INFORMATION RETRIEVAL FROM MOIRE
FRINGE PATTERNS IN THERMAL
DONALD L. G. STURGEON
A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
The author wishes to express his indebtedness to Dr. C. A.
Sciammarella, Chairman of his Supervisory Committee, for proposing
the research subject and for his kind criticism and valuable advice
during this endeavor.
Dr. S. Y. Lu and Dr. E. H. Hadlock, Members of the Supervisory
Committee, are gratefully thanked for their kind cooperation.
The author wishes also to recognize the staunch financial
assistance offered to this work by Dr. W. A. Nash, Chairman of the
Department of Engineering Science and Mechanics.
The National Science Foundation was the sponsor of the research
program of which this dissertation is a part.
Thanks are due to the staff of the Microwave Tube Laboratory,
the Induction Heating Laboratory and the Computing Center for helping
the author use their facilities during different stages of this research.
The fine assistance of Mr. C. Schultz in connection with the programming
code implementation is gratefully acknowledged.
Dr. B. E. Ross, Mr. F. P. Chiang, Mr. N. Lurowist, past and
present members of the Experimental Stress Analysis Laboratory research
team, and Mr. E. Teska, Engineering Aide, are thanked for their ready
The author wishes finally to express his deep gratitude to his
dear wife Mabel, for her help, patience and understanding during the
many lonely hours far away from family and old friends.
TABLE OF CONTENTS
ACKNOWLEDGMENTS . . . . . .
LIST OF TABLES . . . . . .
LIST OF FIGURES ............
LIST OF SYMBOLS . . . . . .
ABSTRACT . . . . . . . .
I INTRODUCTION . . . . .
II THE PHOTO-OPTICAL SYSTEM
2.1 Introduction .
2.2 Optical Analysis
of the System . . .
'er Function of Lenses
'er Function of Films
Experimental Determination of the Modulation
Transfer Function of the Photo-Optical System
Light Intensity Trace of Moire Fringes . .
Correction of Spatial Waveform Distortion
Filtering of a Moire Signal . . . .
III NUMERICAL METHODS FOR THE PROCESSING OF MOIRE DATA . . 31
3.1 Introduction . . . . . . . .. . 31
3.2 Auxiliary Tests . . . . . . . . 3
3.3 Displacement Determination by In-Phase
and In-Quadrature Filters . . . . . . 37
3.4 Strain Determination by Numerical
Differentiation . . .
3.5 Strain Determination by a Hybrid
Computer System . . . . . . ... h9
3.6 Comparative Performance of the
Proposed Techniques . . . . . ... 50
IV THE THERMAL TEST ................... .. 59
4.1 The Experimental Set-Up . . . . . .. 59
4.2 The Operation of the Test . . . . . . 64
TABLE OF CONTENTS (Continued)
THE THERMAL TEST (Continued)
L.3 Theoretical Solutions to the Thermal
Stress Problem . . . . . . .
4.h Experimental Determination of Thermal
Strains and Stresses . . . . .
4.5 Residual Strain and Stress Determination
V CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions . . . . .
5.2 Recommendations . . . .
APPENDIX A . . . . . . . . .
ON LINEAR FILTER THEORY . . . .
A.1 Linear Systems . . . .
A.2 Definition of a Digital Filter
APPENDIX B . . . . . . . . .
ON THE PREPARATION OF METAL MODELS FOR
HIGH TEMPERATURE MOIRE TESTS . . .
B.1 Introduction . . . . .
B.2 Surface Finishing of the Model
B.3 Photo-Engravure of the Grid
B.4 Chemical Etching of the Grid
APPENDIX C . . . . . . . . .
PROGRAMMING CODES . . . . .
APPENDIX D . . . . . . . . .
FIGURES . . . . . . . .
APPENDIX E . . . . . . . . .
DETERMINATION OF FILTER PARAMETERS .
LIST OF REFERENCES . . . . . . .
BIOGRAPHICAL SKETCH . . . . . . .
