Graphics Reference
In-Depth Information
L
L
I
−
R
I
p
(n)
uv
∂R
I
Φ
−
R
Φ
p
(n)
uv
∂R
Φ
p
(n
+
1
)
uv
=
p
(n)
uv
uv
, q
(n)
uv
, q
(n)
+
λ
∂p
+
μ
∂p
l
=
1
l
=
1
L
D
R
D
¯
uv
∂R
D
∂p
p
(n)
q
(n)
+
ν
−
uv
,
¯
,
(5.21)
l
=
1
where
n
denotes the iteration index. A corresponding expression for
q
is obtained
in an analogous manner (cf. also Sect.
3.2.2
). The initial values
p
(
0
)
and
q
(
0
)
uv
must
be provided based on a priori knowledge about the surface or on independently
obtained depth data (cf. Sect.
5.3.3
). The partial derivatives in (
5.21
) are evaluated
at
( p
(n)
uv
uv
, q
(n)
uv
)
, respectively, making use of the phenomenological model fitted to the
measured reflectance and polarisation data. The surface profile
z
uv
is then derived
from the resulting gradients
p
uv
and
q
uv
by means of numerical integration of the
gradient field according to the method suggested by Simchony et al. (
1990
).
Note that for the computation of the derivatives of
R
I
,
R
Φ
, and
R
D
with respect
to the surface gradients
p
and
q
, as required to apply the iterative update rule (
5.21
),
(
5.17
) must be taken into account if the azimuth angles of the light sources are
different from zero.
5.3.1.2 Local Optimisation Scheme
Provided that the model parameters of the reflectance and polarisation functions
R
I
(p
uv
,q
uv
)
,
R
Φ
(p
uv
,q
uv
)
, and
R
D
(p
uv
,q
uv
)
are known and measurements of
intensity and polarisation properties are available for each image pixel, d'Angelo
and Wöhler (
2005b
,
2008
) show that the surface gradients
p
uv
and
q
uv
can be ob-
tained by solving the nonlinear system of equations (
5.14
)-(
5.16
) individually for
each pixel. For this purpose they make use of the Levenberg-Marquardt algorithm
(Press et al.,
2007
). In the overdetermined case, the root of (
5.14
)-(
5.16
) is com-
puted in the least-squares sense. The contributions from the different terms are then
weighted according to the corresponding measurement errors. In the application sce-
nario regarded in Sect.
6.3
, these standard errors have been empirically determined
to
σ
I
≈
10
−
3
I
spec
with
I
spec
as the intensity of the specular reflections,
σ
Φ
≈
0
.
1
◦
,
and
σ
D
≈
0
.
02. The surface profile
z
uv
is again derived from the resulting gradients
p
uv
and
q
uv
by means of numerical integration of the gradient field (Simchony et
al.,
1990
).
It is straightforward to extend this approach to photopolarimetric stereo, because
each light source provides an additional set of equations. Equation (
5.14
) can only
be solved, however, when the surface albedo
ρ
uv
is known for each surface point.
A constant albedo can be assumed in many applications. If this assumption is not
valid, albedo variations strongly affect the accuracy of surface reconstruction. In
Sect.
3.3.2
it is shown that as long as the surface albedo can be assumed to be of
the form (
3.47
), it is then possible to utilise two images
I
(
1
)
uv
and
I
(
2
)
uv
acquired under
different illumination conditions. Equation (
5.14
) can then be replaced by the ratio-
based relation (
3.48
) such that the albedo cancels out (McEwen,
1985
; Wöhler and
Hafezi,
2005
; Lena et al.,
2006
).
Search WWH ::
Custom Search