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