Graphics Reference
In-Depth Information
Sect.
1.3
). The unknown scaling factor may be determined based on a priori knowl-
edge about the scene such as the average pixel scale.
In the framework by Horovitz and Kiryati (
2004
), the locality of the influence
of the depth points on the gradient field is only partially removed. Hence, the ap-
proach by d'Angelo and Wöhler (
2006
,
2008
) incorporating sparse depth informa-
tion into the global optimisation scheme presented in Sect.
5.3.1.1
consists of defin-
ing a depth error term based on the surface gradient field and depth differences
between sparse three-dimensional points. The depth difference between two three-
dimensional points at image positions
(u
i
,v
i
)
and
(u
j
,v
j
)
is given by
(z)
ij
=
z
u
j
v
j
−
z
u
i
v
i
.
(5.29)
The corresponding depth difference of the reconstructed surface gradient field is cal-
culated by integration along a path
C
ij
between the coordinates
(u
i
,v
i
)
and
(u
j
,v
j
)
:
(z)
surf
ij
=
(p dx
+
qdy).
(5.30)
C
ij
In our implementation the path
C
ij
is approximated by a list of
K
discrete pixel
positions
(u
k
,v
k
)
with
k
=
1
,...,K
. While in principle any path
C
ij
between the
points
(u
i
,v
i
)
and
(u
j
,v
j
)
is possible, the shortest integration path, a straight line
between
(u
i
,v
i
)
and
(u
j
,v
j
)
, is used here. Longer paths tend to produce larger
depth difference errors because the gradient field is not guaranteed to be integrable.
Using these depth differences, it is possible to extend the global optimisation
scheme introduced in Sect.
5.3.1
by adding the error term
e
z
which minimises the
squared distance between all
N
depth points according to
N
N
(z)
surf
ij
)
2
((z)
ij
−
e
z
=
2
,
(5.31)
(u
j
,v
j
)
−
(u
i
,v
i
)
i
=
1
j
=
i
+
1
where
2
denotes the Euclidean distance in the image plane in pixel units. The
iterative update rule (
5.21
) then becomes
...
+
λ
∂e
I
∂p
+
μ
∂e
Φ
∂p
+
ν
∂e
D
p
(n
+
1
)
uv
=
p
(n)
uv
∂p
u,v
(z)
ij
−
∂(z)
surf
ij
N
N
(z)
surf
ij
+
2
χ
.
(5.32)
(u
j
,v
j
)
−
(u
i
,v
i
)
2
∂p
i
=
1
j
=
i
+
1
An analogous expression is obtained for
q
. The derivatives of
(z)
surf
ij
with respect
to
p
and
q
may only be nonzero if the pixel
(u
k
,v
k
)
belongs to the path
C
ij
and are
zero otherwise. They are computed based on the discrete gradient field. The deriva-
tive depends on the direction
(d
(k
u
,d
(k
v
)
of the integration path at pixel location
(u
k
,v
k
)
with
d
(k)
u
k
and
d
(k)
=
u
k
+
1
−
=
v
k
+
1
−
v
k
according to
u
v
Search WWH ::
Custom Search