Biomedical Engineering Reference
In-Depth Information
ideal circular aperture with a radius of
r
could
be expressed as
compensate for each other's aberrations, as pre-
viously discussed). Many other image degrada-
tions such as misfocus, sensor vibration (relative
movement), and atmospheric turbulence can
also be modeled with an approximate PSF that
contributes, through convolution, to the overall
PSF. If specific details regarding the degradation
are known, it can sometimes be mitigated
through careful image processing, depending
on the noise in the image
[18]
.
x
a
+
y
a
r
,
1,
A
(
x
a
,
y
a
)
=
(1.6)
0,
x
a
+
y
a
>
r
,
where
(
x
a
,
y
a
)
are spatial coordinates at the
aperture plane. The aperture function sets the
limits of the field distribution (which is usually
determined using the Fraunhofer diffraction
approximation
[10]
).
Previously, we saw that the intensity pattern
at the focal plane of a lens, due only to diffrac-
tion of a circular aperture, resulted in an Airy
disk. It turns out that this intensity pattern at
the focal plane is proportional to the squared
magnitude of the Fourier transform of the aper-
ture function. That is,
h
A
(
x
,
y
)
∝|
F
{
A
(
x
a
,
y
a
)
}|
2
,
where
h
A
(
x
,
y
)
is the PSF at the focal plane of the
circular aperture. Normalized plots of
h
A
(
x
,
y
)
for a circular aperture are shown in
Figure 1.7
;
compare that figure to
Figures 1.3 and 1.4
. If a
point of light
δ(
x
o
,
y
o
)
from the object plane
passes through this aperture to the focal plane,
then the aperture PSF
h
A
(
x
,
y
)
is convolved with
δ(
x
,
y
)
, and by the sifting property of delta func-
tion the result is
h
A
(
x
,
y
)
. Thus, the smallest pos-
sible blur spot due to the aperture is the same
Airy disk as we found before, only we can now
use the power of the Fourier transform and
linear systems theory to extend the analysis.
If the lens is nonideal, it will also contribute
(through convolution) a PSF
h
L
(
x
,
y
)
that devi-
ates from a delta function. The PSF
h
L
(
x
,
y
)
is
determined primarily by the various lens
aberrations that are present. The combined PSF
of the aperture and the lens is thus
h
AL
(
x
,
y
)
=
h
A
(
x
,
y
)
∗
h
L
(
x
,
y
)
, where the * symbol
denotes convolution. The combined PSF
h
AL
(
x
,
y
)
is convolved with an ideal image (from purely
geometrical optics) to obtain the actual image at
the focal plane. If multiple lenses are used, they
each contribute a PSF via convolution in the
same way (unless arranged in such a way as to
1.2.2.2 Optical Transfer Function, Modulation
Transfer Function, and Contrast Transfer
Function
The Fourier transform of the PSF yields
the
optical transfer function
(OTF). That is,
H
(
u
,
v
)
=
F
{
h
(
x
,
y
)
}
, where
(
u
,
v
)
are the spatial-
frequency coordinates at the focal plane. The
PSF is in the spatial domain, and the OTF is in
the spatial frequency domain, both at the focal
plane. For most incoherent imaging systems, we
are most interested in the magnitude of the OTF,
called the
modulation transfer function
(MTF). See
Figure 1.8
for a normalized plot of the MTF due
only to the PSF of a circular aperture (e.g., the
circular aperture PSF shown in
Figure 1.7
).
In this context, modulation
m
is a description
of how a sinusoidal pattern of a particular spatial
frequency at the object plane can be resolved. It
quantifies the contrast between the bright and
dark parts of the pattern, measured at the image
plane. Specifically,
m
=
(max
−
min)/
(
max
+
min
)
.
From
Figure 1.8
, you can see that as the spatial
frequency increases, the ability of the optical sys-
tem to resolve the pattern decreases, until at some
frequency the pattern cannot be discerned. Note
the MTF in
Figure 1.8
is zero at
u
D
/λ
; thus,
D
/λ
is called the
cutoff frequency
f
c
. However, no
real-world optical system can detect a sinusoidal
pattern all the way out to
f
c
; the practical contrast
limit (sometimes called the
threshold modulation
)
for the MTF is not zero but more like 2%, 5%, or
higher, depending on the system and the observer.
The
goodness
of an MTF relates to how high the
modulation level remains as frequency increases,
