Biomedical Engineering Reference
In-Depth Information
How is this pertinent to someone developing
biomimetic vision sensors? For any type of
vision sensor (biomimetic or traditional), the
basic trade-offs of the optics and the spatial sam-
pling remain the same, so knowledge of these
concepts is needed to intelligently guide sensor
development.
FIGURE 1.2 Illustration of the snow fence to be imaged
by the digital camera.
1.2.1.2 Effect of Aperture Size
Another basic concept that is often important
to sensor development is diffraction. No real-
world lens can focus light to an infinitesimally
tiny point; there will be some minimum blur
spot. Figure 1.3 depicts the blur spots due to two
simple optical setups with circular apertures,
where the top setup has a larger aperture than
the bottom one. In the figure, if the object plane
is at optical infinity, then d = f . The diameter of
the aperture is shown as D ; this could be due to
the physical diameter of the lens in a very sim-
ple optical setup, or to the (sometimes variable)
aperture diaphragm of a more complex lens sys-
tem. 2 As light travels from the lens to the image
plane, differences in path length are inevitable.
Where the difference in path length equals
some integer multiple of λ/ 2 , a lower-intensity
(dark) region appears; where the difference in
path length equals some integer multiple of λ ,
a higher-intensity (bright) region appears. With a
circular aperture, the blur spot will take the shape
of what is often called an Airy disk . The angular
separation between the center peak and the first
minimum of an Airy disk, as shown in Figure
1.3 , is θ = 1. 22λ/ D , which confirms the inversely
proportional relationship between the blur spot
diameter and the aperture diameter. The value
of θ is often referred to as the angular resolu-
tion , assuming the use of what is known as the
Rayleigh criterion [7] .
A cross-section of an Airy disk is shown in
Figure 1.4 . Though the angular measure from
the peak to the first minimum of the blur spot
optical axis of the camera is perpendicular to the
fence, so you can neglect any possible angular
distortions.
Solution: The periodic nature of the slats is
not a sinusoidal pattern (it is actually closer to
a square wave), but the spatial period of the
slats is equal to the fundamental frequency of a
Fourier sum that would model the image of the
fence, and the individual slats will be visible
with acceptable fidelity (for this specific appli-
cation) if this fundamental frequency is sam-
pled properly [18] . The sampling theorem
requires a minimum of two samples per cycle;
one complete cycle at the fundamental fre-
quency is a single slat/opening pair. Thus, the
200 mm slat plus a 200 mm opening at the object
plane must span two (or more) pixels at the
image plane for adequate sampling to occur. In
other words, the pixel spacing, mapped to the
object plane, must be 200 mm or less in the hori-
zontal direction (the vertical direction is not as
important for this image). At the image plane,
the pixel spacing is 19. 2mm / 1280 = 15 µ m .
Referring back to Figure 1.1 , we know that
s i = 22. 5 mm since the object plane is at
optical infinity. Using similar triangles,
we get (22. 5 mm/15 µ m) = ( s o /200 mm) , thus
s o = 300 m, which is the maximum distance
allowed from the camera to the snow fence. If
the camera is placed farther away than 300 m,
the fundamental spatial frequency of the snow
fence will alias as described by Eq. (1.3 ), and the
image would likely be unacceptable.
2 A variable circular aperture is often called an iris diaphragm , since it acts much in the same way as the iris of an eye.
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