Biomimetic Vision Sensors - Engineered Biomimicry - page 16

where x d is the size of the photosensitive area in
the x direction ( Figure 1.10 ). While the magni-
tude of a sinc function extends to infinity (so
technically there is no associated cutoff fre-
quency with this MTF), it is common to consider
the MTF of an FPA only up to the first zero of
the sinc, which occurs at 1/ x d . Thus, a larger
photosensitive area may gather more light but
results in the first zero occurring at a lower spa-
tial frequency, which induces more blurring in
the image. Note that the units of spatial fre-
quency in Eq. (1.8) for the MTF of an FPA are
linear units (e.g., cycles/meter); when optical
MTFs were previously discussed, we followed
the common convention of using angular units.
Angular spatial frequency divided by the focal
length of the optical system equals linear spatial
frequency at the focal plane, assuming the small
angle approximation holds.
A commonly used mathematical description
of image sampling is to convolve the point-spread
function of all the nonideal aspects of the imaging
system (optics, sensor array, and associated elec-
tronics) with an ideal continuous image source,
followed by sampling via an ideal basis function
such as a train of delta functions [15, 26] . Thus,
for an image source sampled in the x direction,

then F s = 1/ X s is the spatial sampling frequency.
From the sampling theorem, we can conclude
that any spatial frequencies sampled by the FPA
that are higher than one-half the sampling fre-
quency, or F s /2 = 1/2 X s , will be aliased as
described by Eq. (1.3) . With no dead space
between detectors (i.e., a fill factor of 1), the first
zero of the FPA's MTF occurs at F s , so the FPA
will respond to spatial frequencies well above
F s /2 , which means aliasing is likely. Lower fill
factors exacerbate the potential for aliasing, since
a smaller detector size moves the first zero of the
MTF to a higher spatial frequency.
This may or may not present a problem,
depending on the application. Monochrome
aliasing tends to be less objectionable to human
observers than color aliasing, for example. 9 Real-
world images are often bandlimited only by the
optical cutoff frequency of the optics used to
form the image. This optical cutoff frequency is
often considerably higher than F s /2 , and in that
case aliasing will be present. However, some
FPAs come with an optical low-pass filter in the
form of a birefringent crystal window on the
front surface of the array.
Note that the spatial sampling can be imple-
mented in various ways to meet the require-
ments of the application, as depicted in Figure
1.11 , but the treatment of spatial integration
(which leads to the MTF) and the spatial sam-
pling (with considerations of aliasing) remain
the same. Ideal sampling (as shown in Figure
1.11 a) is a mathematical construct, useful for cal-
culations [as in Eq. (1.9) ] but not achievable in
practice. The most common type of sampling is
top-hat sampling on a planar base, as shown in
Figure 1.11 b, where the fill factor implied by the
figure is 50%. The MTF associated with top-hat
sampling was given in Eq. (1.8) . Some FPAs do

g [ n ]=[ h ( x ) ∗ f ( x )] δ( x − nX s ) ,

(1.9)

n ∈ z

where g [ n ] is the discrete space-sampled image
result, h ( x ) is the combined point-spread func-
tion of all the nonideal effects, f ( x ) is the ideal
continuous space image, and X s is the center-to-
center spacing of the photodetectors. 8 Sampling
in the y direction has a similar form. The top-hat
PSF due the FPA would contribute to the overall
h ( x ) in Eq. (1.9) . But what of the effect of sam-
pling? Since X s is the spatial sampling interval,

8 If X s = x d then the fill factor FF in the x direction is 1, or 100%, meaning there is no dead space between photosensi-
tive areas in the x direction. In general, the two-dimensional fill factor is FF = ( x d y d )/( X s Y s ) .

9 Color images are often formed with a combination of a single FPA and a filter array such as a Bayer mosaic. This results in
a different F s for different colors; typically green has an F s twice that of blue or red, but half of the F s is implied by just the
pixel count.