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integrability constraint has the drawback of a small convergence radius, such that
it needs to be initialised with a surface profile which is already close to the final
solution (Horn,
1989
).
In the single-image shape from shading scenario, the surface albedo
ρ
uv
has al-
ways been regarded as known. If this is not the case, in many applications the as-
sumption of a uniform albedo
ρ
is made. Different albedo values yield different
solutions of the shape from shading problem, since e.g. increasing
ρ
will result in a
surface inclined away from the light source and vice versa. Hence, it is often neces-
sary to make additional assumptions about the surface, e.g. that the average surface
slope is zero or obtains a predefined value. In Sect.
3.3
methods to cope with un-
known and non-uniform surface albedos are described.
3.2.3 Reconstruction of Height from Gradients
Local techniques for computation of height from gradients rely on curve integrals
and are based on specifying an integration path and a local neighbourhood. Accord-
ing to the technique described by Jiang and Bunke (
1997
), reconstruction of height
is started at a given point
(u
0
,v
0
)
of the image, e.g. the centre, for which
z
u
0
,v
0
=
0
is assumed, and the initial paths are forming a cross along image column
u
0
and
image row
v
0
(cf. also Klette and Schlüns
1996
). The image origin is in the upper
left corner. For the upper right quadrant, the height value
z
uv
is obtained according
to
z
u
−
1
,v
+
q
uv
)
(3.31)
1
2
1
2
(p
u
−
1
,v
+
1
2
(q
u,v
+
1
+
z
uv
=
p
uv
)
+
z
u,v
+
1
+
Analogous relations for the remaining three quadrants are readily obtained. In (
3.31
)
deviations of the surface gradient field from integrability are accounted for by aver-
aging over the surface gradients in the horizontal and vertical directions. A drawback
of this method is that the resulting height map
z
uv
depends on the initial location
(u
0
,v
0
)
.
In a more systematic way than in the rather elementary approach by Jiang and
Bunke (
1997
), the three-dimensional reconstruction scheme based on the integrabil-
ity error outlined in Sect.
3.2.2.2
can be used for adapting a surface to the generally
non-integrable surface gradient field obtained by shape from shading. It is desired to
obtain a surface
z
uv
with partial derivatives
}
uv
which come
as close as possible to the values
p
uv
and
q
uv
previously obtained by shape from
shading, which are assumed to be known and fixed. The depth map
z
uv
is chosen
such that the integrability error (
3.25
) is minimised. Accordingly, it is shown by
Horn (
1986
) that (
3.28
) and (
3.29
) directly yield an iterative scheme to determine
the height map
z
uv
(cf. also (
3.30
)):
{
∂z/∂x
}
uv
and
{
∂z/∂y
∂p
∂x
∂q
∂y
.
ε
2
κ
z
(n
+
1
)
uv
z
(n)
uv
=¯
−
uv
+
(3.32)
uv
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