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is desired to derive a three-dimensional reconstruction of linear ridges or cross sec-
tions of craters (Horn,
1989
).
Commonly a constant albedo
ρ
is assumed, and the value of
ρ
is chosen such
that the average surface slope over the region of interest is zero (Wöhler et al.,
2006b
) or corresponds to a given nonzero value. Alternatively, a non-uniform albedo
ρ
uv
is determined based on a pair of images acquired under different illumination
conditions according to Lena et al. (
2006
) (cf. Sect.
3.3.2
). Equation (
3.16
) is then
solved for the surface gradient
p
uv
for each pixel with intensity
I
uv
. For each image
row
v
, a height profile
z
uv
can be readily obtained by integration of the surface
gradients
p
uv
.
3.2.2.2 Single-Image Approaches with Regularisation Constraints
The classical shape from shading approach regarded in this section is based on
the global optimisation of an energy function according to Horn (
1986
,
1989
) and
Horn and Brooks (
1989
). The presentation in this section is adopted from Wöh-
ler and Hafezi (
2005
). The described method involves searching for two functions
p(x,y)
and
q(x,y)
which imply a surface that generates the observed image in-
tensity
I(x,y)
. The original problem formulation is expressed in the continuous
variables
x
and
y
, resulting in a variational framework. Here we will, however, im-
mediately deal with finite sums over the image pixels, the positions of which are
denoted by the discrete variables
u
and
v
, and rewrite the error integrals introduced
by Horn (
1986
) accordingly. Hence, the intensity constraint can be expressed by the
minimisation of an intensity error term
e
i
with
I
uv
−
R
I
(p
uv
,q
uv
)
2
e
i
=
(3.19)
u,v
with
R
I
(p
uv
,q
uv
)
as the reflectance function of the regarded surface. It is straight-
forward to extend this error term to two or more light sources (Sect.
3.3
). This
section, however, concentrates on the single light source scenario. As the corre-
spondingly defined reconstruction problem is ill-posed, we furthermore introduce
a regularisation constraint
e
s
which requires local continuity of the surface. Such
a smooth surface implies that the absolute values of the derivatives
∂p/∂x
,
∂p/∂y
,
∂q/∂x
, and
∂q/∂y
are small, which results in an error term
e
s
with
∂p
∂x
2
uv
+
∂p
∂y
2
uv
+
∂q
∂x
2
uv
+
∂q
∂y
2
.
e
s
=
(3.20)
uv
u,v
This leads to a minimisation of the overall error
e
=
e
s
+
λe
i
,
(3.21)
where the Lagrange multiplier
λ
denotes the relative weight of the two error terms
e
i
and
e
s
. With the approximations
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