Download Methods of measuring the modulation transfer function

OPTICA ACTA,
1971,
VOL. 18, NO. 11,
795-813
Methods of measuring the modulation transfer function
of photographic emulsions
J. C. DAINTY
Applied Optics Section, Physics Department, Imperial College,
London, S.W.7
(Received 15 July 1971; revision received 20 August 1971)
Abstract. The different methods of measuring the modulation transfer function
of photographic emulsions are discussed in four sections; (i) sine wave methods,
(ii) Fourier transformation of the line spread function, (iii) coherent light
processing methods, and (iv) calculation from scattering measurements. Particular attention is paid to the effects of the non-linearities of the photographic
process, and to the effects of film grain noise on the measured transfer function.
A brief discussion of the related microdensitometry is also included.
Introduction
Since the initial work by Luneberg [1] and Duffieux [2] on the application of
Fourier methods to optical systems, the subject of Fourier optics has come to
occupy a major position in the field of modern optics. The basic idea of the
Fourier theory of image formation is that, under certain conditions [3], the performance of a noiseless optical system can be described by a unique optical
transfer functiont (OTF) that simply relates the spatial frequencies in the image
to those in the object. The theory is applicable only to optical systems that are
linear and stationary. Many optical systems are more or less linear in either
intensity (in incoherent illumination) or complex amplitude (in coherent illumination), and are also stationary over regions called isoplanatism patches [5], whose
areas depend on the aberrations of the system. Measurement of the OTF has been
reviewed by Murata [6], and Rosenhauer and Rosenbruch [7].
When applying Fourier methods to photographic emulsions, the stationarity
condition is satisfied over large areas, as it may be assumed that photographic
emulsions are statistically isotropic. The isotropy of the emulsion also means
that the two-dimensional OTF is rotationally symmetrical, and that, in the absence
of noise due to film granularity, the imaginary part is zero; it is therefore adequate
to specify only the modulus of the OTF, the modulation transfer functiont
(MTF). Unfortunately, most photographic emulsions have non-linear inputoutput characteristics, and, unlike optical systems, may not be approximated by
a linear system under many practical conditions of use. In certain cases, for
example when studying photographic granularity, the transmittance or density
differences in the developed emulsion may be small enough to give effective
linearity of the input-output characteristic. It is not normally possible to make
this assumption when considering the emulsion transfer function; the solution
1.
t Where possible, the nomenclature follows that recommended by the International
Commission for Optics (see reference [4]).
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J. C. Dainty
is to consider the effective exposure inside the emulsion, rather than the transmittance which results from the effective exposure, for it is the development stage
that accounts for most of the non-linearity, not the overall effects of the light
scattering process. This means that the emulsion is studied as a light diffuser,
not as a recording medium. The effective exposure is found in practice by
measuring the transmittance of the developed emulsion and using the macroscopic transmittance-exposure curve; this procedure assumes the absence of
adjacency effects (see [8], p. 521) which are invariably present, and which contribute another source of non-linearity.
These non-linearities of the photographic process result in the fact that
there is no unique MTF for a given emulsion/developer combination. The
MTF depends critically on the method of measurement. We therefore devote
§ 2 of this review to a discussion of the non-linearities, and then describe the
methods of measuring photographic MTF in detail in § 3. The general methods
described here are applicable not only to silver halide emulsions, but also to colour
films and non-silver materials. Since microdensitometers are generally used
at some stage in the measurement, some aspects of their use are mentioned in § 4.
The MTF of an emulsion does not, by itself, describe the ability of the emulsion
to record fine detail. It is of most value when combined with the Wiener spectrum
of photographic granularity [9, 10] in quantities such as the detective quantum
efficiency [11] and the information capacity [12], both of which place the performance of photographic detectors on an absolute scale. These concepts are
discussed in a review article by Perrin ([8], Chap. 23), and also described (with
examples) in a tutorial paper by Schade [13].
2. Application of MTF to photographic emulsions
2.1. The photographicemulsion as a recording medium
When a photographic emulsion is to be considered as a link in a communication
system, the relevant output quantity is usually transmittance. In general, the
transmittance-exposure curve is non-linear, with the result that a sine wave
input gives an output that consists of harmonics of the input in addition to the
fundamental frequency; this has been studied experimentally by Wilczynski [14].
