Graphics Reference
In-Depth Information
This section describes the integration of an additional error term according to
Herbort et al. (
2011
) and Grumpe et al. (
2011
) into the variational shape from shad-
ing approach of Horn (
1989
), which has been described in detail in Sect.
3.2.2.2
.
This error term
e
DEM
measures the deviation between the gradients of the actively
scanned depth data
z
DEM
uv
on spatial scales larger than the extension of the image
pixels (the index DEM stands for 'digital elevation model') and is defined as
f
σ
DEM
∂z
DEM
∂x
f
σ
DEM
(p
uv
)
2
e
DEM
=
−
uv
uv
f
σ
DEM
∂z
DEM
∂y
f
σ
DEM
(q
uv
)
2
,
+
−
(5.36)
uv
where, in contrast to the continuous formulation by Grumpe et al. (
2011
), the dis-
crete notation in terms of the pixel coordinates
u
and
v
is used. In (
5.36
), the function
f
σ
DEM
denotes a low-pass filter which is in principle arbitrary but is implemented by
Grumpe et al. (
2011
) as a Gaussian filter of half width
σ
DEM
. The overall error term
g
then corresponds to
δe
DEM
,
(5.37)
with
γ
and
δ
as weight factors, where
e
i
and
e
int
are defined by (
3.19
) and (
3.25
),
respectively. According to Grumpe et al. (
2011
), setting the derivatives of
g
with
respect to the surface gradients
p
and
q
to zero results in an iterative update rule
according to
g
=
e
i
+
γe
int
+
z
(n)
γ
I
uv
−
R
uv
z
(n)
,z
(n
y
∂R
uv
∂p
1
p
(n
+
1
)
z
(n)
=
+
uv
x
x
x
f
σ
DEM
∂z
DEM
∂x
f
σ
DEM
z
(n
x
γ
ij
δ
+
−
ij
z
(n)
(5.38)
∂f
σ
DEM
(p
uv
)
∂p
×
ij
x
∂p
(n
+
1
)
∂x
∂q
(n
+
1
)
∂y
ε
2
κ
z
(n
+
1
)
z
(n)
=¯
−
uv
+
uv
uv
uv
z
(n)
with
uv
as the average over the
κ
nearest neighbours of the regarded pixel and
as the lateral extent of the pixels. Equation (
5.38
) is an extension of the iterative
update rule introduced by Horn (
1989
) for determining an integrable surface. An
expression for
q
(n
+
1
)
¯
uv
analogous to the first part of (
5.38
) is obtained in a straight-
forward manner. Non-uniform lateral pixel extensions or unequal lateral extensions
of the pixels in the horizontal and vertical image directions (as is the case for certain
map projections) must be taken into account by replacing
by appropriately chosen
pixel-specific values.
The sum over
u
and
v
in the first part of (
5.38
) corresponds to the partial deriva-
tive
∂e
DEM
/∂p
. At this point it is favourable to introduce the
K
×
L
filter matrix
F
σ
DEM
, leading to the expression
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