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Fig. 4.15
Focal length
f
2
according to (
4.21
) required
to obtain the same relative
depth resolution
|
h
−
1
|
as a
lens of
f
1
=
25 mm at a
distance of
z
0
=
0
.
5m(
open
circle
)
setting
c
1
=
c
2
yields the focal length
f
2
according to
L
1
K
−
L
2
2
±
−
1
L
2
4
f
2
=
M
+
.
(4.21)
Only the solution with the plus sign before the square root yields positive values
for
f
2
. Close i
n
spection of (
4.21
) reveals that the value of
f
2
is approximately pro-
portional to
√
k
independent of the chosen value of
h
as long as
|
h
−
1
|
1. Hence,
for constant f-stop number
κ
, constant relative variation
of the object depth
z
,
and constant pixel size, the requi
re
d focal length and thus also the aperture of the
lens are largely proportional to
√
z
0
. Figure
4.15
depicts the focal length
f
2
accord-
ing to (
4.21
) required to obtain the same relative depth resolution
|
h
−
1
|
|
h
−
1
|
as a lens
of
f
1
=
0
.
5 m. Hence, the depth from defocus approach
is favourably used in the close-range domain as long as standard video cameras and
lenses are used, since otherwise the field of view would become too small and the
required lens aperture too large.
25 mm at a distance of
z
0
=
4.3 Reconstruction of Depth from Focus
In contrast to depth from defocus, depth from focus uses images taken by a sin-
gle camera at different focus settings to compute depth. The classical depth from
focus approach is introduced by Nayar (
1989
). He points out that the PSF of an
optical system not being in focus yields a low-pass filtered version of the origi-
nal image, where the high spatial frequencies of the original image are suppressed
most strongly. Hence, for determining the influence of the PSF, the acquired image
of the analysed surface has to display variations of the pixel grey levels on small
spatial scales, which is inherently the case for rough surfaces and can be achieved
by the projection of light patterns for smooth surfaces. The values of the princi-
pal distance
b
and the focal length
f
are determined based on a calibration pro-
cedure, such that the value of
z
0
is directly given by (
4.3
). If the depth deviates
from the value
z
0
in (
4.3
) for which the scene point appears best focused while
b
and
f
are kept constant, the image of the scene point becomes blurred. Nayar
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