The effect is shown in figure 1, where a sinusoidal input exposure is transferred
through the transmittance-exposure curve; the output is not sinusoidal, and
Fourier analysis shows the amplitude of the harmonics present. Thus when
transmittance is the output parameter of interest, Fourier techniques are only
generally applicable if the input signal differences are small. An exception to this
is when a reversal film is developed to a gamma of minus one. Over the straight
line part of the conventional photographic density-log exposure curve, we have
D=y . logo E+c
where
D = photographic density =-log 10 (transmittance, T),
E=relative exposure,
c = constant,
y = slope or contrast.
(1)
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T=c' . E-Y
(2)
Taking antilogs,
where c' is another constant. Therefore transmittance is directly proportional
to exposure, if y= - 0.
This problem of linearity is in fact usually side-stepped by restricting considerations to low contrast signals. This being the case, it is usually also practical
to retain density as the output of the process. As a general rule of thumb, it is
found that peak-to-peak exposure fluctuations which result in an image density
difference of less than 04 to 0-5 give rise to spurious harmonics of an acceptably
low degree [77]. The effects of the spurious harmonics are discussed by Lamberts
[15, 16]; when the input signal differences are large, non-linear analysis has to be
used [17].
J
INPUT
-s
FREQUENCY
Figure 1.
The transfer of a sine wave exposure through the transmittance-exposure curve.
The output contains harmonics of the input.
The restriction to low contrast is usually justified in so far as it includes many
of the threshold resolution and fine detail problems which may be usefully tackled
using Fourier methods. It predictably falls down when working out, for example,
the maximum information capacity of microfilm, which involves using the
full density range of the film. In recording applications, it is often convenient
to multiply the normalized MTF by the slope, y, which is a measure of the
amplification of the photographic process. The combined function is called the
contrasttransfer function (CTF),
(3)
CTF=y. MTF.
Gamma is itself a function of the exposure level, and therefore we can represent
the complete contrast transfer function in a three-dimensional plot, such as that
shown in figure 2. This shows the variation of the CTF of the photographic
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process with both exposure and spatial frequency. Often it is more useful to
plot two-dimensional sections of this, and two examples are given in figure 3.
The upper plot shows CTF against log exposure, for contours of equal spatial
frequency. The density-log exposure curve is also drawn. The lower part
shows lines of equal CTF plotted on a spatial frequency-log exposure diagram.
Figure 2.
The relationship between CTF, log exposure, and spatial frequency.
from Shaw [77].)
(Redrawn
2.0
2.0
°~~~~~~~~
D
CTF
1. 0
4t
0
11,0
2
1.0
2.0
REL. LOG E
6B0
3.0
-....
C ,
z 60
0.4
1.2
0
IC,
tu
To
0O
1.0
REL
2.0
LOG E
3.C
Figure 3. The upper diagram shows the variation of CTF with log exposure for contours
of constant spatial frequency. The lower diagram shows the variation of spatial
frequency with log exposure for contours of equal CTF. (Redrawn from Shaw [77].)
Measurement of photographic MTF
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The latter is a more useful mode of expression, since there are often practical
criteria for the lowest acceptable contrast. The area inside the particular contour
then shows the permitted frequency-log exposure working region.
2.2. The photographicemulsion as a light scattering material
In some applications, the photographic material may be more profitably
considered as a light scattering medium rather than a light recording medium.
The output modulation is considered in terms of the effective exposure distribution
inside the emulsion at the exposure stage. This concept, first used by Frieser
[18], partially overcomes the problem of non-linearity. The formation of the
photographic image is considered in two stages: (a) exposure in the emulsion
layer, with so-called photographic turbidity ([8], p. 507) due to reflection,
refraction, diffraction and scattering of light, and (b) development.
Frieser and Metz [19] have shown that, in general, the exposure in the emulsion
is not linearly related to the incident exposure, although in most cases the linearity
is sufficient for practical purposes. Langner and Muiller [20] also emphasize
that, even in the absence of adjacency effects, the effective exposure in the emulsion
layer should be found using micro-characteristic curves of the correct spatial
frequency, and they cite a method [21] for determining the effective exposure.
Generally, however, it is assumed that this stage is more or less linear.
The second stage is not linear because of the statistical nature of latent image
formation and development ([8], p. 87), and the effective exposure technique is
an attempt to get round this stage. The MTF is found by comparing the incident
exposure modulation with the apparent exposure modulation derived from the
transmittance distribution in the image using the macro-characteristic curve.
By this means the non-linearity between density or transmittance and exposure is
eliminated.
