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u
k
,v
k
=
d
(k
u
p
u
k
v
k
∂(z)
surf
ij
∂p
(5.33)
u
k
,v
k
=
∂(z)
surf
ij
d
(k
v
q
u
k
v
k
.
∂q
The update of the surface gradient at position
(u, v)
is then normalised with the
number of paths to which the corresponding pixel belongs. Error term (
5.31
) leads
to the evaluation of
N(N
1
)/
2 lines at each update step and becomes prohibitively
expensive for a large number of depth measurements. Therefore, only a limited num-
ber of randomly chosen lines is used during each update step. Due to the discrete
pixel grid, the width of each line can be assumed to correspond to one pixel. It is
desirable for a large fraction of the image pixels to be covered by the lines. For
randomly distributed points and square images of size
w
−
w
pixels, we found that
about 70 % of all image pixels are covered by the lines when the number of lines
corresponds to
×
10
w
.
This method is termed 'shape from photopolarimetric reflectance and depth'
(SfPRD). It is related to the approach by Wöhler and Hafezi (
2005
) described in
Sect.
5.2
combining shape from shading and shadow analysis using a similar depth
difference error term, which is, however, restricted to depth differences along the
light source direction. The method proposed by Fassold et al. (
2004
) directly im-
poses depth constraints selectively on the sparse set of surface locations with known
depth. As a consequence, in their framework the influence of the depth point on the
reconstructed surface is restricted to its immediate local neighbourhood. Horovitz
and Kiryati (
2004
) reduce this effect based on a surface which interpolates the sparse
depth points. In their framework, the influence of the three-dimensional points on
the reconstructed surface is better behaved but still decreases considerably with in-
creasing distance. In contrast, our method effectively transforms sparse depth data
into dense depth difference data as long as a sufficiently large number of paths
C
ij
is taken into account. The influence of the depth error term is thus extended across a
large number of pixels by establishing large-scale surface gradients based on depth
differences between three-dimensional points.
∼
5.3.4 Experimental Evaluation Based on Synthetic Data
To examine the accuracy of the three-dimensional surface reconstruction methods
described in Sects.
5.3.1
and
5.3.3
in comparison to ground truth data and to re-
veal possible systematic errors, the algorithms are tested on synthetically generated
surfaces.
The evaluation by d'Angelo and Wöhler (
2005c
) regards the integration of depth
from defocus into the SfPR framework based on the synthetically generated surface
shown in Fig.
5.15
a. For simplicity, to generate the locally non-uniform blur, for the
PSF radius
σ
the relation
σ
according to Sect.
4.2.1
is assumed (Pentland,
1987
; Subbarao,
1988
). To obtain a visible surface texture in the photopolarimetric
∝|
z
−
z
0
|
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