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by Wöhler and Hafezi
2005
and Lena et al.
2006
), and a much more general class
of reflectance characteristics.
3.3.2.1 Ratio-Based Photoclinometry of Surfaces with Non-uniform Albedo
Similar to the scenario of single-image photoclinometry (cf. Sect.
3.2.2.1
), illumi-
nation of the scene along the image rows (
q
s
=
0) is assumed. For oblique illumina-
tion, the dependence of the reflectance map on
p
is much more pronounced than the
dependence on
q
, provided that it has no strongly specular component. Hence, we
again approximate the surface gradient
q
perpendicular to the direction of incident
light with zero values. The presentation in this section is adopted from Wöhler and
Hafezi (
2005
) and Lena et al. (
2006
).
Two images of the surface acquired under different illumination angles are re-
quired to separate intensity variations due to topographic relief from those due to
albedo variations. The images have to be pixel-synchronous, which is achieved by
an image registration step (Gottesfeld Brown,
1992
). We do not have to restrict our-
selves to Lambertian reflectance; instead we assume a reflectance map of the form
ρ
uv
R
I
(
s
,p
uv
,q
uv
).
(3.47)
Photoclinometry is then performed along image rows by extending (
3.16
)asde-
scribed by McEwen (
1985
) and determining
p
uv
such that
I
(
1
)
R
I
(ρ
uv
,
s
,p
uv
,q
uv
)
=
R
I
(
s
1
,p
uv
,q
uv
)
R
I
(
s
2
,p
uv
,q
uv
)
uv
I
(
2
)
uv
=
.
(3.48)
The surface gradients
q
uv
are still kept zero, and the albedo
ρ
uv
cancels out, such
that (
3.48
) directly yields the surface gradient
p
uv
individually for each pixel. It
is generally not possible to obtain an analytical solution for
p
uv
; thus numerical
techniques like the Newton method have to be used.
When the images are not absolutely calibrated radiometrically, their average pixel
grey values must be adjusted according to the different illumination angles, relying
on the assumption that on the average the surface is flat. As long as
R
I
(p, q)
is not
strongly nonlinear in
p
and
q
, it is sufficient to normalise
I
(
1
)
by multiplying its
uv
pixel grey value with the factor
R
I
(
s
1
,
0
,
0
)
I
(
2
)
u,v
uv
.
R
I
(
s
2
,
0
,
0
)
I
(
1
)
u,v
uv
The non-uniform surface albedo
ρ
uv
is then recovered by
L
I
(l)
1
L
uv
R
I
(
s
l
,p
uv
,q
uv
)
ρ
uv
=
.
(3.49)
l
=
1
In the next step, the albedo map
ρ
uv
is inserted into one of the single-image shape
from shading schemes described in Sect.
3.2.2
, preferably relying on the image ac-
quired under the more oblique illumination conditions in order to extract the relief
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