The effective exposure technique, using a macro-characteristic curve, assumes
that the development of a grain depends only on the latent image present in that
grain, and is not dependent in any way on the distribution of latent image in
surrounding grains. This assumption is very important, and is only true in the
absence of adjacency effects which tend to occur during development [22-26].
These effects are due to exhaustion, diffusion and inhibition occurring during
development. One of the more important effects is shown in figure 4 for three
As
0.1
I
I
D 76
MICRODOL
HIGH DEFINITION
Figure 4. Step function responses to an input contrast of 25: 1 for three developers.
'High definition ' developer is designed to encourage the adjacency effect. (Redrawn
from Kelly [30].)
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developers. Active developer diffuses from the low density side of the step to
increase the density of the high density side, and exhausted developer, loaded
with development inhibiting bromide ions, diffuses from the high density side
to reduce the density on the low density side. Chemical diffusion effects are
most marked in developers of low concentration and high pH [27]. If this effect
is absent, then most workers agree that the effective exposure technique using the
macro-characteristic curve allows Fourier techniques to be meaningfully applied
to the light scattering properties of photographic emulsions. This means that
the emulsion can be characterized by a single MTF that is independent of development and exposure time [23], and that MTF's found by different methods are
identical [22, 28] and independent of input modulation.
Whilst adjacency effects may be substantially reduced by a suitable emulsion/
developer combination, they are a normal and (usually) desirable part of the
photographic process. The ' acutance ' developers which give increased sharpness in black and white films are designed to encourage these effects, and adjacency
effects are also used to advantage in colour film processing [29]. In the presence
of such effects, transfer functions are obtained which are variable with respect
to the exposure configuration (line, edge, or sine wave), and also to development
time and developer composition. Both Powell [28] and Eyer [22] have shown
experimentally that when chemical diffusion effects are present, the difference
between the effective exposure MTF's measured using sine wave test objects and
line or edge test objects may be large, and they conclude that there is no unique
transfer function for the photographic process.
input
exosure
AL
OT A
DIFFUSION
SENSITOMETRY
DIFFUSION
output
If the chemical
diffusion effects in development are
linearly related to the
Kdensi ty
transfer
expotra
function to~~~~~~_an
represent this.
Figure 5.
Kelly [30] has used the concept
Schematic diagram illustrating Kelly's analysis of photographic image formation.
final image is related to the input exposure modulation.
Ction
the in
This is
relevant
quantity
or density),
it isdiffusion,
possible to
introduce
shownFigure
5. Output
Schematically
diagram
in (transmittance
illustrating Kelly's
analysis then
ofphotographical
resultion.g
in
an extra transfer function to represent this. Kelly [30] has used the concept
ofanthe
effective
' chemical
hemicalxposure
transfer
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difstribution
effectsinina system
development
where the
are output
linearlydensity
relatedmodulato thesopic
reldensity-exposure curve, (distransmittance
this effecantivetydensity), then
or iture
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5. second
The first
stage In
is optical
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' in figure
function
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stage.
the third
stage,
density
the
modulaoutputin
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exposure
distribution
in
the
emulsion.
Using
the
macroscopic
diffusion stage,
finalatent
the
image distribution is convrelated
exposure
to modulation.
the inputThis is
density-exposure curve, this effective distribution of exposure is converted
to
a latent image distribution in the second stage. In the third stage, the chemical
diffusion stage, the latent image distribution is converted to a density distribution.
Only the second stage is non-linear. Whilst the optical diffusion MTF tends to
blur the image, the chemical diffusion MTF tends to enhance higher spatial
frequencies. Kelly's analysis assumes that the magnitude of the chemical
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diffusion is linearly related to the density of the developed image; it is more
likely to be related to the mass of developed silver, and the relation between
mass of developed silver and developed density is not necessarily linear ([8], p. 73).
This three-stage analysis has recently been extended by Nelson [31], and the
whole question of the effect of non-linearities discussed by Higgins [32].
Some typical effective exposure MTF's are shown in figures 6 and 7. Figure
6 shows the MTF's of three particular film/developer combinations. As a
I
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II .. I
I I I
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40
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I I I
1,0 -
0.8 0.6M.T.F
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4
I i I I I
6
8 10
SPATIAL
I
20
FREQUENCY
I I I I II
60 80 100
I
200
I/mm
Figure 6. Some typical photographic MTF's, plotted on a log-log scale. The films were
Kodak Panatomic-X, D.76 (1: 1), 6 min.
all exposed to 'green' light. Kodak Royal-X, D.K. 50, 5 min.
