\oyal Institution of 
'ten calculated before the lens 
rers supply transfer functions 
er functions for image move- 
applied in the “ sine-wave 
ing the literature, one might 
las been made and that the 
are nearing solution. How- 
what impact all this activity 
titioner of aerial photogram- 
d by the questionnaire, is : 
lalysis and the transformation 
cies remain almost unknown 
earch institutes and industrial 
aains the standard image- 
lown and used by practical 
lere is a strong feeling among 
e to the image quality of aerial 
that the sine-wave ideas are 
they seem to have led to con- 
n. Moreover, it cannot be 
n of sine-wave techniques is 
better equipment, since some 
enses have been designed and 
do not use these methods, 
thy effort to expose the funda- 
•s to have got out of touch with 
ppraisal would be too severe, 
istify a closer examination of 
le limitations as well as the 
j approach. Although the 
i very well covered in the 
iven here for orientation and 
iring the last four years has 
widely known among photo- 
spetition is justified. 
d Transfer Functions 
¡search has recognised that 
:h as sharpness are the result 
idamental properties of lenses, 
model ” is developed from the 
which we see as individual 
ions can be mathematically 
3f sinusoidal wave-trains at 
appropriate combinations of amplitude and frequency.* 
A test target containing such “ spatial frequencies ” is shown 
in Figure 1. The primary reason for adopting the sinusoidal 
intensity-distribution is its fundamental nature ; other shapes, 
such as sharp-edged bars, change their shape with deterior 
ating image quality, whereas the sine-wave retains its shape 
while the amplitude decreases. For conceptual simplicity 
and instrumental convenience, sine-wave targets are normally 
one-dimensional, as in Figure 1. 
M _ I MAX- I MINI 
I MAX+ 1 MIN 
Figure 1. Sinusoidal Target and Definition of 
Modulation 
Any object, when reduced to image scale by a hypothetical 
perfect camera, has a characteristic spatial frequency spectrum 
determined by its shape and intensity distribution. The 
simplest spectrum is given by a pure sinusoidal frequency, 
which naturally exhibits only one spectral line, while an 
actual line or bar has a continuous spectrum extending 
theoretically to infinity, of the form sinnx/nx, the first zero 
falling at a frequency 1 /w where w is the line width (Figure 2). 
Since no camera is perfect, the object spectrum is more or 
less changed, affecting the sharpness and contrast of the 
recorded image. Lenses, emulsions and other components 
of the aerial photographic system have characteristic attenua 
tions which can be measured with a multi-frequency target 
such as Figure 1. The contrast of the target is expressed 
.. , , . „ , _ , I max— I min, 
as modulation, defined as - —— (Figure 1) and is 
I max+I min 
normally kept constant at all target frequencies. Modulation 
10 D -1, 
may also be expressed as 
photographic density. 
10 D +1 
where D is the normal 
When a sinusoidal target is imaged by a lens, the image 
modulation is in general less than in the target and progres 
sively diminishes with increasing frequency. For a perfectly 
Figure 2. Frequency Spectra for Lines of 
1 mm. and 1/10 mm. Width 
corrected lens, whose performance is limited only by diffrac 
tion, the relative image modulation falls off in a definite way 
which depends only on the aperture and the wavelength of 
the light, and reaches zero at a definite spatial frequency. 
The curve showing the relative modulation transfer as a 
function of spatial frequency always has the same shape 
when normalised for aperture and wavelength, and is the 
graph of what is now known as the “ Modulation Transfer 
Function,”f otherwise expressed as “ Transfer Function ” 
or “ MTF.” An approximate rule is that for green light the 
cut-off frequency for a perfect lens is 1,800 divided by the 
relative aperture (f/no.). Lenses in general are not perfect 
and the MTF’s of photogrammetric lenses fall far below the 
theoretical aperture-limited curves. Figure 3 shows a curve 
typical of a good photogrammetric lens, with the aperture- 
limited curve for comparison ; the rapid decline in modula 
tion transfer at low frequencies, followed by a slower run-out 
towards the aperture-limited cut-off, is characteristic of 
many photographic lenses. 
Photographic emulsions also reduce the modulation of 
images falling upon them, because of scattering at the silver 
halide grains. Typical emulsion MTF’s are also shown in 
Figure 3. In obtaining such curves, the targets are imaged 
by a lens which is essentially perfect over the frequency range 
concerned, e.g., a well-corrected lens such as a microscope 
objective and corrections are applied for its MTF if necessary. 
The emulsion MTF refers only to the optical image within 
the sensitive layer during exposure. For example, the 
function for Plus X Aerographic shows appreciable modula 
tion transfer at frequencies around 100 lines per millimetre, 
but this does not mean that such high frequencies will always 
appear in the developed image. 
* For the synthesis of a square wave, see Ref. (1). 
t From a photographer’s point of view the displaced “ Contrast Transfer 
Function ” was apt and expressive.