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VI.
Image sampling
For most image-formation systems, the output image
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(the
optical image
),
can be
completely described by a four-dimensional function n(x, y, t, l) that gives the mean
quantum irradiance as a function of space (x, y), time (t) and wavelength (
λ
). This is
usually a complete description because, in most situations, the image noise is that of an
inhomogeneous Poisson process (which is completely determined by its mean or
"intensity" function).
All the image information available for performing a given visual
task is carried in such four-dimensional functions.
In biological vision systems, and in virtually all artificial (computer) vision systems,
the images formed by the optical system must (because of hardware/wetware limitations)
eventually be coded into a discrete representation in space, time, and wavelength.
This
image sampling process is a crucial step in visual processing that can, and often does,
result in significant information loss.
The loss results because it is often impossible to
sample all four dimensions with sufficiently high resolution.
Thus, compromises must
always be struck.
In the biological vision systems, the first stage of sampling is
carried out by the
photoreceptors, in artificial systems, usually by some other two-dimensional array of
elements (e.g., a CCD array or an array of photodiodes).
The principles of sampling are
much the same in biological and artificial systems.
A fundamental principle of sampling
is captured by the so-called Wittaker-Shannon sampling theorem which is useful for
understanding the information loss due to discrete sampling.
A.
The sampling theorem
Below are statements of the Whittaker-Shannon sampling theorem for one and two
dimensions.
Similar statements hold for higher dimensions.
The sampling theorem (one-dimensional case):
If a one-dimensional function, f(x), is
limited to frequencies below w
c
cycles per unit value of x, then the function can be
completely reconstructed by taking 2w
c
evenly spaced samples per unit value of x.
The sampling theorem (two-dimensional case):
If a two-dimensional function, f(x,
y), is limited to frequencies below u
c
cycles per unit value of x (in the x direction),
and to frequencies below v
c
cycles per unit value of y (in the y direction), then the
function can be completely reconstructed by taking 4u
c
v
c
samples per unit area on the
x, y plane.
In other words, what the sampling theorem says is that almost any smooth continuous
function can be represented exactly, with perfect precision, by the values of the function
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For example, there would be two images in a binocular optical system.