- Kodak Plus-X, D.76, 6jimin. - - A
M.T.F
iREEN
BLUE
RED
D
SPATIAL
FREQUENCY
L.Imm
Figure 7. Variation of MTF with the colour of the exposing illumination for Kodak
Shellburst Linagraph film, developed for 8 min in D.19. The film was exposed
to a tungsten lamp (2854°K) with the following filters: green-Ilford 408 ; blueIlford 306; red-Ilford 205.
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J. C. Dainty
general rule, films with a higher sensitometric speed tend to give lower values of
modulation transfer factor at a particular spatial frequency; referring to figure 6,
it is worth noting that Royal-X film is some forty times faster than Panatomic-X.
Figure 7 shows the MTF's of one film/developer combination exposed to different
colour light sources. The variation of MTF obtained by exposing one particular
film/developer combination to different illuminants may often be as large as the
variation obtained for different film/developer combinations (see, for example,
[80]).
The choice of whether linear or logarithmic axes should be used depends on
the particular application. A logarithmic spatial frequency scale has the advantage that low frequencies, often of most importance, are expanded in scale.
A logarithmic MTF scale has the advantage that log MTF curves for each part
of a photographic system need only be added together to give the total system
response. However, as mentioned earlier, the photographic film may not interact linearly with other parts of a system, and some care has to be taken when
applying curves such as those of figures 6 and 7.
3. Measurement of photographic MTF
3.1. Introduction
The various methods of measuring the MTF of photographic emulsions
may be divided into four sections.
(i)
Sine wave methods.
(ii) Fourier transformation of the line spread function.
(iii) Coherent light processing methods.
(iv) Calculation from scattering measurements.
The division is somewhat arbitrary, and in particular §§ (i) and (iii) overlap;
§ (iv) is mainly of theoretical interest, although it may provide practical information on the development process. Most of the methods described may be used
to measure either the transmittance or effective exposure MTF. Since the
second of these is more likely to satisfy the condition of linearity, the methods are
described with particular reference to the effective exposure MTF. As the
measurement of MTF is invariably carried out in the presence of adjacency
effects, it will depend on the particular method of measurement, and also on the
exposing and processing conditions, some of which are listed below.
(a) Spectral distribution of exposing light source. The degree of light
scatter in the emulsion depends on the wavelength of the light, but there
is no systematic trend.
(b) Mean exposure level. This affects the distribution in depth and the
quantity of latent image, thus influencing the rate of exhaustion of
developer (and hence the amount of adjacency effect).
(c) Exposure modulation or contrast.
of adjacency effect.
This directly influences the amount
(d) Initial developer composition.
(e) Time, temperature, and degree of agitation of developer.
Measurement of photographic MTF
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3.2. Sine wave methods
A sinusoidally varying intensity distribution of known spatial frequency
u, and modulation b/a of the form
I(x) = a + b . cos(27Tux)
(4)
is exposed onto the emulsion. After processing, the transmittance distribution
of the emulsion is measured on a microdensitometer, and the effective exposure
distribution is derived using the macroscopic transmittance-exposure curve.
The ratio of the effective exposure modulation to the input exposure modulation
is the modulation transfer factor for that spatial frequency. The method is
shown diagrammatically in figure 8. There are a number of problems associated
with the measurement relating in particular to the production of a sinusoidally
varying exposure distribution of known modulation.
INPUT
EXPOSURE
MODULATION
I
I
EMULSION
EFFECTIVE EXPOSURE
OUTPUT MODULATION
Figure 8. Determination of the effective exposure MTF using sine wave imagery. The
ratio of the output effective exposure modulation to the input exposure modulation
is the modulation transfer factor for that spatial frequency.
A relatively straightforward method of producing the required exposure
distribution is to photograph a variable area test chart using either a cylindrical lens
[23, 33-35], or a slit pupil [36], or other 'smearing' techniques [37]. In those
methods that involve imaging of a chart, the measured MTF depends to some
extent on the numerical aperture of the imaging system [38, 39]. A method
using a scanning microdensitometer was employed by Shaw [12]. A long
narrow slit is imaged onto the film, and rotating polarizers with a half-wave
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plate between them provide a time varying sinusoidal intensity of known modulation. The film is driven across the slit image to translate the time variation to
a distance variation. Some more specialized scanning methods are described
by Kapany and Pike [40] and others [41, 42]. Interference fringes produced by
a Michelson interferometer [43, 44] or by the Lloyds mirror method [45] have
also been used. Finally, there is the possibility of using suitably defocused
Moire patterns as test objects [46].
