ISO 15529:2010

INTERNATIONAL ISO
STANDARD 15529
Third edition
2010-08-01

Optics and photonics — Optical transfer
function — Principles of measurement of
modulation transfer function (MTF) of
sampled imaging systems
Optique et photonique — Fonction de transfert optique — Principes de
mesure de la fonction de transfert de modulation (MTF) des systèmes
de formation d'image échantillonnés




Reference number
ISO 15529:2010(E)
©
ISO 2010

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ISO 15529:2010(E)
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ISO 15529:2010(E)
Contents Page
Foreword .iv
Introduction.v
1 Scope.1
2 Normative references.1
3 Terms, definitions and symbols .1
3.1 Terms and definitions .1
3.2 Symbols.4
4 Theoretical relationships.5
4.1 Fourier transform of the image of a (static) slit object.5
4.2 Fourier transform of the output from a single sampling aperture for a slit object scanned
across the aperture .6
4.3 Fourier transform of the average LSF for different positions of the slit object.8
5 Methods of measuring the MTFs associated with sampled imaging systems .8
5.1 General .8
5.2 Test azimuth.9
5.3 Measurement of T of a sampled imaging device or complete system .9
sys
5.4 Measurement of the MTF of the sampling aperture, T .15
ap
6 Method of measuring the aliasing function, the aliasing ratio and the aliasing potential.15
Annex A (informative) Background theory.17
Annex B (informative) Aliasing in sampled imaging systems.20
Bibliography.25

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ISO 15529:2010(E)
Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 15529 was prepared by Technical Committee ISO/TC 172, Optics and photonics, Subcommittee SC 1,
Fundamental standards.
This third edition cancels and replaces the second edition (ISO 15529:2007) which has undergone a minor
revision to include measurement and test procedures for aliasing of sampled imaging systems.
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ISO 15529:2010(E)
Introduction
One of the most important criteria for describing the performance of an imaging system or device is its MTF.
ISO 9334 covers the conditions to be satisfied by an image system for the MTF concept. These conditions
require that the imaging system be linear and isoplanatic.
For a system to be isoplanatic, the image of a point object (i.e. the point spread function) must be independent
of its position in the object plane to within a specified accuracy. There are types of imaging systems where this
condition does not strictly apply. These are systems where the image is generated by sampling the intensity
distribution in the object at a number of discrete points, or lines, rather than at a continuum of points.
Examples of such devices or systems are: fibre optic face plates, coherent fibre bundles, cameras that use
detector arrays such as CCD arrays, line scan systems such as thermal imagers (for the direction
perpendicular to the lines), etc.
If one attempts to determine the MTF of this type of system by measuring the line spread function of a static
narrow line object and calculating the modulus of the Fourier transform, one finds that the resulting MTF curve
depends critically on the exact position and orientation of the line object relative to the array of sampling points
(see Annex A).
This International Standard specifies an “MTF” for such systems and outlines a number of suitable
measurement techniques. The specified MTF satisfies the following important criteria:
⎯ the MTF is descriptive of the quality of the system as an image-forming device;
⎯ it has a unique value that is independent of the measuring equipment (i.e. the effect of slit object widths,
etc., can be de-convolved from the measured value);
⎯ the MTF can, in principle, be used to calculate the intensity distribution in the image of a given object,
although the procedure does not follow the same rules as it does for a non-sampled imaging system.
This International Standard also specifies MTFs for the sub-units, or imaging stages, which make up such a
system. These also satisfy the above criteria.
A very important aspect of sampled imaging systems is the “aliasing” that can be associated with them. The
importance of this is that it allows spatial frequency components higher than the Nyquist frequency to be
reproduced in the final image as spurious low frequency components. This gives rise to artefacts in the final
image that can be considered as a form of noise. The extent to which this type of noise is objectionable will
depend on the characteristics of the image being sampled. For example, images with regular patterns at
spatial frequencies higher than the Nyquist frequency (e.g. the woven texture on clothing) can produce very
visible fringe patterns in the final image, usually referred to as moiré fringes. These are unacceptable in most
applications if they have sufficient contrast to be visible to the observer. Even in the absence of regular
patterns, aliasing will produce noise-like patterns that can degrade an image.
A quantitative measure of aliasing can be obtained from MTF measurements made under specified conditions.
This International Standard defines such measures and describes the conditions of measurement.

