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Fig. 4.5
Enlarged section of
the right diagram of Fig.
4.3
(
f
=
20 mm and
κ
=
2
.
4),
showing that
c
is
approximately proportional to
|
z
−
z
0
|
for small
|
z
−
z
0
|
respectively, according to
ω
u
+
ω
v
2ln
I
2
(ω
u
,ω
v
)
I
1
(ω
u
,ω
v
)
Σ
2
=−
(4.10)
ω
u
,ω
v
(cf. also Subbarao,
1988
), where
ω
u
,ω
v
denotes the average over the coordinates
ω
u
and
ω
v
in frequency space. Only the range of intermediate spatial frequencies
is regarded in order to reduce the influence of noise on the resulting value of
Σ
.If
the amplitude spectrum of the examined image window displays a very low value at
(ω
u
,ω
v
)
, the corresponding amplitude ratio tends to be inaccurate, which may result
in a substantial error of
Σ
. Hence, we first compute
Σ
according to (
4.10
), identify
all spatial frequencies
(ω
u
,ω
v
)
for which the term in brackets in (
4.10
) deviates by
more than one standard deviation from
Σ
, and recompute
Σ
after neglecting these
outliers.
For a given value of
Σ
, the corresponding value of
(z
...
z
0
)
is ambiguous, since
two depth values
z
1
<z
0
and
z
2
>z
0
may correspond to the same value of
Σ
. In our
experiments this twofold ambiguity was avoided by placing the complete surface to
be reconstructed behind the plane at distance
z
0
, implying
z>z
0
. One would expect
Σ
−
z
0
, since ideally the small-aperture image and the large-aperture
image are identical for
z
→∞
for
z
→
z
0
. It was found empirically, however, that due to the
imperfections of the optical system, even for
z
=
z
0
an image window acquired
with a larger aperture is slightly more blurred than the corresponding image window
acquired with a smaller aperture. This remains true as long as the small aperture is
sufficiently large for diffraction effects to be small. As a consequence,
Σ
obtains a
finite maximum value at
z
=
z
0
and decreases continuously for increasing
z
.
The geometric optics-based approach of Pentland (
1987
) and Subbarao (
1988
)
implies that the radius
c
of the circle of confusion is proportional to the value of
b
,
such that the PSF radius in image space (being proportional to
Σ
−
1
) is assumed to
be proportional to the radius
c
of the circle of confusion, implying
Σ
−
1
=
→
z
0
. For the lenses, CCD sensors, and object distances regarded in our experiments
(cf. Sect.
6
), it follows from the models of Pentland (
1987
) and Subbarao (
1988
) that
Σ
−
1
→
0for
z
of some millimetres and
for small radii of the circle of confusion of a few pixels (cf. Fig.
4.5
). D'Angelo and
is proportional to
|
z
−
z
0
|
for small values of
|
z
−
z
0
|
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