18
Nyquist theorem
It is known from Part 1 that the information con-
tent of the signal is limited by the resolving power
of the microscope optics. A realistic estimate for
the resolving power is the full width at half maxi-
mum intensity (FWHM
lat
) of a point image (see
equation 3).
To avoid a loss of information during the scanning
process, it is necessary to stick to the Nyquist
theorem. The optimal pixel spacing in scanning a
periodic signal, as defined by the Nyquist theorem,
is half the period of the feature spacing to be
resolved, or two pixels per resolvable structure
detail. Together with the resolving power defined
above, this results in a maximum pixel spacing of
d
pix
= 0.5 x FWHM
lat
.
With a two-point object (see explanation on page
17), the pixel spacing needed to separate the
two Airy discs in the digitized image is 0.25 AU
(figure 13).
If the number of sampling points per feature size is
smaller than that given by the Nyquist theorem
(undersampling), part of the information will be
lost. This is evident in Figure 14c especially by the
unresolved fine features.
A greater number of sampling points per feature
size (oversampling) means a greater number of
readings without a gain in information; simultane-
ously, the time per pixel becomes shorter. Thus,
the volume of data to be processed is inflated, and
the noise of the measurement signal increases (see
page 20)
Under unfavorable conditions, also artefacts may
result out of the digitization process (aliasing). As
a rule, this is the case if the feature spacing in the
specimen is equal, or nearly equal, to the pixel
spacing.
Fig. 13 The graph illustrates the scanning of a two-point object
with the minimum number of sampling points needed to avoid
a loss of resolution (spacing of sampling points 0.25 AU).
Fig. 14 Oversampling, correct
sampling and undersampling
of a continous signal.
0.2
0.4
1
0.8
0.6
Relative intensity
0.25
0.5
1.25
1
0.75
1.5
1.75
a)
Oversampling
AU
Pixels
intensity
50
100
150
0
50
100
150
337_Zeiss_Grundlagen_e 25.09.2003 16:16 Uhr Seite 21