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5.3.3.1 Fusion of SfPR with Depth from Defocus
To obtain a dense depth map of the surface, we employ the two-image depth from
defocus method described in Sect.
4.2.2
. Once the characteristic curve
σ(z
−
z
0
)
which relates the PSF radius
σ
to the depth offset
(z
−
z
0
)
is known (cf. Fig.
4.6
),
it is possible to extract a dense depth map from a pixel-synchronous pair of images
of a surface of unknown shape, provided that the images are acquired at the same
focus setting and with the same apertures as the calibration images. The resulting
depth map
z
DfD
uv
, however, tends to be very noisy as illustrated in Fig.
4.7
(vari-
ables representing results obtained by depth from defocus are marked by the index
'DfD'). It is therefore favourable to fit a plane
z
DfD
uv
to the computed depth points,
since higher order information about the surface is usually not contained in the noisy
depth from defocus data. This procedure reveals information about the large-scale
properties of the surface (d'Angelo and Wöhler,
2005c
). Approximate surface gra-
dients can then be obtained by computing the partial derivatives
p
DfD
uv
˜
z
DfD
uv
=
∂
˜
/∂x
and
q
DfD
z
DfD
=
˜
uv
/∂y
.
In many cases there exists no unique solution for the surface gradients
p
uv
and
q
uv
within the SfPR framework, especially for highly specular reflectance func-
tions. This applies both to the global (Sect.
5.3.1.1
) and to the local (Sect.
5.3.1.2
)
optimisation schemes. Therefore, the obtained solution tends to depend strongly on
the initial values
p
(
0
)
∂
uv
uv
and
q
(
0
)
uv
. As we assume that no a priori information about
the surface is available, we initialise the optimisation scheme with
p
(
0
)
p
DfD
uv
=
uv
and
q
(
0
)
q
DfD
uv
, thus making use of the large-scale surface gradients obtained
by the depth from defocus analysis. The ambiguity of the solution of the global
optimisation scheme is even more pronounced when no a priori knowledge about
both the surface gradients and the albedo is available. In such cases, which are of-
ten encountered in practically relevant scenarios, an initial albedo value is com-
puted according to (
5.24
) based on the initial surface gradients
p
DfD
uv
=
uv
and
q
DfD
uv
.
We found experimentally that it is advantageous to keep this albedo value con-
stant during the iteration process as long as no additional constraints can be im-
posed on the surface, since treating the albedo as a further free parameter in
the iteration process increases the manifold of local minima of the error func-
tion.
The depth from defocus data are derived from two images acquired with a large
and a small aperture, respectively. In practise, it is desirable but often unfeasible
to use the well-focused image acquired with small aperture for three-dimensional
reconstruction—the image brightness then tends to become too low to obtain rea-
sonably accurate polarisation data. The surface reconstruction algorithm thus may
have to take into account the position-dependent PSF. We incorporate the depth from
defocus information into the global optimisation scheme, since it is not possible to
introduce PSF information (which applies to a local neighbourhood of a pixel) into
an approach based on the separate evaluation of each individual pixel. The error
terms (
3.19
), (
5.18
), and (
5.19
) of the SfPR scheme described in Sect.
5.3.1
are
modified according to
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