LIST OF TABLES
1 Relative Errors in the Strains of the Disk
Under Compression ................... .. 57
2 Film Performance as a Function of Temperature . ... 61
3 Yield Stress of 30h Stainless Steel as a Function
of Temperature ...................... 67
LIST OF FIGURES
1 Modulation Transfer Function Measurement . . . .. 112
2 H & D Characteristic Curve of a Film . . . . .. 113
3 Spatial Waveform Distortion Due to the Film . . .. 11
4 Unfiltered Transmission Trace of a Sinusoidal Exposure 115
5 The FunctionsSin x and Log Sin x . . . . . .. 116
6 Density Trace of a Sinusoidal Exposure in the Toe and
Straight Line Portion of the H & D Curve . . . .. 117
7 Correction of Waveform Distortion by the Non-Linear
Wedge Method ... . . . . . . ... . 118
8 Non-Linear Density Wedge Experiment . . . ... 119
9 Fine Structure of Moire Fringes . . . . .... 120
10 Coherent and Incoherent Light Systems
a) Coherent, b) Coordinates, c) Incoherent . . .. 121
11 Coherent Optical Filtering by Removal of the Bias Term
of the Grid . . . . . . . .... ..... 122
12 Incoherent Optical Filtering by Decreasing the
Aperture of the Iris . . . . . . . .... 123
13 Incoherent Optical Filtering by Defocusing ..... . 124
14 Incoherent Optical Filtering by Modification of the
Pupil Function with a Dirac Comb . . . . .... 125
15 Incoherent Optical Filtering by Increasing the
Aperture of the Scanning Slit . . . . .... 126
16 The Scanning Microdensitometer . . . . . .. 127
17 The Electro-Optical System. . . . . . . . 128
LIST OF FIGURES (Continued)
18 Ring Under Diametral Compression (Moire of U) .... 129
19 Ring Under Diametral Compression (Moire of V) .... 130
20 Transfer Functions of Ideal Filters . . . ... 131
21 Transfer Functions of a Combination of Tukey Filters . 132
22 Transfer Functions of an Ormsby Filter . . . .. .. 133
23 Digital Determination of the Displacement Curve Along
the Longitudinal Axis of a Traction Sample ...... 134
24 Digital Determination of the Displacement Curve Along
the Horizontal Axis of the Disk . . . . .... 135
25 Digital Determination of the Strains Between Two
Fringes in a Traction Sample . . . . . .... 136
26 Digital Determination of the Strains Along the
Horizontal Axis of the Disk. Comparison of Results 137
27 Analog Filtering and Differentiating Circuits with
Their Corresponding Outputs . . . . . .... 138
28 Analog Determination of Strains. Comparison of
Results . . . . . . . . ... . . .139
29 General View of the Experimental Set-Up . . . .. 140
30 Close View of the Model, Induction Coils, Insulator
and Cooling System . . . . . . . . .. 141
31 Temperature Distribution in Ring No. 1 . . . .. 142
32 Temperature Distribution in Ring No. 2 . . . .. 143
33 Moire Patterns of Ring No. 1, 5.5 in. OD and
2.5 in. ID . . . . . . . . ... . . 144
34 Moire Patterns of Ring No. 2, 5.5 in. OD and
0.5 in. ID .. . . . . . . . . . 145
35 Experimentally Measured Strains in Ring No. 1 ... 146
LIST OF FIGURES (Continued)
36 Experimentally Measured Strains in Ring No. 2 ... 147
37 Temperature Variation of the Mechanical Properties
of AISI 304 Stainless Steel . . . . . . .. 148
38 Comparison of Experimental Stresses in Ring No. 1 with
Those Predicted by Elastic Theory . . . . .. 149
39 Comparison of Experimental Stresses in Ring No. 2 with
Those Predicted by Elastic Theory . . . . . 150
40 Comparison of Experimental Stresses for a Material with
v = 0.5 with Hilton's Solution . . . . . . 151
41 Total Residual Strains in Ring No. 2 by the Moire
Method . . . . . . . . ... . . . 152
42 Boring-Out Test Set-Up . . . . . . . . 153
43 Residual Elastic Stresses in Ring No. 2 . . . .. 154
44 Residual Elastic Strains in Ring No. 2 . . . . 155
14 Residual Plastic Strains in Ring No. 2 . . . . 156
46 1OOX Magnification of an Attempt to Engrave 750 Lines
per Inch on an Insufficiently Polished Aluminum Sample 157
47 40X Magnification of a Photoengraved Cross Grid of
300 Lines per Inch . . . . . . . . . 158
48 Diagram Showing Underetching Effect . . . . . 159
49 40X Magnification of an Etched Cross Grid of 300 Lines
per Inch After Removal of the Resist . . . ... 160
LIST OF SYMBOLS
(w xw y
in x y
Defocal length or interior radius of a ring or a constant
Weight functions of a linear in-phase filter
Real part of H(f)
Amplitude distribution over the output plane
Fourier transform of A
Amplitude distribution over the input plane
Fourier transform of Ai
Exterior radius of a ring or a constant
Weight functions of a linear in-quadrature filter
Contrast of the object test target
Contrast of the image at spatial frequency w
Contrast of the object at spatial frequency w
Diameter of the aperture
Photographic density unit
Photographic film density in the x direction
Negative film density in the x direction
Dynachem Photo Resist
HA(U xu )
Impulse response of a linear system
Amplitude response to a point source
Fourier transform of hA(x,y)
Intensity response to a point source
Fourier transform of h1(x,y)
Filter transfer function
LIST OF SYMBOLS (Continued)
Modulus of elasticity or film exposure
Modulus of elasticity for average temperature
Negative exposure variation along the x direction
Highest frequency present in the raw signal
Center frequency of the desired signal
Cut-off frequency of the desired signal
Highest frequency of the desired signal
Roll-off frequency interval
Complex amplitude distribution over the surface of
Complex conjugate of f(a,B)
Shear modulus for average temperature
LIST OF SYMBOLS (Continued)
1in( x' y)
o ( x' wy)
Infinite series approximation to H(f)
Finite series approximation to H(f)
Summation variable, summation limit or v/1
Intensity variation in the x direction
Intensity amplitude coefficient
Intensity of a point source
Intensity distribution over the input plane
Fourier transform of Iin(x,y)
Intensity distribution over the output plane
Fourier transform of I (x,y)
Signal plus noise intensity function
Maximum of a sinusoidal intensity function
Minimum of a sinusoidal intensity function
Digitalized version of I (x)
Finite series approximation to I
Filtered version of I
In-quadrature intensity function
