come blurred or degraded at
ical imagery, by image move-
:se stages an ideal point object
blurred image, the “ spread
verall performance we must
which can only be done with
hematical process of convolu-
important cases the spread
directly but is derived by
the transfer function. Some
dw or atmospheric turbulence,
ransfer function, but for any
ive transfer function can be
ng all the functions together,
rail transfer function of the
)le is given in Figure 4. The
of Transfer Functions
1 6-inch photogrammetric lens,
emulsion, and for 25 microns
t, are shown separately and
s information about the system
on of each component, clearly
frequency passband. By using
ication is simplified, slide-rule
procedure becomes very useful
where weaknesses lie, rapidly
verge his system towards the
pparently a great advance on
; information about just one
md can only be combined with
>y arbitrary reciprocal formulae.
to obtain an overall transfer
itself this tells us nothing about
in, since it is merely a statement
Lsfer. We cannot even rank
tions alone ; to say that one is
specify the purpose for which it
nowledge of the input spectrum
cannot be said too often that
by electing to use transfer functions we have moved into the
“ frequency domain,” and cannot apply our results directly
to images as they exist in the “ space domain.” 3 Although
the term “ spatial frequency ” has long been familiar to
photographers through the “ lines per millimeter ” of resolv
ing-power, it has not been used in a strict sense, and there was
no clear differentiation between the two domains. Transfer
functions can operate only on frequencies, and any
conclusions drawn from their use apply to frequency only.
Failure to appreciate this leads to much confusion.
Aerial scenes contain details of all sizes, and the correspond
ing spectra extend to all frequencies, but it is neither possible
nor necessary to reproduce the complete spectrum. At
first sight we might consider making a frequency analysis of
typical aerial scenes and nominating certain bandwidths as
representing typical tasks, with a weighting of the frequency
spectrum according to the probability of each frequency’s
occurrence. There are reasons why this would not be a very
profitable approach. First, any statistically determined
spectrum would inevitably be incorrect on many occasions ;
for example, the relative importance of a particular frequency
band could change in passing from built-up areas to desert
and back again. Secondly, any object with sharp edges has
a wide spectrum and the decision where to cut would always be
arbitrary. Thirdly, we cannot easily think in frequency
terms, and what primarily interests us is the size of objects in
aerial scenes, not their frequency content. A more rational
approach would therefore be to state the smallest sized
objects which we need to reproduce clearly, preferably in the
shape of lines, whose spectra can be readily calculated, and
to make no weighting for the probability of occurrence. This
is a subject on which a great deal of argument can take place.
It would not be profitable to continue the discussion here,
beyond pointing out that the problem is complicated by the
continuously-falling shape of photographic transfer functions,
and that small-scale aerial photography generally needs all
the bandwidth the designer can give it. This discussion may
have illustrated how, in using transfer functions, continual
mental adjustment is necessary between the space and spatial
frequency domains.
Assuming that an input spectrum flat to a certain frequency
is to be reproduced, we are still faced with the problem of
comparing different transfer functions in some way which
corresponds to the images we actually see, or converting the
information they contain into more familiar terms. This is
often done by using the function to predict resolving-power,
which may seem to be going a long way round to obtain a
result which could have been measured directly in much less
time. However, it does enable the designer, who has had the
benefit of using transfer functions throughout the growth of
his system, to sum up its final performance by a figure on
a commonly understood scale, pending the general adoption
of some better standard.
To predict resolving-power with the transfer function we
have to make use of the H and D curve of the emulsion and
its granularity, as well as the target contrast or modulation.
The target modulation, reduced by the transfer function, is
recorded as a density difference, which is approximately given
by the product of the reduced modulation and the gamma or
local slope of the H and D curve. This may be regarded as
a “ signal.” The granularity is the random density fluctua
tion observed when a uniformly exposed and developed area
of emulsion is scanned by a small aperture, typically 24 microns
in diameter. The visual effect is roughly proportional to
spatial frequency, since the scanning aperture of the eye is
effectively reduced when smaller details are observed under
greater magnification. Since granularity is a signal generated
by the system without an input it may be termed “ noise.”
When the signal is much greater than the noise it is easily
seen, if much less it is not seen, when it is about equal the
target is just resolved.
The model as described so far has included all the system
transfer functions, including the emulsion, and now only
requires a “ threshold ” to take account of the granularity.
An alternative is to take all the emulsion factors out of the
model and lump them together in one “ Emulsion Threshold,”
which includes the gamma, transfer function and granularity.
Such thresholds can be determined by exposing targets of
varying frequency and contrast on to the emulsion, using a
high quality lens and correcting for its modulation transfer
where necessary. The resolving-power is read for each
contrast and a curve plotted which expresses the image
contrast necessary for resolution at any frequency. By
intersecting these curves with the system transfer functions,
resolving-power for any target contrast can be shown, and
considerable insight obtained into the behaviour of different
optical systems with various emulsions. This approach
relies on the fact that aerial emulsions are commonly
processed in a standard way, and that resolution does not
depend critically upon gamma.
Figure 5 shows threshold curves for Tri X and Panatomic X
12 J tO 20 •jO 100
SPATIAL FREQUENCY CYCLES PER M.M.
Figure 5. Transfer Functions of a Lens and
Emulsion Thresholds
emulsions, also transfer functions for a 6-inch wide angle lens
on axis, and at the edge of the field for tangentially oriented
targets. The upper lens curves correspond to the actual
image modulations for high contrast targets, the lower for
targets of 0-21 modulation. (Since the scales are logarithmic,
the modulation in the image of a low contrast target is
obtained simply by sliding the curves downwards until the
modulation-transfer at near-zero-frequency coincides with
the value of the target modulation on the ordinate scale.)
This diagram illustrates how resolving-power can be predicted
from the intersections of the thresholds and the transfer
functions. Reading from these we find the following data :—
5