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INTERNATIONAL STANDARD ISO 15529:2010(E)

Optics and photonics — Optical transfer function — Principles
of measurement of modulation transfer function (MTF) of
sampled imaging systems
1 Scope
This International Standard specifies the principal MTFs associated with a sampled imaging system, together
with related terms, and outlines a number of suitable techniques for measuring these MTFs. It also defines a
measure for the “aliasing” related to imaging with such systems.
This International Standard is particularly relevant to electronic imaging devices such as digital still and video
cameras and the detector arrays they embody.
Although a number of MTF measurement techniques are described, the intention is not to exclude other
techniques, provided they measure the correct parameter and satisfy the general definitions and guidelines for
MTF measurement as set out in ISO 9334 and ISO 9335. The use of a measurement of the edge spread
function, rather than the line spread function (LSF), is noted in particular as an alternative starting point for
determining the OTF/MTF of an imaging system.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 9334, Optics and photonics — Optical transfer function — Definitions and mathematical relationships
ISO 9335, Optics and photonics — Optical transfer function — Principles and procedures of measurement
ISO 11421, Optics and optical instruments — Accuracy of optical transfer function (OTF) measurement
3 Terms, definitions and symbols
3.1 Terms and definitions
For the purposes of this document the terms and definitions given in ISO 9334 and the following apply.
3.1.1
sampled imaging system
imaging system or device, where the image is generated by sampling the object at an array of discrete points,
or along a set of discrete lines, rather than a continuum of points
NOTE 1 The sampling at each point is done using a finite size sampling aperture or area.
NOTE 2 For many devices “the object” is actually an image produced by a lens or other imaging system (e.g. when the
device is a detector array).
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ISO 15529:2010(E)
3.1.2
sampling period
a
physical distance between sampling points or sampling lines
NOTE Sampling is usually by means of a uniform array of points or lines. The sampling period may be different in two
orthogonal directions.
3.1.3
Nyquist limit
maximum spatial frequency of sinewave that the system can generate in the image, equal to 1/(2a)
NOTE See also 3.1.9.
3.1.4
line spread function (LSF) of the sampling aperture of a sampled imaging system
L (u)
ap
variation in sampled intensity, or signal, for a single sampling aperture or line of the sampling array, as a
narrow line object is traversed across that aperture, or line and adjacent apertures or lines
NOTE 1 The direction of traverse is perpendicular to the length of the narrow line object and in the case of systems
which sample over discrete lines, is also perpendicular to these lines.
NOTE 2 L (u) is a one-dimensional function of position u in the object plane, or equivalent position in the image.
ap
3.1.5
optical transfer function (OTF) of a sampling aperture
D (r)
ap
Fourier transform of the line spread function, L (u), of the sampling aperture
ap
D rL=⋅u exp−i2πur du
() ( ) ( )
ap ap
∫
where r is the spatial frequency
3.1.6
modulation transfer function (MTF) of a sampling aperture
T (r)
ap
modulus of D (r)
ap
3.1.7
reconstruction function
function used to convert the output from each sampled point, aperture or line, to an intensity distribution in the
image
NOTE The reconstruction function has an OTF and MTF associated with it denoted by D (r) and T (r) respectively.
rf rf
3.1.8
MTF of a sampled imaging system
T (r)
sys
product of the aperture MTF, T (r), and the MTF of the reconstruction function, T (r), with the MTF of any
ap rf
additional input device (e.g. a lens) and output device (e.g. a CRT monitor) which are regarded as part of the
imaging system
NOTE When quoting a value for T it should be made clear what constitutes the system. The system could, for
