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results described later, the value of
c
is computed for a Baumer CCD camera with
square pixels of size 4
.
65
m).
It is assumed by Subbarao (
1988
) that the PSF radius
σ
is related to
c
by
μ
=
σ
γc
(4.5)
with a camera-specific value of
γ
, and it is pointed out that an observed image
region
I
uv
is obtained from the corresponding focused image region
I
(
0
)
by
I
=
I
(
0
)
, where '
G
σ
∗
' denotes a convolution operation. According to Chaudhuri and
Rajagopalan (
1999
), under the assumption of a Gaussian PSF the difference between
the squared PSF widths for two images of the same object acquired under different
focus settings corresponds to
∗
2
ω
u
+
ln
I
1
(ω
u
,ω
v
)
σ
1
−
σ
2
=−
I
2
(ω
u
,ω
v
)
,
(4.6)
ω
v
where
I
2
(ω
u
,ω
v
)
are the amplitude spectra obtained for the two
focus configurations. Note that in frequency space, the convolution becomes an
element-wise multiplication. Subbarao (
1988
) points out that, in principle, only one
pair of amplitudes
I
1
(ω
u
,ω
v
)
and
I
2
(ω
u
,ω
v
)
is required to compute
(σ
1
−
σ
2
)
,but
I
1
(ω
u
,ω
v
)
and
averaging over a larger domain of the amplitude spectrum is favourable.
For two images blurred due to different focus settings and thus (in the general
case) different principal distances
b
1
and
b
2
, different focal lengths
f
1
and
f
2
, and
different lens apertures
r
1
and
r
2
, Subbarao (
1988
) and Chaudhuri and Rajagopalan
(
1999
) establish a linear relation between
σ
1
and
σ
2
according to
γr
1
b
1
1
r
1
b
1
r
2
b
2
1
b
1
−
1
f
2
+
1
b
2
σ
1
=
ασ
2
+
β
with
α
=
and
β
=
f
1
−
(4.7)
and derive from (
4.7
) the relation
σ
1
−
σ
2
=
α
2
1
σ
2
+
β
2
.
−
2
αβσ
2
+
(4.8)
The value of
σ
2
is then obtained from (
4.8
) when the value of
(σ
1
−
σ
2
)
determined
according to (
4.6
) is inserted into (
4.8
).
Pentland (
1987
) suggests an approach to determining the amount of defocus in
the image which assumes that intensity transitions along object borders are ideal step
functions in the real world, which are blurred by the optical system of the camera.
For this purpose, Pentland (
1987
) analyses Laplace-filtered blurred images in order
to determine the PSF radius.
As an example, two intensity profiles extracted orthogonal to an object boundary,
displaying different amounts of defocus, are shown in Fig.
4.4
. According to Krüger
and Wöhler (
2011
), the intensity profile
I(u)
generated by blurring an ideal edge
with a Gaussian PSF can also be represented by a function of the form
a
erf
u
u
0
√
2
σ
−
=
+
I(u)
b
(4.9)
with
u
as the pixel coordinate orthogonal to the boundary,
a
as the amplitude of
the edge,
b
as an offset parameter,
u
0
as the position of the steepest brightness
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