q
u
−
1
,v
)
∂q
∂y
1
2
(q
u,v
+
1
−
uv
=
q
u,v
−
1
)
and the average values
p
uv
=
(p
u
+
1
,v
+
p
u
−
1
,v
+
p
u,v
+
1
+
p
u,v
−
1
)/
4
¯
(3.23)
q
uv
=
(q
u
+
1
,v
+
q
u
−
1
,v
+
q
u,v
+
1
+
q
u,v
−
1
)/
4
we obtain an iterative update rule for
p
uv
and
q
uv
by setting the derivatives of
e
with
respect to
p
uv
and
q
uv
to zero, which yields the expressions
p
(n)
λ
I
uv
−
R
I
¯
uv
∂R
I
∂p
p
(n
+
1
)
p
(n)
p
(n)
q
(n)
=¯
+
uv
,
¯
uv
uv
uv
, q
(n)
uv
p
(n)
(3.24)
λ
I
uv
−
R
I
uv
∂R
I
∂q
q
(n
+
1
)
q
(n)
p
(n)
q
(n)
=¯
+
¯
uv
,
¯
uv
uv
uv
, q
(n)
uv
(Horn,
1986
). The initial values
p
(
0
)
uv
and
q
(
0
)
uv
must be provided based on a priori
knowledge about the surface. The surface profile
z
uv
is then derived from the slopes
p
uv
and
q
uv
by means of numerical integration, as outlined in detail in Sect.
3.2.3
.
Wöhler and Hafezi (
2005
) suggest that the albedo
ρ
uv
is set to a uniform value,
which may be updated using (
3.16
) and (
3.18
) in each iteration step based on a
certain number of selected pixels (e.g. all pixels of a certain image column)—hence,
the iterative update rule (
3.24
) not only determines the surface gradients
p
uv
and
q
uv
but also the albedo
ρ
by minimisation of error function (
3.21
). Section
3.3
describes
how a non-uniform albedo
ρ
uv
is taken into account.
The three-dimensional reconstruction algorithm proposed by Horn (
1989
), which
generates an integrable surface gradient vector field, is described as follows. It si-
multaneously yields the surface gradients
p
uv
and
q
uv
and the depth
z
uv
. Here, the
assumption of a smooth surface according to (
3.20
) is replaced by the departure
from integrability error expressed by the error term
∂z
∂x
p
uv
2
∂z
∂y
q
uv
2
.
e
int
=
uv
−
+
uv
−
(3.25)
uv
Accordingly, the shape from shading problem corresponds to a minimisation of the
overall error term
f
=
e
i
+
γe
int
.
(3.26)
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