Because of the inconvenience of making sinusoidally modulated exposures,
a number of workers use a square wave chart of the type described by Sayce
[47]. A modulation transfer factor, '(u), is measured from the square wave
exposure, the frequency, u, being that of the fundamental; a calculation due
originally to Coltman [48] gives the true MTF, T(u), as
(u)= - (T
(U)+ i T(3u)-
Tr(5u)
...
(5)
A typical method of measuring the MTF may use a square wave chart. A
Sayce chart is imaged onto the emulsion using a high quality lens of aperture
f/4 and of known on-axis MTF (for the wavelength distribution of interest).
Exposures of fixed time and varying intensity are made, and the resulting images
are scanned in a microdensitometer. The image with the highest modulation
at 50 cycles/mm is scanned using a 2 x 500 ,. slit and 025 numerical aperture
objective. The microdensitometer trace is measured, the data transferred
through the transmittance-exposure curve, and the apparent MTF found using
Coltman's formula. The apparent MTF is divided by the combined transfer
functions of the exposing lens and the microdensitometer. This method is
unsatisfactory in several respects, in particular in the method of correction for
the microdensitometer (see § 4).
Another problem is the granularity of the photographic emulsion which
makes the measurement of MTF inaccurate, and which also introduces a systematic
error (this is discussed in more detail in § 3.3). When the measurement of MTF
is made using sine wave images consisting of several cycles of the same frequency,
the microdensitometer may be fitted with a multiple slit to improve the signal-tonoise ratio [35]. An automatic instrument that permits rapid measurement of
the MTF and minimizes grain noise and problems of microdensitometry has
been described [49].
3.3. Fouriertransformation of the line spreadfunction
For a linear photographic system, the MTF is the modulus of the Fourier
transform of the line spread function
A+ W
r(u)=
f
L(x) exp (-2 riux) dx
(6)
The non-linearity of the photographic process has already been discussed in
some detail, and we need only note that, in practice, different methods of measurement will yield different MTF's. The effective exposure line spread function is
usually obtained by scanning the image of an edge. The edge trace is converted
from transmittance units to effective exposure units through the transmittanceexposure curve, and differentiated to give the line spread function. The main
advantage of the method is that it is readily adaptable to digital recording and data
Measurement of photographic MTF
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processing [50, 51], and it allows the MTF to be measured either from laboratory
exposures or from normal operational exposures (as edges are commonly occurring objects). Scott et al. have also described a numerical method of extracting
the MTF from edge gradient data which simply involves the taking of finite sums
and differences [52].
The preparation of suitable test charts presents no problems, and the microdensitometry is also fairly straightforward (see § 4). The main problem in
automatic methods is the computational procedure which is further complicated
by the film granularity. A block diagram for the method used by Jones [50,53,54]
is shown in figure 9. The stages are:
(1) The microdensitometer edge data is converted from arbitrary units,
usually proportional to density, into transmittance units, and convoluted with
a microdensitometer correction function (see § 4).
(2) The resultant corrected transmittance distribution is converted to
effective exposure using the macroscopic transmittance-exposure curve. Jones
[53] uses a polynomial least squares fit to the transmittance-exposure curve to
find the effective exposure. Polynomial curve fitting over large regions of the
characteristic may result in inaccuracies at the low exposure regions where a
very small transmittance difference corresponds to a large exposure difference;
linear interpolation over closely spaced sensitometric data may be sufficiently
accurate and is simpler.
(3) The data is smoothed and differentiated in a single step by convolution
with a suitable function [53].
(4) The Fourier transformation is conventional.
(2)
Transmission
to exposure
Figure 9.
The use of edge trace data to measure the MTF.
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In the measurement of the MTF from the line spread function (and, to a
lesser extent, in other methods of measurement) noise due to film granularity
introduces a certain amount of error. Figure 10 gives an estimate of the precision
obtained by Jones [50]. The importance of the phase of the measured OFT in
estimating the reliability of a particular MTF measurement has been emphasized
by both Blackman [55] and Yeadon et al. [56]. The interaction of the 'true '
MTF and the noise is illustrated in figure 11. The magnitude of a true value of
the modulation transfer factor for a particular spatial frequency is given by S
(drawn arbitrarily at zero phase). The noise spectral component vector,
$~~~~~~~~~~~~~~~
0
U
Io
0.8
-ov
SYMBOL EDGE
~0
OA~~~~~~
4
z
4
0
0.6
4
g
0
0.4
a
o}
0
0 0.2
0o
;0~~~~~~~~~
d ..
