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=
an identical slope at
u
u
0
of the fitted sigmoidal profile and the profile according
to (
4.9
) (cf. Sect.
1.4.8
for representations of an ideal edge blurred by a Gaussian
PSF). Krauß (
2006
) represents the dependence of 1
/σ
on
b
for the utilised lens as
being proportional to a Lorentz function of the form
ψ
1
(b)
2
ψ
1
+
ψ
2
(4.13)
+
with
ψ
1
and
ψ
2
as empirical parameters.
4.2.3.1 Definition of the Depth-Defocus Function
In order to obtain a relation between the depth of an object and the PSF radius
σ
according to Kuhl et al. (
2006
), the image plane is assumed to be fixed while the
distance
z
of the object varies by the amount
z
, such that
z
0 refers to an
object in best focus. But since neither
z
nor
z
is known, the functional relation
needs to be modelled with respect to
b
according to
1
b
0
+
b
+
=
1
z
=
1
f
.
(4.14)
A value of
b
0 refers to a defocused object point. Solving (
4.14
)for
b
and
inserting
b
in (
4.12
) yields the 'depth-defocus function'
=
exp
zf
z
−
f
−
b
0
2
1
φ
1
1
φ
2
S
(z)
=
−
+
φ
3
.
(4.15)
4.2.3.2 Calibration of the Depth-Defocus Function
Stationary Camera
Barrois and Wöhler (
2007
) propose a depth from defocus
calibration approach that requires a stationary camera calibrated geometrically, e.g.
with one of the methods described in Sect.
1.4
—for all experiments described here,
the semi-automatic approach by Krüger et al. (
2004
) described in Sect.
1.4.7
is em-
ployed. Two pixel-synchronous images of the calibration pattern shown in Fig.
4.9
are acquired at a small and a large aperture, respectively, as described in Sect.
4.2.2
.
The calibration pattern consists of a random noise pattern on the left, which is
especially suitable for estimating the PSF radius
Σ
in frequency space based on
Fourier-transformed image windows according to (
4.10
), and a chequerboard pat-
tern of known size on the right. The pose of the chequerboard pattern is obtained
at high accuracy by extracting the corner points and applying bundle adjustment
(cf. Sect.
2.1
). Hence, the coordinates of each corner point are known in the camera
coordinate system. The position of the random dot pattern with respect to the che-
querboard pattern and thus the depth
z
of each pixel on the random dot pattern are
also known. A window of size 32
32 pixels is extracted around each pixel on the
random dot pattern. Fitting (
4.15
) with
×
1
/Σ(z)
to the measured
(Σ, z)
data
points yields a depth-defocus function like the one shown in Fig.
4.10
.
S
(z)
=
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