Digitalized version of I*(x)
Finite series approximation to I (x)
Filtered version of (I* )k
LIST OF SYMBOLS (Continued)
j Summation variable
J1 Bessel function of the first order
k Parameter defining extent of parabolic transition in
an Ormsby filter, summation limit 2n/X, or data
KPR Kodak Photo Resist
L Radius of a lens or a linear operator
z Initial length
zf Final length
m Summation limit
IMI Real part of a complex transmittance
MTF Modulation transfer function
Mo(W) Modulation transfer function of the photo-optical system
MF(w) Modulation transfer function of a film
ML(w) Modulation transfer function of a lens
N Summation limit
N(x) Noise content of a signal
n Summation variable
p Master grid pitch or summation variable
r Summation variable radial coordinate of a lens disk
s Summation limit
LIST OF SYMBOLS (Continued)
T Transmittance or temperature
T Transmittance of a negative
T Transmittance of a positive
T (uw) Maximum of a sinusoidal transmittance function at
T (w) Minimum of a sinusoidal transmittance function at
U(x) Displacement function in the x direction
u Object distance
uk Digitalized version of U(x)
uk Filtered version of uk
v Image distance
w Weights of a smoothing filter
x Cartesian coordinate or raw input to analog circuit
X Filtered output from analog circuit
X Filtered and derived output from analog circuit
x Particular value of x
y,z Cartesian coordinates
a Summation variable or angular coordinate, analog filter
parameter or thermal expansion coefficient
ar Thermal expansion coefficient for average temperature,
angular coordinate or analog filter parameter
o Frequency location of a maximum in the transfer function
of an annular aperture
LIST OF SYMBOLS (Continued)
o' c' t
Slope of the straight line portion of the characteristic
curve of a film
Slope of the straight line portion of the characteristic
curve of a negative film
Slope of the straight line portion of the characteristic
curve of a positive print
Distance between fringes
Input to a linear system
Eulerian strain or a variable
Digitalized raw value of the strain
Filtered version of ek
Eulerian elastic strain component
Eulerian plastic strain component
Eulerian total strain
Eulerian circumferential strain
Eulerian radial strain
Average strain in a traction sample
Limit of integration
Argument of H(f)
Spatial or light wavelength
Center, cut-off and shortest spatial wavelength in the
Mean value of a random variable
Poisson's ratio for the average temperature
LIST OF SYMBOLS (Continued)
Argument function of the coordinate x
Yield stress of stainless steel
Normal circumferential stress
Normal radial stress
Output of a linear system
Fourier transform of c(x)
Input to a linear system
Fourier transform of t(x)
Complex intensity function
Sampled version of y(x) and *(x)
Finite series approximation to pn
Spatial frequency in a given direction
Spatial frequency in the radial direction
Abstract of Dissertation Presented to the Graduate Council
in Partial Fulfillment of the Requirements for the
Degree of Doctor of Philosophy
INFORMATION RETRIEVAL FROM MOIRE FRINGE
PATTERNS IN THERMAL STRAIN FIELDS
Donald L. G. Sturgeon
Chairman: Dr. C. A. Sciammarella
Major Department: Engineering Science and Mechanics
This research deals with the implementation of a linear filter-
ing system capable of yielding the strains in a solid body from the
information on the displacement field contained in the light intensity
trace of moire fringes.
An analysis of the photo-optical device used to produce and
record the fringes illustrates quantitatively the filtering abilities
of the lens-film combination. Solutions to the non-linear sensitometric
characteristics of the film are applied to moire data.
Several experiments of optical filter synthesis that increase
substantially the quality of recorded patterns are reported.
The retrieval of the meaningful phase modulation of the light
intensity trace from the spurious amplitude and phase variations due to
noise is successfully accomplished by the implementation of numerical
filters in quadrature. This is used to determine new bounds on auto-
matic fractional moire fringe interpolation.
The numerical differentiation of the empirically determined
displacement curve is critically reviewed and a significant increase
in the accuracy of this process is obtained by the use of several
Necessary techniques of model grid engravure able to withstand
high temperatures are developed.
The above improvements on the moire method are used to measure
thermal strains in stainless steel rings, under steady state thermal
loading, up to a maximum temperature of 16000F.
The moire method has the inherent capability of performing
large field strain determinations in such important areas of solid
mechanics as elasticity, viscoelasticity, plasticity and fracture,
under mechanical, thermal, static and dynamic loading conditions.
In some of these fields the experimental stress analyst has a
choice of several applicable methods, in others, moire today offers
the only possible solution.
It is unnecessary to review here the history of the development
of the moire method as an experimental stress analysis tool. Complete
chronological listings of work performed on this subject are available
Though in principle the method can yield the information which
we desire on the strain field in the body, not always does this come
in a convenient and direct form. A certain degree of sophistication
in the photo-optical instrumentation involved is therefore necessary
if the required degree of sensitivity and accuracy in the quantitative
results is to be achieved.
In this connection it is important at this early stage of our
presentation to stress as fundamental the difference between the dis-
crete and the continuous approach to the information contained in the
moire fringes as was explained by Sciammarella [ 3].
The foundation of what will be said in this entire work will be
found in the moire light intensity-displacement law derived by
Sciammarella  and tested by Ross, Sciammarella and Sturgeon .