0
0
20
40
60
80
100
120
SPATIAL FREQUENCY(CYCLES PER MM)
Figure 10.
Repeated MTF determinations for the same edge exposure.
(From Jones [50].)
Figure 11. Signal and noise interaction for a signal to noise ratio of three. If all values
of are equiprobable, then the average value of (an estimate of the MTF) is greater
than S (the true MTF). (Redrawn from Blackrman [55].)
N. exp (i3) is added to this signal vector to yield the resultant, . exp (i/) . -ris the
measured MTF. If no signal and noise correlation exists, all values of are
equiprobable, and the mean value of is greater than S. Thus one effect of noise
is to yield a measured MTF which is greater than the true MTF, the error
decreasing as the signal to noise ratio increases. The noise also introduces
fluctuations in the measured MTF, as a result of the random variations in the
phase of the noise, and, to a lesser extent, of variations in the amplitude of the
noise. Improvement of the reliability of a single MTF measurement may be
Measurement of photographic MTF
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1.0-
Measured
'r(u)
Smoothed
0.5-
2b0b
Figure 12.
e
4
u
0
I/mm
Comparison of a smoothed and unsmoothed MTF.
The smoothed MTF
is obtained by convolving the unsmoothed OTF with a suitable function.
from Blackman [55].)
(Redrawn
made by using a suitable smoothing function [55, 56]. This point is stressed
in correspondence between the two authors [57, 58]. Figure 12 shows the
effect of applying a smoothing function to the (complex) OTF; it can be seen
that, in this particular case, there is not a large difference between the smoothed
and unsmoothed MTF, and, in general, the smoothing procedure is only used
when the granularity is large.
3.4. Coherent optical processing methods
In this section, three methods of measuring the MTF of photographic
emulsions are described. The second of these is useful for measuring high
resolution emulsions of the type used in holography. In holography, of course,
the effect of the MTF on the reconstructed image is completely different from
that in conventional photography; the MTF of the emulsion affects the brightness
of the reconstruction [59]. In the first two methods of measurement the emulsion
is exposed to a spatially periodic intensity distribution and the diffracted light
from the resultant transmittance grating is measured in a modified spectrometer.
The principle of the first method is to compare the light intensities in the
Fraunhofer diffraction patterns of a square wave grating and a photographic
copy of the grating. For a given diffraction order, the ratio of the intensities is
simply related to the modulation transfer factor for a particular frequency,
subject to certain assumptions. The method is described by several authors
[10, 45, 60-63]. In practice, a Fraunhofer diffraction spectrometer of the type
described by Thiry [61] is used. A square wave grating of period about 5 cycles/
mm, and a bar to space ratio of unity, is contact printed onto the emulsion. The
exposed and developed emulsion is immersed in a refractive index matching
fluid in a sample holder to reduce the effect of phase variations in the emulsion
layer. When the phase variations are eliminated, the amplitude transmittance
is equal to the square root of the intensity transmittance. The main advantage
of the method, apart from its speed of operation, is the elimination of the effects of
granularity. The spectrum of the granularity is fairly white, whereas the periodic
image has a line spectrum. However, the method is based on a linear relationship
between amplitude transmittance and exposure, and this is only true for small
transmittance differences. The system has been analysed for a linear D-log E
curve [62], but the results are somewhat complicated. Any given exposed
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emulsion can easily be tested for linearity. The intensity spectrum should
only contain the odd harmonics; the presence of even harmonics in the measured
spectrum indicates a departure from linearity.
The second method is generally used for holographic and high resolution
emulsions [64, 65]. Two beam interference is used to produce a sinusoidal
exposure distribution of known spatial frequency on the emulsion. The
diffracted light from the processed emulsion is measured and it can be shown
[59, 64] that, all other factors being constant, the diffracted intensity is proportional to the square of the MTF. For holographic purposes, the square
root is not usually taken, and it is more useful to plot the diffraction efficiency as a
function of offset angle (or spatial frequency).