sys
example, be just a detector array and associated drive/output electronics, or could be a complete digital camera and CRT
display.
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ISO 15529:2010(E)
3.1.9
Fourier transform of the image of a narrow slit produced by the imaging system
F (r)
img
F rL=⋅u exp−i2πur du
() ( ) ( )
img img
∫
where the line spread function of the system, L (u), is the variation in sampled intensity, or signal, across the
img
image of a narrow slit object generated by the complete system
NOTE L (u) is different for different positions of the slit object relative to the sampling array.
img
3.1.10
aliasing function of a sampled imaging system
A (r)
F, sys
half the difference between the highest and lowest value of |F (r)| [i.e. the modulus of F (r)] as the image
img img
of the MTF test slit is moved over a distance equal to, or greater than, one period of the sampling array
Fr −Fr
() ()
img img
max min
Ar =
()
F,sys
2
NOTE 1 It is the limiting value of this difference as the width of the test slit approaches zero (i.e. its Fourier transform
approaches unity).
NOTE 2 A (r) is a measure of the degree to which the system will respond to spatial frequencies higher than the
F, sys
Nyquist frequency and as a result generate spurious low frequencies in the image.
3.1.11
aliasing ratio of a sampled imaging system
A (r)
R, sys
ratio A (r)/|F (r)| , where |F (r)| is the average of the highest and lowest value of |F (r)| as the
F,sys img av img av img
image of the MTF test slit is moved over a distance equal to, or greater than, one period of the sampling array
NOTE A (r) can be considered as a measure of the noise/signal ratio where A (r) is a measure of the noise
R, sys F, sys
component and |F (r)| as a measure of the signal.
img av
3.1.12
MTF of an imaging pick-up subsystem
T (r)
imp
product of the aperture MTF, T (r), with the MTF of the lens, T (r), where the MTF of the lens includes the
ap lens
effect of any optical anti-aliasing filters that are part of the system and which form the image on the sampling
array
3.1.13
aliasing potential of a sampled imaging system
A
P, imp
ratio of the area under the imaging pick-up MTF, T (r), from r = 0,5 to r = 1, to the area under the same
imp
curve from r = 0 to r = 0,5, where the spatial frequency (r) is normalized so that 1/a becomes unity
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ISO 15529:2010(E)
3.2 Symbols
See Table 1.
Table 1 — Symbols used
Symbol Parameter Units
A (r) Aliasing function associated with the complete imaging system 1
F, sys
A Aliasing potential associated with the imaging subsystem 1
P,imp
A (r) Aliasing ratio associated with the complete imaging system 1
R,sys
a Sampling period mm, mrad, degrees
−1 −1 −1
1/(2a) Nyquist spatial frequency limit mm , mrad , degree
D (r) Optical transfer function of a sampling aperture 1
ap
D (r) Optical transfer function of the optical system including any anti-aliasing 1
lens
filters
D (r) Optical transfer function of the reconstruction function 1
rf
F (r) Fourier transform of L (u) 1
av av
F (r) Fourier transform of the final image of the slit object 1
img
F (r) Fourier transform of L (u) 1
in in
F (r) Fourier transform of the slit object 1
slt
L (u) Line spread function of a sampling aperture 1
ap
L (u) Line spread function obtained by averaging the LSF associated with 1
av
different positions of the slit object relative to the sampling array
L (u) Line spread function associated with the complete imaging system 1
img
L (u) Line spread function of the combination of slit object, optical system 1
in
including any anti-aliasing filters and sampling aperture
−1 −1 −1
r Spatial frequency mm , mrad , degree
T (r) Modulation transfer function of a sampling aperture 1
ap
T (r) Modulation transfer function of an imaging pick-up subsystem 1
mp
i
T (r) Modulation transfer function of the optical system including any 1
lens
anti-aliasing filters
T (r) Modulation transfer function of the reconstruction function 1
rf
T (r) Modulation transfer function of a sampled imaging system 1
sys
u Local image field coordinate mm, mrad, degrees

