The Scientist and Engineer's Guide to Digital Signal Processing400
the input image contributes a scaled and shifted version of the point spread
function to the output image. As viewed from the output side, each pixel in
the output image is influenced by a group of pixels from the input signal. For
one-dimensional signals, this region of influence is the impulse response flipped
left-for-right. For image signals, it is the PSF flipped left-for-right and top-
for-bottom. Since most of the PSFs used in DSP are symmetrical around the
vertical and horizonal axes, these flips do nothing and can be ignored. Later
in this chapter we will look at nonsymmetrical PSFs that must have the flips
taken into account.
Figure 24-3 shows several common PSFs. In (a), the pillbox has a circular top
and straight sides. For example, if the lens of a camera is not properly focused,
each point in the image will be projected to a circular spot on the image sensor
(look back at Fig. 23-2 and consider the effect of moving the projection screen
toward or away from the lens). In other words, the pillbox is the point spread
function of an out-of-focus lens.
The Gaussian, shown in (b), is the PSF of imaging systems limited by random
imperfections. For instance, the image from a telescope is blurred by
atmospheric turbulence, causing each point of light to become a Gaussian in the
final image. Image sensors, such as the CCD and retina, are often limited by
the scattering of light and/or electrons. The Central Limit Theorem dictates
that a Gaussian blur results from these types of random processes.
The pillbox and Gaussian are used in image processing the same as the moving
average filter is used with one-dimensional signals. An image convolved with
these PSFs will appear blurry and have less defined edges, but will be lower
in random noise. These are called smoothing filters, for their action in the
time domain, or low-pass filters, for how they treat the frequency domain.
The square PSF, shown in (c), can also be used as a smoothing filter, but it
is not circularly symmetric. This results in the blurring being different in the
diagonal directions compared to the vertical and horizontal. This may or may
not be important, depending on the use.
The opposite of a smoothing filter is an edge enhancement or high-pass
filter. The spectral inversion technique, discussed in Chapter 14, is used to
change between the two. As illustrated in (d), an edge enhancement filter
kernel is formed by taking the negative of a smoothing filter, and adding a
delta function in the center. The image processing which occurs in the retina
is an example of this type of filter.
Figure (e) shows the two-dimensional sinc function. One-dimensional signal
processing uses the windowed-sinc to separate frequency bands. Since images
do not have their information encoded in the frequency domain, the sinc
function is seldom used as an imaging filter kernel, although it does find use
in some theoretical problems. The sinc function can be hard to use because its
tails decrease very slowly in amplitude ( ), meaning it must be treated as1/x
infinitely wide. In comparison, the Gaussian's tails decrease very rapidly
( ) and can eventually be truncated with no ill effect.e
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