The third coherent light processing method makes use of the ' white ' noise
of a laser produced speckle pattern [79]. The statistical properties of speckle
patterns have been discussed extensively in the literature (see, for example,
reference [78]); the only point of relevance here is that the power spectrum of the
intensity fluctuations in a speckle pattern may be made essentially constant over
spatial frequencies of interest for the measurement of MTF's of photographic
emulsions (0 to 200 cycles/mm). For a linear photographic emulsion, the
power spectrum of the output, Nout(u, v) is related to the power spectrum of the
input, Nin(u, v), by
Nout(u, v)=Nin(u, v) . v(u, v)l:
(7)
where (u, v) is the OTF of the emulsion (neglecting, for the moment, film
granularity).
If Nin(u, v) is constant for all (u, v) of interest, then the output power spectrum
is simply the square of the film MTF. The power spectrum of the complex
amplitude fluctuations of the film may be conveniently measured in a coherent
optical system. In practice, two measurements are made, one with the film
exposed to a uniform laser beam with no speckle (to give the film granularity),
and the other with the film exposed to a speckle pattern. The difference between
the two measurements is the square of the MTF. Some precautions have to be
taken to ensure linearity between complex amplitude and exposure [79]. The
method is claimed to be simple and rapid, since the entire MTF can be obtained
from a single test exposure.
3.5. Calculationfrom scattering measurements
This method is not really a practical method of measuring MTF's of photographic emulsions; it is included because of its conceptual importance.
When considering the photographic emulsion as a light diffusing medium
it should in principle be possible to calculate the MTF from physical characteristics such as absorption, scattering, thickness, and so on. A simple theoretical
calculation for very thin layers was made by Bowker [66], but in real emulsions,
where the MTF depends on the thickness of the emulsion, there are multiple
scattering effects. Random-walk (Monte Carlo) techniques have been applied
to this problem [67, 68], and require knowledge of experimental absorption and
scattering data.
Figure 13 shows some of the possible paths of a photon entering a photographic layer. It may pass straight through (i). It may be absorbed at the
first collision (ii), after one collision and scattering (iii), or after colliding and
809
Measurement of photographic MTF
being scattered many times (iv). In (v) there is back surface halation, in (vi)
there is front surface halation, and in (vii) and (viii) the photon leaves the layer
without being absorbed. In applying the random walk method, it is assumed
that the incident light beam can be considered as particles which are followed
separately as they are scattered from grain to grain. If a large number of photons
(about 2000) are considered to enter at one point in the emulsion, and the number of
absorptions at a radius r is counted using a random walk calculation, then the
plot of the number of absorptions at radius r against the radius will be the point
spread function. The one-dimensional MTF is the Hankel transform of the
point spread function [69]. Six pieces of data are required:
(1) the thickness of the emulsion layer,
(2) the thickness of the base,
(3) the refractive indices of the gelatine and the base,
(4) the probability that a photon will be absorbed on collision,
(5) the probability distribution for the scattering angle of a photon that
collides with a grain but it not absorbed,
(6) the probability distribution for the distance travelled by a photon between
collisions.
(i)
(ii)(iii)iv)
, . ,o.
..o,\i
0
*'
': .t
00
(vii) (vii)
(v) (vI
....
i,\.,
'
.
*0
.0
o
EMULS
0 EMULSION
BASE
I
I
/
Figure 13.
Some possible paths of a photon entering a photographic emulsion.
scale. (Redrawn from Wolfe et al. [68].)
Not to
The computation of the point spread function from this data is not difficult
(details are given in [68]). The difficulty lies in the measurement of the two
probability distributions.
A few results have been given in which the MTF found using this method
is compared with the MTF measured by the sine wave method in the absence
of adjacency effects [68]. A comparison is shown in figure 14; the MTF
found using this method is greater than that found using the more conventional
method, No adequate explanation has been given for this, but clearly the method
gives some quite fundamental information on the scattering of light within the
emulsion layer.
j. C. Dainty
810
1.
od
SPATIAL
FREQUENCY
cycles/mm
Figure 14. Calculated MTF compared with a conventional measurement (in the absence
of adjacency effect) for an emulsion of thickness 10-8 . (Redrawn from Wolfe
et al. [68].)
4.
Microdensitometry
The more important methods of measuring the MTF of photographic
emulsions involve the use of some form of microdensitometer. It is therefore
important to know the limitations and performance of the instrument used, and
some general aspects of microdensitometer performance are discussed by Charman
[70] and Galburt et al. [71]. There are two features of particular importance;
the linearity of the instrument, and the method of correcting measured results
for the microdensitometer degradation.
