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:
Xo and Yo are the sampling distances
or intervals, d(*,*)
is the ideal impulse function, and we have used eq. . (At some point, of
course, the impulse function d(x,y) is converted to the
discrete impulse function d[m,n].) Square sampling
implies that Xo =Yo. 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 finite
sampling aperture p(x,y) into account, we can
modify the sampling model as follows:
combined effect of the aperture and sampling are best understood by
examining the Fourier domain representation.
is the sampling frequency in the x direction and
is the sampling frequency in the 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. .
prevent the possible aliasing (overlapping) of spectral terms
that is inherent in eq. two conditions must hold:
Bandlimited A(u,v) -
Nyquist sampling frequency -
uc and vc are the 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:
NA is the numerical aperture of the lens and
is the shortest wavelength of light used with
the lens . If the lens does not meet one or more of these assumptions
then it will still be bandlimited but at lower cutoff frequencies than
those given in eq. . When working with the F-number (F) of the
optics instead of the NA and in air (with index of refraction
= 1.0), eq. becomes:
The aperture p(x,y) described above will have only a marginal effect on the final signal if the two conditions eqs. and are satisfied. Given, for example, the distance between samples Xo equals Yo and a sampling aperture that is not wider than Xo, the effect on the overall spectrum--due to the A(u,v)P(u,v) behavior implied by eq.--is illustrated in Figure 16 for square and Gaussian apertures.
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 Xo contains +/-3
(99.7%) of the Gaussian. The rectangular
apertures have a width such that one occupies 95% of the sampling
interval and the other occupies 50% of the sampling interval. The 95%
width translates to a fill factor of 90% and the 50% width
to a fill factor of 25%. The fill factor is
discussed in Section 7.5.2.
Figure 16: Aperture spectra P(u,v=0) for frequencies up to half the Nyquist frequency. For explanation of "fill" see text.
"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.
Assuming square sampling, Xo = Yo and the unbiased algorithm for estimating area which involves simple pixel counting, the CV (see eq. ) of the area measurement is related to the sampling density by :
in D dimensions:
S is the number of samples per object diameter.
In 2D the measurement is area, in 3D volume, and in D-dimensions
Again assuming square sampling and algorithms for estimating length based upon the Freeman chain-code representation (see Section 3.6.1), the CV of the length measurement is related to the sampling density per unit length as shown in Figure 17 (see .)
CV of length measurement for various algorithms.
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:
Ne is the number of even chain codes, No
the number of odd chain codes, and Nc the number of
corners. The specific formulas are given in Table 7.
Table 7: Length estimation formulas based on chain code counts (Ne, No, Nc)
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 (bias) and precision (CV). In a case of uncertainty, one should choose the higher of the two sampling densities (frequencies).