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ISO 15529:2010(E)
4 Theoretical relationships
4.1 Fourier transform of the image of a (static) slit object
4.1.1 General case
The stages of image formation in a generalized sampled imaging system are illustrated in Figure 1. The
values of the relevant parameters used in this International Standard are specified in Clause 3.

Key
1 slit object F (r)
slt
2 lens OTF, D (r)/MTF, T (r)
lens lens
3 sampling apertures OTF, D (r)/MTF, T (r)
ap ap
4 reconstruction function OTF, D (r)/MTF, T (r)
rf rf
Figure 1 — Image formation by a sampled imaging system
For a sampled imaging system we have:
⎡⎤
FrF=−rk/ea⋅xpi2πφk/a⋅Dr (1)
() ( ) ( ) ()
( { })
img ∑ in rf
⎣⎦
k
where
F rF=⋅r D r⋅D r; (2)
() () ( ) ()
in slt lens ap
k is an integer (i.e. k = 0,1,2,3.);
φ is a phase term describing the position of the slit relative to the sampling array.
NOTE More information on the mathematical relationships involved in imaging with sampled systems can be found in
References [2] and [3], and in most textbooks dealing with Fourier transform methods.
4.1.2 Special cases
4.1.2.1 General
The relationships listed in this clause are given without derivation (a brief explanation of their derivation can be
found in Annex A).
4.1.2.2 Cut-off spatial frequency of |F (r)| is less than or equal to the Nyquist frequency 1/(2a)
in
For this condition and for spatial frequencies less than the Nyquist frequency, the system behaves as a
non-sampled system and we have:
F rF=⋅r T r (3)
() () ()
img in rf
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ISO 15529:2010(E)
where
F rF=⋅r T r⋅T r (4)
() () () ( )
in slt lens ap
so that
TT=⋅T⋅T=F ()r F ()r (5)
sys lens ap rf img slt
4.1.2.3 Cut-off spatial frequency of |F (r)| is less than or equal to twice the Nyquist frequency
in
(i.e. 1/a)
For this condition and for spatial frequencies less than twice the Nyquist limit, we get a maximum and
minimum value for |F (r)| as the position of the slit image relative to the sampling apertures of the array is
img
varied. The two values are given by:
⎡⎤
FrF=+rFr−1/a⋅Tr (6)
() () () ()
img in in rf
⎣⎦
max
and
⎡⎤
FrF=−rFr−1/a⋅Tr (7)
() () () ()
img in in rf
⎣⎦
min
from which it can be shown that the system MTF is given by:
Fr +Fr
() ()
{ img img }
max min
Tr()=⋅F (r)T ()r F (r)= for r < 1/(2a) (8)
sys in rf slt
2Fr()
slt
and
Fr −F r
() ()
{ img img }
max min
Tr =
() for r > 1/(2a) (9)
sys
2Fr
()
slt
It should be noted that in theory, the position of the slit, relative to the sampling array, where one obtains
|F (r)| and where one obtains |F (r)| , can be different for each value of the spatial frequency r. This
img max img min
can however only occur if L (u) is asymmetrical so that there is a significant (non-linear) variation of the
in
associated phase transfer function with spatial frequency. In practise the effect will be small and one can
assume that the relevant slit positions are the same for all spatial frequencies.
4.2 Fourier transform of the output from a single sampling aperture for a slit object
scanned across the aperture
In this case we define a line spread function L (u) which is the signal obtained from a single sampling
in
aperture as a function of the position u of a slit in object space (see Figure 2). The modulus of the Fourier
transform of the line spread function is given by:
F rF=⋅r T r⋅T r (10)
() () () ( )
in slt lens ap
and the MTF of aperture is given by:
Fr
()
in
Tr = (11)
()
ap
F rT⋅ r
() ()
slt lens
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ISO 15529:2010(E)
Note that MTF of the reconstruction function does not appear in these equations and by re-arranging
Equation (11) we have
F r
()
in
Tr⋅=T r T r= (12)
() () ()
ap lens imp
F r
()
slt
where T (r) is the MTF of the imaging pick-up subsystem (provided any anti-aliasing filters are included in
imp
the optical train) and is the basis of a measurement of aliasing potential.

a)  Schematic measurement arrangement
Key
1 slit object F (r)
slt
2 lens OTF, D (r)/MTF, T (r)
lens lens
3 sampling apertures OTF, D (r)/MTF, T (r)
ap ap
4 output from single sampling aperture [see Figure 2 b)]

