## Image Sampling |
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Converting
from a continuous image a(x,y) to its digital
representation b[m,n] requires the process of
sampling. In the ideal sampling system a(x,y) is
multiplied by an ideal 2D impulse train:
where
Y are the sampling distances
or intervals, _{o}(*,*)
is the ideal impulse function, and we have used eq. . (At some point, of
course, the impulse function d(dx,y) is converted to the
discrete impulse function [dm,n].) Square sampling
implies that X =_{o}Y. Sampling with
an impulse function corresponds to sampling with an infinitesimally
small point. This, however, does not correspond to the usual situation
as illustrated in Figure 1. To take the effects of a _{o}finite
sampling aperture p(x,y) into account, we can
modify the sampling model as follows:
The
combined effect of the aperture and sampling are best understood by
examining the Fourier domain representation.
where
x direction and
_{s}
= 2
/Y
is the sampling frequency in the _{o}y direction. The aperture p(x,y)
is frequently square, circular, or Gaussian with the associated P(
,
). (See Table 4.) The periodic nature of the
spectrum, described in eq. is clear from eq. .
## Sampling Density for Image Processing
To
prevent the possible
where
v are the _{c}cutoff frequencies
in the x and y direction, respectively. Images that are
acquired through lenses that are circularly-symmetric, aberration-free,
and diffraction-limited will, in general, be bandlimited. The lens acts
as a lowpass filter with a cutoff frequency in the frequency domain (eq.
) given by:
where
## Sampling aperture
The aperture Y
and a sampling aperture that is not wider than _{o}X, the
effect on the overall spectrum--due to the _{o}A(u,v)P(u,v)
behavior implied by eq.--is illustrated in Figure 16 for square and
Gaussian apertures.The
spectra are evaluated along one axis of the 2D Fourier transform. The
Gaussian aperture in Figure 16 has a width such that the sampling
interval fill factor of 90% and the 50% width
to a fill factor of 25%. The fill factor is
discussed in Section 7.5.2.
## Sampling Density for Image Analysis
The
"rules" for choosing the sampling density when the goal is
image analysis--as opposed to image processing--are different. The
fundamental difference is that the digitization of objects in an image
into a collection of pixels introduces a form of spatial quantization
noise that is not bandlimited. This leads to the following results for
the choice of sampling density when one is interested in the measurement
of area and (perimeter) length. ## Sampling for area measurements
Assuming square sampling, Y and the unbiased algorithm for estimating area
which involves simple pixel counting, the _{o}CV (see eq. ) of the
area measurement is related to the sampling density by :
and
in
where
## Sampling for length measurements
Again assuming square sampling
and algorithms for estimating length based upon the Freeman chain-code
representation (see Section 3.6.1), the
The
curves in Figure 17 were developed in the context of straight lines but
similar results have been found for curves and closed contours. The
specific formulas for length estimation use a chain code representation
of a line and are based upon a linear combination of three numbers:
where
N
the number of odd chain codes, and _{o}N the number of
corners. The specific formulas are given in Table 7. _{c}
N,_{o} N)_{c}## Conclusions on sampling
If one is interested in image
processing, one should choose a sampling density based upon classical
signal theory, that is, the Nyquist sampling theory. If one is
interested in image analysis, one should choose a sampling density based
upon the desired measurement accuracy ( |