We accept therefore that the displacement component in the
direction normal to the master grid at a point in the deformed body
is related in a continuous form to the light intensity at that point
by the expression
U(x) = arc cos p (1.1)
where the coordinate x is measured in the direction normal to the
It has been shown [h] that by the proper use of (1.1) it is
possible to increase the sensitivity and accuracy of the moire method
using coarse grids of 300 lines per inch far beyond the bounds set by
the discrete theory . Furthermore, it was shown that the process-
ing of moire data could be made almost totally objective when instru-
mentation was designed to perform the operations made by hand in the
discrete approach .
The principal goal of this research was to implement an experi-
mental setup capable of performing quantitative strain determinations
with sufficient accuracy in metals at high temperatures by the moire
Since a careful systems analysis of the entire moire process
was thought to be indispensable to achieve our purpose, several aux-
iliary tests under mechanic loading had to be performed whenever these
helped to clarify aspects of our analysis. Traction tests on bars and
compression tests on disks were made for this purpose.
The optical synthesis of moire data, not previously attempted,
seemed to offer interesting possibilities to filter out the inevitable
photographic noise present in all moire patterns obtained under any-
thing but the ideal conditions of illumination and surface character-
istics. This matter was therefore pursued and definite improvements in
the quality of the moire patterns were obtained.
Previous to this research, an electronic band-pass filter able
to operate between 0.02 cps and 2000 cps, had been used to filter both
high and low spurious spatial frequencies from the moire light intensity
trace . By means of an initial mismatch the frequency of the moire
signal before and after deformation was made to fall within the range of
However, the inclusion of the electronic filter caused large
oscillations of long duration on the analog voltage output of the photo-
reading device every time a negative was introduced or removed from the
electro-optical circuit. The decay time of these oscillations, caused
by the response of the instrument to a step load, was found to be in
excess of one hour.
Furthermore, the displacement information due to the deforma-
tion of the solid body had to be obtained as the difference between
the final and initial pattern. When strong mismatch was present, in
conjunction with small deformations, this difference between two large
quantities of approximately the same magnitude impaired the accuracy
of the procedure.
Since the amplitude variations were not corrected by the elec-
tronic filter, a numerical normalization technique was necessary in
the digital program that handled the moire data . The errors
involved in this step were found to be considerable and are discussed
fully in .
As an alternative to the above procedure, numerical filtering
with the aid of a digital computer was introduced here as a means of
retrieving the meaningful spatial phase modulation of the light inten-
sity trace from the spurious variations of phase and amplitude. This
was possible by the implementation of in-quadrature filters that not
only removed all the unwanted frequency components of the signal with
no transient perturbations, but also produced an output completely
independent of the amplitude of the moire light intensity trace. This
was instrumental in establishing the practical bounds on the fractional
fringe interpolation that is possible by application of the continuous
The differentiation process that enables us to obtain the
strains from the displacement field was also given a good deal of atten-
tion. A critical evaluation of several differentiation procedures of
experimental data was made. This permitted a considerable increase in
the accuracy of the strain determination which is perhaps the single
most undesirable feature of the moire method as applied to experimental
We mentioned at the beginning of this introduction several pos-
sible applications of the moire method in solid mechanics; none is more
challenging than the measurement of strains in metals subjected to
thermal loading. In this case the absolute value of the strains and
of their difference across the body are small, so both the sensitivity
and accuracy of the method of measurement are factors of primary
Therefore, the ability of moire to produce good results under
these conditions is a suitable test of excellence for the entire system
performance. Such results were obtained in this research.
The extension of the moire method to high temperatures, at which
conventional photoengravure techniques of model line printing fail
because the photosensitive materials used burn off, required the estab-
lishment of new methods of grid manufacture. Several such methods were
devised for aluminum, copper, mild and stainless steel.
By the use of these grids, moire fringe patterns were photo-
graphed up to a maximum temperature of 16000F.
THE PHOTO-OPTICAL SYSTEM
The photo-optical system used to produce and record moire
patterns, has the ability to perform as a linear low-pass filter of
The system is composed of the lens and the film, both of which
have individually this ability. Thus, they operate together as two
linear filters in series.
The filtering effect can be thought of as the process by which
the higher spatial harmonics of the Fourier decomposition of the object
are unable to pass onto the recorded image. This has a degrading effect
on the image, since the infinite expansion that represents the object
in all its detail, is interrupted after a given finite number of terms.
Moreover, we will see that the effect of the system is to damp,
with progressive vigor as frequencies.increase, the corresponding ampli-
tude coefficients of the Fourier expansion. This produces an increasing
loss of contrast as the spatial wavelengths get shorter.
In what follows, we will understand by system analysis the proc-
esses by which we record passively the action of a given system on the
We will call system synthesis the processes by which we act on
the system to change its frequency response characteristics to suit
2.2 Optical Analysis of the System
Due to the filtering effect mentioned above lens and film
manufacturers have sought to devise means of evaluating the image form-
ing capabilities of lenses and films. One such technique, based on the
Modulation Transfer Function (MTF), offers special advantages for the
study of the photo-optical system used to record moire data.
The method consists of obtaining the image formed by the photo-
optical system of an object made up of parallel lines with a sinusoidal
intensity distribution. A typical test object is manufactured by the
Eastman Kodak Company and can be purchased commercially.
The MTF can be defined numerically as the ratio of the contrast
in the image to the contrast in the object for the same spatial
We define as is usual the contrast C of the object test target
under coherent or incoherent illumination, as the ratio of the ampli-
tude of the sinusoidal intensity distribution to the mean background
I max I min .
