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ground plane is known. For geometrical reasons an 'aperture problem' occurs, since
structures that run parallel to the direction of incident light cannot be evaluated.
To extract the bottom of a surface feature, we make use of the fact that a surface
region with a slope towards the light source appears brighter than a flat surface
region. We thus segment and analyse with the BCC analysis algorithm all regions
from image
I
1
(
I
2
) which are brighter than a given threshold
θ
1
(
θ
2
) and which are
illuminated in one image while unlighted in the other. For the pixels of these regions,
the relations
I
1
>θ
1
(
I
2
>θ
2
) and
I
1
/I
2
>θ
0
(
I
2
/I
1
>θ
0
) must hold, respectively.
The contour lines of these image regions are obtained by BCC analysis and are used
as an initialisation to the active contour algorithm described by Williams and Shah
(
1992
), which then adapts them more accurately to the outline of the bottom of the
surface feature.
The ridges and bottoms in the two images are merged based on the mutual dis-
tance of their centres, measured in pixel coordinates in image
I
1
and
I
2
, respectively,
to form complete surface features. As it is not always evident which object in im-
age
I
1
belongs to which object in image
I
2
, the algorithm is implemented such that
it may suggest combinations based on their mutual distance that can either be ac-
cepted or rejected by the user. Figure
3.1
d shows the result of three-dimensional
reconstruction by shadow analysis.
3.1.2 Shadow-Based Surface Reconstruction from Dense Sets
of Images
Kender and Smith (
1987
) analyse the shadows cast by a surface on itself to recon-
struct its shape, where moving light sources and thus implicitly an infinite number
of images acquired under different illumination conditions are assumed. The az-
imuthal direction of illumination is parallel to the image rows. The method regards
the two-dimensional version of the shape from shadow problem; i.e. it reconstructs
intersections of the surface with planes perpendicular to the
xy
plane, which are
of the form
z(x)
. Several constraints for the depth values of shadowed and illumi-
nated pixels are formulated. For the surface parts between the starting point
x
i
and
the end point
x
e
of a shadow, the surface is located below the straight line through
the points
z(x
i
)
and
z(x
e
)
, while illuminated pixels must be located above that line.
Kender and Smith (
1987
) show that the surface shape can be fully recovered when
the shadows are observed for the continuous range of all possible illumination di-
rections. Also for a finite number of observations, many images are necessary to
obtain a good reconstruction accuracy. To extend the reconstruction to the full sur-
face
z(x,y)
, Kender and Smith (
1987
) propose an iterative relaxation method.
Hatzitheodorou (
1989
) introduces a spline-based representation of the surface
in order to reconstruct the three-dimensional surface shape from a set of images
displaying shadows acquired under different illumination conditions. It is shown
that the illumination angle yields information about the elevation difference be-
tween the starting point and the end point of a shadow and about the surface slope
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