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as accurately as possible. The surface gradients
p
uv
determined based on (
3.48
)are
used as initial values for the corresponding iterative update rule (
3.24
)or(
3.30
).
Accordingly,
p
uv
hardly changes in the course of the iteration process, while
q
uv
obtains values consistent with the smoothness constraint (
3.20
) or the integrability
constraint (
3.25
).
3.3.2.2 Ratio-Based Variational Photometric Stereo Approach
Another approach to cope with a non-uniform surface albedo
ρ
uv
is to return to the
single-image shape from shading schemes in Sect.
3.2.2
and to replace the single-
image intensity error term (
3.19
) by the modified ratio-based error term
I
(
1
)
1
2
R
I
(
s
2
,p
uv
,q
uv
)
uv
e
i
=
˜
R
I
(
s
1
,p
uv
,q
uv
)
−
(3.50)
I
(
2
)
uv
u,v
as proposed by Wöhler and Hafezi (
2005
), who infer the ratio-based iterative update
rule
p
(n)
λ
I
(
1
)
1
I
(
1
)
p
(n)
q
(n)
R
I
(
s
2
,
R
I
(
s
2
,p
uv
,q
uv
)
R
I
(
s
1
,p
uv
,q
uv
)
¯
uv
,
¯
uv
)
∂
∂p
p
(n
+
1
)
p
(n)
uv
uv
I
(
2
)
=¯
+
uv
)
−
uv
uv
I
(
2
)
p
(n)
q
(n)
R
I
(
s
1
,
¯
uv
,
¯
q
(n)
uv
uv
,
¯
uv
uv
(3.51)
p
(n)
λ
I
(
1
)
1
I
(
1
)
R
I
(
s
2
,
p
(n)
q
(n)
R
I
(
s
2
,p
uv
,q
uv
)
R
I
(
s
1
,p
uv
,q
uv
)
¯
uv
,
¯
uv
)
∂
∂q
uv
uv
I
(
2
)
q
(n
+
1
)
q
(n)
=¯
+
uv
)
−
.
uv
uv
I
(
2
)
p
(n)
q
(n)
R
I
(
s
1
,
¯
uv
,
¯
uv
, q
(n)
uv
uv
uv
The non-uniform albedo
ρ
uv
cancels out and can be recovered by (
3.49
) after deter-
mination of the surface gradients
p
uv
and
q
uv
. The properties of the results obtained
with the ratio-based iterative scheme (
3.51
) are comparable to those obtained by
single-image shape from shading analysis, except that one is not restricted to sur-
faces of uniform albedo. For more than two light sources and images (
L>
2), the
error term (
3.50
) can be extended to a sum over all
L(L
1
)/
2 possible pairs of
images, which reveals surface gradients in all directions if the light sources are ap-
propriately distributed.
A drawback of the presented method is the fact that it must be initialised with a
surface which is already close to the final solution; otherwise the algorithm diverges
or becomes stuck in local minima. Hence, as long as the albedo variations are not
strong, it is shown by Wöhler and Hafezi (
2005
) that it is advantageous to combine
the two shape from shading approaches described in this section as follows. First, the
surface profile is reconstructed using the multi-image intensity error (
3.43
), resulting
in values for both
p
uv
and
q
uv
and a uniform albedo
ρ
. As a second step, the iterative
update rule (
3.51
) is initialised with the results of the first step and then started. This
procedure changes the surface profile only at the locations of albedo variations. The
albedo map
ρ
uv
is then obtained according to (
3.49
). As a third step,
p
uv
and
q
uv
are recomputed according to (
3.43
). These three steps are repeated until
p
uv
,
q
uv
,
and
ρ
uv
converge towards a self-consistent solution. Experimental results obtained
with this approach are described in detail in Sect.
6.3
.
−
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