The microdensitometer should satisfy the conditions of linearity (in intensity)
and stationarity if it is to have a unique OTF. The second condition is easy to
achieve as only small field angles are used, but the condition of linearity is not so
easy to attain. For perfectly incoherent illumination (and hence exact linearity)
we require that the ratio
numerical aperture of condenser be infinite.
be ifinte.
numerical aperture of objective
Clearly, this is impossible, and the exact choice of this ratio for ' effectively
incoherent' illumination depends partly on the spatial frequency range of
interest. The problem is discussed in some detail by Swing [72]. The ratio
should always be greater than 10, and typically about 150. It is worth adding
that under these conditions, a good microdensitometer is essentially diffraction
limited [73].
When an image is scanned in a microdensitometer, it is degraded by an amount
determined by the shape of the scanning aperture, the numerical aperture and
aberrations of the objective, focussing errors, and aperture misalignment [74].
If a noiseless linear photographic emulsion is scanned in an MTF measurement,
then one can allow for the degradation using the equation,
MTF (emulsion) =
MTF (measured)
MTF (microdensitometer)
M'NTF (microdensitometer)'
(8)
Measurement of photographic MTF
811
However, the measured MTF is usually in terms of effective exposure, whereas
it is the transmittance distribution of the emulsion that is observed by the
instrument. The use of equation (8) for real non-linear emulsions tends to
result in overcorrection at higher spatial frequencies. In a noiseless system,
the correction should be applied in transmittance units as suggested by Jones
and Coughlin [75]. The measured transmittance distribution should by convoluted with a correction function in distance space; the correction function is
the inverse Fourier transform of the reciprocal of the microdensitometer transfer
function. It turns out to be a highly oscillating function, and this correction
procedure is not always easy to carry out. In the presence of noise due to film
granularity, even this correction procedure is not strictly correct [57, 76].
When noise is present, a correction function designed on probabalistic criteria
using a prioriknowledge about the measurement would be more accurate.
Conclusions
It is clear from the preceding sections that the measurement of photographic
MTF is not always straightforward. If the measurement is carried out in the
absence of adjacency effects using low contrast exposures, then the results given
by each method will be similar. It has been emphasized however that these
conditions may not be encountered in practice. Adjacency effects may be a
useful part of a particular photographic system, and in many applications high
contrast exposures may be used. Under these conditions there is no unique
MTF for a particular film/developer combination, and the different methods
yield different results, The concept of MTF is still of value under these conditions for comparisons of emulsions, but a little care has to be taken in generalizing
the results.
5.
ACKNOWLEDGMENTS
I am particularly grateful to Dr. R. Shaw (Ciba-Geigy Photochemie) and
Dr. W. T. Welford (Imperial College) for their helpful comments on the original
manuscript. I also wish to thank the following for allowing me to reproduce
their results; Dr. R. A. Jones (figure 10), Eastman Kodak Company (figure 6),
and J. Charlton and T. Hutchinson (figure 7). This review was written whilst
the author was financially supported by the Science Research Council.
On discute, en quatre chapitres, les diverses mrthodes de mesure de la fonction de
transfert de modulation des mulsions photographiques: (1) les mathodes des ondes
sinusoidales; (2) la transformed de Fourier de la fonction image d'un trait; (3) les m6thodes
utilisant la lumiere cohfrente et (4) le calcul partir de mesures de diffusion. On porte
une attention particulire aux effets des nonlinfaritfs du processus photographique et aux
effets du bruit du grain du film sur la fonction de transfert mesurfe. On a inclus aussi
une breve discussion de la microdensitomftrie qui est relive ceci.
Die verschiedenen Verfahren zu Messung der Modulationstibertragungsfunktion
photographischer Emulsionen lassen sich in vier Gruppen einteilen: (1) Verfahren mit
Sinusgittern, (2) Fouriertransformation der Streufunktion fr eine Linie, (3) Verfahren
mit koharenter Beleuchtung und (4) die Berechnung aus Streuungsmessungen. Eine
besondere BerUicksichtigung findet das nichtlineare Verhalten des photographischen
Prozesses und der Einfiuss des Filmkomrns auf die Messung der Ubertragungsfunktion. In
diesem Zusammenhang wird auch auf die Mikrodensitometrie eingegangen.
O.A.
3 I
812
J.
C. Dainty
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