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Fig. 5.13
Initialisation of the shape from shading algorithm by shadow analysis according to
Sect.
5.2.3
.(
a
)The
images to the left
in the middle are evaluated with respect to shadow, the
image
to the right
with respect to shading. The reconstructed surface part is marked by a
white rectangle
.
(
b
) Surface patch between the two shadow lines (
hatched
) along with the initial surface profile
derived from shadow analysis
from shading algorithm and compute a reliable value for the surface albedo
ρ
.The
basic idea is developed based on the example of lunar surface reconstruction with
a single light source used for the shape from shading analysis, but the proposed
method can also be used with shape from shading algorithms that rely on multiple
light sources. The shape from shading analysis is performed by making use of error
function (
5.10
) described in Sect.
5.2.2
, which takes into account the fine structure
of the shadow along with the shading information. The approach described in this
section is applicable to arbitrary reflectance functions
R(p,q)
, but for simplicity we
again assume Lambertian reflectance.
Figure
5.13
shows a lunar crater (Theaetetus) at sunrise (illumination from the
right) at solar elevation angles
μ
(
1
)
and
μ
(
2
)
17
.
9
◦
12
.
2
◦
shadow
=
shadow
=
and at sunset
165
.
0
◦
. The third image
is used as the shading image; the eastern (right) half of the crater is reconstructed as
indicated by the rectangular boxes in Fig.
5.13
a. To achieve this, we again follow an
iterative approach.
(illumination from the left) at a solar elevation angle of
μ
=
1. Initially, it is assumed that the depths of the ridges casting the shadows are con-
stant and identical. The iteration index
m
is set to
m
0. Based on the depth
differences with respect to the ridges, the three-dimensional profile
=
˜
z
m
(u, v)
of
the small surface patch between the two shadow lines can be derived from the
shadow lengths measured according to Sect.
5.2.1
(cf. Fig.
5.13
b).
2. The surface profile
z
m
(u, v)
directly yields the slopes
p
(
0
)
and
q
(
0
)
˜
uv
in the hori-
zontal and vertical image directions, respectively, for all pixels belonging to the
surface patch between the shadow lines. The known values of
p
(
0
)
uv
uv
and
q
(
0
)
uv
are
used to compute the albedo
ρ
, and they serve as initial values inside the surface
patch between the two shadow lines for the shape from shading algorithm. These
values are kept constant throughout the following steps of the algorithm. Outside
the region between the shadow lines, the initial values of
p
(
0
)
and
q
(
0
)
are set to
uv
uv
zero.
3. Using the shape from shading algorithm with the initialisation applied in the
previous step, the complete surface profile
z
(m)
is reconstructed based on the
uv
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