I max + I min (
where the mean value is sufficiently large so the luminance is always
If we now define the spatial frequency of the sinusoidal test
object as W = 2rn/, where X is the wavelength, then the MTF is
M(w) = C (2.2)
where C (w) and Co(u) are the contrast of the image and the object at
The MTF is then a measure of the relative degradation in ampli-
tude of the image of a sinusoidal object.
Under incoherent illumination, a condition common in moire work,
the optical response of the lens and the film may be regarded as two
linear operations in series. This requires, however, that the exposure
level of the photograph be adjusted so that the entire amplitude of the
sinusoidal input will lie within the straight line portion of the H & D
characteristic curve of the film.
Under the above conditions of linearity, the MTF of the photo-
optical system can be expressed as
Mo(u) = ML( ) MF() (2.3)
where Mo(w), ML(u) and MF(u) are the transfer functions of the photo-
optical system, the lens and the film, in that order.
The MTF technique offers the following advantages for moire
a) The test targets used in the method resemble exactly the
actual objects, in fact, the sinusoidal test targets can be considered
as perfect, noiseless, undistorted moire fringes.
b) The performance of the system is evaluated over a large area
in the object plane.
c) The MTF can not only be used to predict the response of the
entire system, but can also be used to examine each element in the
photo-optical process to determine what degradation of the image is
introduced at each stage.
d) The MTF of a system can be obtained from those of its com-
ponent elements by simply multiplying the corresponding MTF curves,
ordinate by ordinate, for each abscissa value.
e) The knowledge of the MTF curves of the system, as will be
explained later, enables us to determine the largest maximum displace-
ment that can be measured by the system.
f) For the critical work of reproduction of moire line masters,
the MTF offers again quantitative information regarding the contrast
that can be expected between dark and clear lines in the copy.
g) Finally, the MTF curve adequately represents the optical-
filtering capabilities of the system.
Notwithstanding the above, Hempenius  has found that photo-
graphic systems analysis based only on the MTF characteristics may be
in some cases significantly in error and that the effect of grain should
be taken into account. This is not believed to be the case in moire
2.3 Modulation Transfer Function of Lenses
Camera lenses degrade the arriving light signal by aberration
and by diffraction.
We will first consider the diffraction effects, assuming for
this purpose that we possess an ideal aberration-free lens.
Between the spatial wavelengths large enough to be passed with-
out attenuation of amplitude and the one that represents the limit of
resolution, the MTF of an ideal lens varies as shown in Figure la.
We will now turn to the aberration losses, which depend on the
lens construction. Photographic lenses must be made to operate at
various aperture settings and the best any manufacturer can do is offer
several different compromises between the various types and amounts of
aberrations: distortion, astigmatism, curvature of the field, etc.
However, the optimum chromatic correction available should be preferred
for both moire master and fringe photography.
Regarding the other aberration losses, unfortunately no recom-
mendations of such a general nature can be given. Kelly  reports
tests on a 50 mm f/2.8 photographic lens that show that geometrical
aberrations are responsible for most degradation at f/2.8 while at f/8
most degradation is due to diffraction. Hence the aperture setting of
the iris has an important effect on the MTF.
This situation obliges us to resort to actual testing of the
lens we intend to use for a future optical filtering application.
Modification to the curve of Figure la due to aberration must be deter-
mined experimentally for the magnification and aperture setting we
intend to use if this information is not supplied by the lens manufac-
A test of this nature is described in Section 2.5 in detail.
2.4 Modulation Transfer Function of Films
The Modulation Transfer Function of a film decreases with
increasing spatial frequency and resembles, in a general way, that of
a lens. The damping of high frequency amplitude coefficients reflects
the effect of the diffusion of light within the emulsion on the micro-
structure of the image.
However, all films do not possess the same limiting curve of
maximum resolution as is the case with lenses. Films with a MTF
approximately equal to one over a broad range of spatial frequencies
are manufactured for spectroscopy and document reproduction.
The MTF for the Kodak Royal Ortho film used in the photo-
optical test described in Section 2.5 can be seen in Figure lb.
Complete information on the MTF of numerous photographic films
and plates is available from the manufacturers .
2.5 Experimental Determination of the Modulation
Transfer Function of the Photo-Optical System
Tests were conducted to determine the MTF of a 12 in., f/10,
Alphax copying lens, and of this lens together with Kodak Royal Ortho
The lens and film were used later in the thermal test of
Chapter IV. The magnification tested was one to one and the aperture
setting was f/10.
Since the MTF will vary across the image plane of a photo-
graphic object, from its highest value near the optical axis to a lower
value near the edges, test targets were placed in the center and at one
edge of the object plane.
The test targets manufactured by the Eastman Kodak Company vary
sinusoidally in transmittance, with harmonic distortion of less than
2%, and range in spatial frequency from 3/8 to 42 cycles per millimeter.
The modulation of transmittance is approximately 65% for the target
placed in the center of the field and 30% for the one placed at the edge.
The targets also possess a grey scale for calibration purposes. The
optimum performance of the lens was found to be restricted to a circle
with center at the optical axis and radius 0.16 the focal distance.
