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in the direction of incident light for the starting point of a shadow. Another con-
straint taken into account is (similar to Kender and Smith
1987
) that the surface
between the starting point and the end point of a shadow must be located below
the connecting straight line. According to the experimental evaluation, the method
of Hatzitheodorou (
1989
) does not require a continuously moving light source but
is able to perform a three-dimensional surface reconstruction based on images in
which the surface is illuminated under four to six different illumination angles in
the horizontal and the vertical image direction, respectively.
The previously described shadow constraints are formulated as graphs by Yu and
Chang (
2002
). They develop a formal framework to constrain the shape of a sur-
face based on multiple images and show that the surface shape can be reconstructed
based on the resulting shadow graph when the set of images acquired under differ-
ent illumination conditions is infinitely large. To allow a three-dimensional surface
reconstruction from a moderate but still considerable number of about 10 images,
Yu and Chang (
2002
) propose a technique to integrate shadow and shading infor-
mation, for which it is required that the images be illuminated from a large variety
of known directions. For a small number of pixels, reasonably good initial depth
values should be given, such that the shadow constraints allow one to infer depth
bounds for all pixels. Depth values obtained by shape from shading are then made
consistent with the bounds derived from shadow information based on a constrained
optimisation procedure.
The shape from shadow approach is used in combination with silhouette informa-
tionbySavareseetal.(
2002
) to estimate the three-dimensional shape of an object.
They apply a technique termed space carving for an initial estimation of the three-
dimensional shape of the object relying on a set of silhouette images acquired from
different viewpoints. For a refinement of the shape of the object, a second set of
images is required where point light sources are used for illumination. Based on the
deviations between the self-shadows on the surface which would result from the ini-
tially estimated shape and those observed in the images, an adjustment of the object
shape is performed to minimise the deviations from the image data. This combined
method reveals the object shape much more accurately than shape from silhouettes
alone, but it still requires many images to recover fine shape details.
The combination of shadow analysis with other three-dimensional surface recon-
struction methods is described in Sect.
5.2
.
3.2 Shape from Shading
This section describes methods for three-dimensional reconstruction of object sur-
faces which are based on the pixel grey values of a single image. Different surface
reflectance models are introduced in Sect.
3.2.1
. Many intensity-based methods rely
on the determination of the surface gradients (cf. Sect.
3.2.2
). Early approaches
aiming for a reconstruction of height profiles along image rows (photoclinometry)
emerged in the domain of remote sensing (Wilhelms,
1964
) (cf. Sect.
3.2.2.1
). Un-
der certain assumptions about the surface it is also possible to determine a full height
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