b)  Illustration of output
Key
X position of slit object
Y output from single sampling aperture
Figure 2 — Output from a single sampling aperture as slit object is scanned
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ISO 15529:2010(E)
4.3 Fourier transform of the average LSF for different positions of the slit object
If the LSF of the sampled imaging system is measured for many different positions of the slit object relative to
the sampling array and the average value of these L (u) is taken after adjustment to a common slit position,
av
then the Fourier transform of this average LSF is given by:
F rF=⋅r T r (13)
() () ()
av in rf
and the system MTF is given by:
F r
()
av
TT=⋅T⋅T= (14)
sys lens ap rf
F r
()
slt
5 Methods of measuring the MTFs associated with sampled imaging systems
5.1 General
5.1.1 Range of application
The relationships outlined in Clause 4 can be the basis for suitable measurement techniques.
There are however many different types of sampled imaging system and each can require the use of a
different experimental arrangement for implementing these techniques. The main purpose of this International
Standard is to specify these relationships and indicate in general terms how they can be applied to measuring
the relevant parameters. This International Standard does not describe in detail the measurement techniques
for each type of sampled imaging system, but does illustrate the application of a particular method with some
specific examples. Most of the measurement techniques and equipment for measuring the MTF of appropriate
non-sampled imaging systems may be adapted for testing sampled imaging systems by the methods specified
in this International Standard.
5.1.2 Additional measurement considerations
This International Standard shall be used in conjunction with the ISO 9334, ISO 9335 and ISO 11421. These
define terms used in the present International Standard, and provide guidelines which have not been repeated
here, for achieving the accurate measurement of MTF.
Many sampled systems include electro-optic devices that may behave in a non-linear fashion under certain
operating conditions. It is important to adjust light levels, etc., so that the MTF measurements are made with
the system functioning as far as possible in a linear mode.
5.1.3 Specifying the relevant MTF
In general the most useful of the MTFs specified in this International Standard will be that of the system
(i.e. T ) which, as a minimum, will describe the combined effect of the sampling aperture and the
sys
reconstruction function, but may also include other components of a system such as lenses and displays.
When quoting MTF values for sampled imaging systems or devices, it is generally assumed that the values
refer to T unless otherwise specified. When quoting such values care should be taken to avoid any
sys
ambiguities over what constitutes the system.
5.1.4 Test conditions
It is necessary to follow the guidelines set out in ISO 9335 and quote all relevant test conditions associated
with a particular measurement of MTF. These will include the spectral response of the measurement system,
field positions, focusing criterion, etc.
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ISO 15529:2010(E)
5.2 Test azimuth
5.2.1 Detector arrays and raster scan devices
For the purposes of this International Standard the MTF of a sampled system that includes a detector array,
must refer to a test azimuth. Normally this will be either perpendicular to the row of elements along which the
signal is read out, or parallel to them, but may also have other orientations. This also applies to systems that
include devices (such as mirror scanners, CRT displays, vidicon tubes, etc.) where an image is generated by
a linear raster scan, although in most such cases the system will behave as a sampled system only in the
direction perpendicular to the scan lines.
It is important to note that the orientation of the test azimuth can, in some cases, have significant implications
for the detailed manner in which some of the measurement techniques described in 5.3 and 5.4 are
implemented. This is particularly so when measurements are being made directly on a video output signal
from the system under test. There are two points to note in this case. The first is that the LSF corresponding to
a slit object perpendicular to the rows along which the array is read out will appear directly as such in the
video signal, but for a slit in the orthogonal direction, the LSF shall be constructed from the video signal on
sequential video lines. The second point is that the reconstruction function will be different for the two
azimuths and in fact in the latter case it will approximate to a delta function [i.e. T (r) = 1 for all frequencies].
rf
Methods 5.3.3 and 5.3.4 are in general only applicable to test azimuths in the direction of the rows or columns.
5.2.2 Fibre-optic face plates, channel multipliers and similar devices
For this type of device where, in effect, the output from each sampling aperture generates the corresponding
image point directly, test method 5.3.3 allows any test azimuth to be used for measuring the MTF, provided
the azimuth used is specified unambiguously with the result of a measurement. In practise it is usual to use
azimuths that correspond closely to recognised axes of symmetry in the pattern of sampling apertures.
5.3 Measurement of system MTF, T (r) of a sampled imaging device or complete system
sys
5.3.1 Measurement with cut-off spatial frequency of |F (r)| less than the Nyquist frequency —
in
Applicable to most types of device
Provided |F (r)| has a cut-off spatial frequency that is less than or equal to the Nyquist frequency, then we
in
see from Equation (5), that T (r) can be determined by calculation from the measured Fourier transform of
sys
the image of a static slit object. The effect of any components which make up the measurement system, such
as a lens, can be excluded from the value of T provided we know its MTF (this can usually be determined
sys
by a separate measurement).
The technique is applicable to almost all types of system. The method of measurement for a particular type of
system will be the same as that used for a non-sampled imaging system (see ISO 9335) with the important
proviso that |F (r)| [see Equation (5)] fall to zero by the Nyquist frequency.
in
The cut-off spatial frequency for |F (r)| can in many cases be appropriately adjusted by selecting a suitable
in
width for the slit object. However, this may not always be
...