Since the focussing of the camera has a great influence on the
MTF the exact focus will provide the highest limit of resolution. For
every position other than the true focus the lens performs with a differ-
ent MTF. The further the image plane is removed from the focal plane,
the poorer the resolution becomes .
This immediately suggests a way of filtering out excessive low
frequency passed by the lens by slightly defocussing the optical system.
This also obliges us to perform the lens test for the same posi-
tion of the image plane with respect to the true focal plane as that
at which we will operate during the actual running of the moire test.
This is not an easy matter since both tests may be performed at differ-
ent locations and with different optical equipment except for the lens
The recommended technique  of taking several photographs at
positions close to the true focus and selecting the optimum between them
is too lengthy for most experimental stress analysis applications where
the object may have to be moved several times during the test.
We have used, to select the optimum focussing in both the lens
and thermal tests, the visual inspection with the aid of a 60X micro-
scope of the image of 12 lines per millimeter produced on a finely
ground glass manufactured by us for this purpose.
To record the image we selected Kodak Royal Ortho film with an
ASA speed of 400 because of its known MTF, H & D characteristic, and
proven ability to record moire patterns at elevated temperatures.
Care was exercised to expose the sinusoidal test patterns entirely in
the straight line portion of the sensitometric curve.
The MTF of the film is represented in Figure Ic. The MTF of
the lens-film combination was obtained by scanning both the test object
and the negative with a Joyce and Loebl microdensitometer. By scanning
the grey scales of the target, a calibration curve can be obtained
relating the densitometer reading to the corresponding density .
The scans of the sinusoidal patterns were then analysed to
determine the average values of the maximum and minimum densities for
a given frequency.
The modulation was then obtained by the use of the formulas 
AD = log T min (w) log T max (w) (2.h)
C(w) = T max (w) T min (w) (2)
T max (w) + T min (w)
The plot of the modulation transfer function Mo(w) correspond-
ing to the lens-film combination that was obtained from the experiment
can be seen in Figure ld.
According to (2.3), if we have measured Mo(w) and we are given
Mp(w), the MTF of the lens alone can be obtained. The results have
been plotted in Figure le.
The product of the MTF of the ideal lens and the film differs
considerably from the measured values of the combination. This is par-
ticularly so as the wavelengths get shorter. The vigor of this damping
is undoubtedly due to incorrect focussing and aberrations rather than
to the latter alone. This emphasizes the strong dependence of the
transfer characteristics of the lens on the correct focussing .
Figure If shows the results of testing the lens-film combina-
tion at one edge of the image plane. Comparison of Figure ld with
Figure If shows how the system performs better for smaller angular
apertures. The damping of the very long wavelengths indicates loss of
focus by lack of parallelism of the image plane with respect to the
2.6 Light Intensity Trace of Moire Fringes
The light intensity variation along a line in a moire pattern
I(x) = 1I + Il cos 2n p(x) (2.6)
can be obtained from the photographic negative used to record the
pattern. A photo-reading device as the one presented in  can be
used to obtain the trace of the transmittance of the negative, or
a scanning microdensitometer can be employed to record the density varia-
tions of the film.
The incident exposure on the film can then be retrieved from the
transmittance or from the density trace by performing the operations
En(x) = I(x).t = T n()vn (2.7)
En(x) = I(x).t = ILog-1 Dn(x) n (2.8)
as the case may be.
It has been the custom in moire work to regard the negative,
obtained by photographing the fringes, as a suitable representation of
the original exposure pattern. The clear fringes in the model appeared
as dark fringes in the negative and vice versa, but no consequence was
attached to this since the curve of displacements was plotted with
respect to an arbitrary origin.
A look at the characteristic curve of a film shows immediately
that the output of a film to incident exposure is transmittance or
density of silver deposit. Furthermore, this relationship is non-
linear and independent of the spatial frequency of the input. There-
fore, what is recorded on film of an arriving spatial sinusoidal varia-
tion of intensity is a harmonically distorted waveform of spatial den-
sity or transmittance, Figure 2.
The amount of this distortion is illustrated in Figure 3 where
exposure rather than the logarithm of the exposure has been plotted in
the abscissas. In Figure h we show the strong distortion of an actual
transmittance trace of a sinusoidal light intensity variation corre-
sponding to a moire pattern of fringes.
Since the application of the continuous light intensity dis-
placement law makes use of the light intensity of all portions of the
spatial waveform, the undistorted reproduction of the signal is manda-
tory and Equations (2.7) or (2.8) should be applied always.
It will be seen in Section 2.7 that it is possible to perform
photographically these operations.
It is interesting to note that there is no space shift of the
maximums and minimums of the intensity curve with respect to those of
the transmittance curve. Therefore, the usual method of hand analysis,
that only takes these points into account for the plot of the displace-
ment curve, will be free of the errors due to photographic reproduction
The continuous trace of incident exposure will contain, in addi-
tion to the phase modulated spatial curve containing the information on
the displacement in the body given by (2.6), spurious amplitude and phase
variations due to noise. These disturbances are visible in Figure U.
We wish to extract (2.6) from the noise in which it is submerged
before proceeding with the determination of displacements to increase
the accuracy of the procedure.
The amplitude variations are caused by the non-uniform illumina-
tion of the object, by changes in the reflectance of the model surface,
and by the progressive vigor with which the amplitude coefficients of
the Fourier decomposition of the image are damped by the filtering
action of the photo-optical system.
The more frequent causes of short wavelength variations are the
image of the master and model grid lines on the moire fringe, the
scratches and pits in the model surface, and the dust that can settle
over the model during testing. The long wavelength variations, on the
other hand, are often due to non-uniform illumination and surface
reflectance or transmission.
Since these perturbations in the trace are of both longer and
shorter wavelengths than those of the useful signal, a band-pass filter
is necessary to eliminate them.
Optical band-pass filters cannot be easily synthesized; however,
the photo-optical system used to produce and record the moire fringes
has inherent low-pass filter capabilities of which we can take advantage.
This will allow us to implement,optically at least, one part of our
Both long wavelength and amplitude perturbations may be removed
by other than optical means, and, to do this, digital in-quadrature
filters have been used successfully as will be seen in Chapter III.
It remains for us to develop in this chapter the techniques to
photographically correct the wave distortion due to the non-linear film
response and to substantially reduce optically the noise content of the
moire fringe light intensity trace.
2.7 Correction of Spatial Waveform Distortion
The so-called straight line portion of the characteristic curve
of the film is a linear relationship between the density and the logar-
ithm of the exposure. The logarithm of a sinusoidal exposure variation
is shown in Figure 5.
Therefore, a sinusoidal exposure within the straight line portion
of the sensitometric curve will produce a spatial density variation lin-
early proportional to the logarithm of the arriving sinusoidal spatial
waveform. However, if the exposure cannot be kept entirely within this
straight section of the sensitometric curve and the toe of this curve
is used to record the arriving signal, further distortions are produced
as shown in Figure 6.
If we wish to eliminate photographically the spatial harmonic
distortion due to the non-linear response of the negative film, the
following two processes have been proven effective.
a) Negative-positive film combination
A combination of negative and positive films can be chosen such
that the products of their gamma coefficients is equal to one .
For the straight line portion of the characteristic curve of
the negative, the following holds
D = v log E + C (2.9)
where D is the density of the developed film, E is the exposure,
Yn the slope of the sensitometric curve which in general will be posi-
tive and larger than one, and C1 is a constant. The subscript n refers
to the negative.
The transmission of the negative is given by
T = C E n (2.10)
where C2 is a constant. If we print this negative on a positive film
with a v such that
n p =l (2.11)
T = C2 E en p = C2 E (2.12)
p 2 n 2 n
where T is the transmission of the positive and C2 is a constant.
Thus a film linearly proportional in transmission to the exposure of
the object has been obtained.
This process cannot always be applied in moire work because of
the diversity of films that are used as negatives depending on the
applications and the very few pairs of films whose product of gammas
is equal to one.
In the thermal test reported in Chapter IV, for example, six
different films were used to record the fringe patterns, since varia-
tions in the ASA speed became necessary as temperatures increased.
Since the value of y depends on the developing time, some suit-
able pairs of films can be obtained if the stringent requirements on
processing can be enforced.
b) Non-linear density wedge method
In this method we make use of a microdensitometer as the Mark
III C manufactured by Joyce Loebl and Company of Gatehead, England.
This machine measures density on the negative by balancing it with
a known density located on a linear density wedge.
The principle of operation is based on a true double-beam light
system in which two beams from a single light source are switched alter-
nately to a single photo-multiplier.
One beam of light is passed through a point on the film, the
other through a calibrated reference wedge of linear density variation.
If the two beams arriving at the photo-amplifier are of different inten-
sity, a servo motor is activated that causes an optical attenuator to
reduce this difference to zero. In this way a continuously null balanc-
ing system is obtained in which the position of the optical alternator
is made to record the density at any point of the specimen.
Under the above mentioned conditions the machine will perform
linear density measurements. However, since we are interested in
retrieving the exposure sinusoidal variation, the density wedge of the
machine can be modified to achieve this.
Figure 7 shows the density trace of a sinusoidal exposure, the
sensitometric characteristic curves of the linear wedge and of a non-
linear wedge with the corresponding output functions.
The linear wedge produces a true density measurement at each
point which results in a distorted exposure curve. The non-linear wedge
will produce an output identical to the exposure input, if the sensito-
metric curves of the film and wedge are identical.
To accomplish this practically, we have printed on the type of
film being used as a negative, the appropriate linear density wedge.
The developing process -was identical for both the negative and the non-
linear wedge so the resulting y coefficient of both sensitometric
curves was the same. The results obtained by this method are shown
in Figure 8.
The linear wedges photographed on film were placed between two
glass plates held together with Canada Balsm. The resulting non-linear
wedges were placed in the travelling carriage of the microdensitometer,
which then performed linear measurements of exposure for this particular
The advantage of this procedure over the negative-positive film
combination is that the appropriate non-linear wedge characteristic
curve can always be obtained for any negative. Both share the dis-
advantage of requiring strict film processing control.
2.8 Filtering of a Moire Signal
Several successful optical filtering operations have been
reported in the communication sciences and optical literature [14, 15,
16], however, moire work has not previously benefitted from these use-
We will describe in what follows several experiments in which
the quality of moire fringes has been substantially increased by
As we said in Section 2.6, an optical filter of the low-pass
type can be readily implemented, in fact, the tests reported in Sec-
tion 2.5 indicate that a perfect lens has high frequency filtering
ability. However, the cut-off frequency of a real lens-film combina-
tion may not be sufficiently short to eliminate the undesirable disturb-
ances of the light intensity trace of the moire fringes. Therefore,
additional modifications of the transfer characteristics of the optical
system are necessary.
The purpose of this section is to establish the practical means
of achieving these modifications in moire work. It will be seen that
strong disturbances are not always completely eliminated optically but
that the noise level is substantially reduced in all cases.
We should keep in mind during this discussion the structure of
the image of a moire fringe shown in Figure 9a. The phase variations of
the average light intensity trace indicated in this figure carry the
information regarding the displacements in the body. Our purpose is to
obtain this average by optically erasing the image of the individual
grid lines and the random high frequency variations due to other sources
of noise. Failure to do this will result in an intolerable amount of
noise in the analog voltage output of our photo-reader that is to be
automatically processed to yield the displacements and strains.
It is necessary to distinguish in our treatment between the
coherent and incoherent illumination over the object plane. In prac-
tice, conditions of illumination are such that some degree of coherence
is always present, but, for simplicity, only the extreme cases mentioned
above will be assumed here to exist.
a) Coherent illumination
Coherent light waves add in amplitude and linear relations hold
between the amplitudes at different points in the system. Furthermore,
these systems possess the property that a Fourier transform relation
exists between the front and rear focal plane of a lens.
Under these conditions the amplitude distribution A over the
output or image plane is given by Figure lOa.
Ao(x3,y3) = Ain(lyl)hA(x3 -x, y3-)dx1 dyI (2.13)
where Ain is the amplitude distribution over the input or object plane
and hA is the amplitude response to a point source. In the spatial
frequency domain the above equation is equivalent to
o x' y A( xwy in( x' y (.1
where A ,HA,'i are the Fourier transforms of A HA and Ain
HA is therefore the amplitude transfer function of the optical
system that operates on the object amplitude spectrum to give the image
Due to the Fourier transform relation that holds between the
front and rear focal plane of a lens, Figure lOa, the function HA is
displayed at F2, thus affording a means of controlling the Fourier com-
ponents that make up the image by inserting an appropriate mask in
The mask can in general be a complex transmittance M = IM(ei ,
thus affording control over the amplitude by varying the optical den-
sity of the mask and over the phase by varying the thickness. One such
complex filter function has been implemented recently by Vander Lugt .
In dynamic and high temperature moire investigations it is often
necessary to produce a fringe pattern between the image of the model's
grid and a master grid placed in the back of a photographic camera.
The moire pattern thus recorded on film contains not only the image
of the fringes but also the contact copy of the line master. These
lines produce an undesired high frequency disturbance in the light
intensity trace of the fringes, Figure 9b.
When the master is in the object plane, a well-focused photo-
graphic camera will also record the image of the individual lines
together with the moire fringes, if the aperture is not sufficiently
reduced to perform a type of filtering that will be discussed in (b).
In this case again, the imagesof the grating lines are a source of
noise, Figure 9a.
To remove this disturbance, the negative of the moire pattern
was inserted in plane P1 of Figure 10a. The spectrum of the image of
the grating lines was displayed at P2 and a completely opaque mask
wide enough to interrupt the central order of the Fourier decomposition
was placed at P2. The remaining orders were collected by the lens L3
to produce an image almost completely free of the grating lines at P3'
A 5OX magnification of the appearance of the microstructure of
the image before the bias removal is shown in the upper insert of Fig-
ure 11, while the lower insert corresponds to a O5X magnification after
The strong damping is due to the fact that the film, as the eye,
is unable to register negative intensities and records only a pattern
proportional to the square of the amplitude of the arriving disturbance.
b) Incoherent illumination
Incoherent light waves add in intensity and linear relations
hold between the intensities at different points in the system.
The intensity distribution over the image or output plane
Io(x2'Y2) is given by Figure 10c.
Io(x2'Y2) = .f lin(x1,1)h(x2-x1 Y2-Yl)dxdyl (2.15)
where l. (x1,y1) is the intensity distribution over the object or
input plane and h is the intensity response to a point source. In the
spatial frequency domain the above equation is equivalent to
To (x ,Wy) = in(uW ,y) (x ,ui ) (2.16)
o' x' y in' x' y ix' y
where T in', are the Fourier transforms of I I. and HI,
O in 0 in I'
HI is then the intensity transfer function that operates on the
object intensity spectrum to yield the image intensity spectrum.
It can be shown  that
C ( *c I w w
H(W, = f(,) f -, do dO (2.17)
where f(c,B) is the complex amplitude distribution over the surface
of the aperture and f (,P) its complex conjugate. The angular
coordinates a and B are shown in Figure lOb. C is a constant and
K = 2n/X, where X is the wavelength of the radiation.
Therefore, control over the amplitude and phase of f(ce,B)
affords a means of achieving various filtering effects.
For a perfect lens f(a,B) is real and equal to one over the
entire aperture and zero elsewhere. The application of (2.10) to a
perfect lens results in the curve of Figure la and indicates that a
perfect lens is limited in resolution by its aperture. The limit of
resolution being equal to d/X. This immediately suggests a simple
way of damping the high frequencies present in the object by reducing
the size of the iris of the optical instrument.
Application of this filter effect to moire fringes produced the
results shown in Figure 12 where the microdensitometer tracesof two
fringes corresponding to the same traction test are presented. The
master grid was in the object plane when the photograph was taken.
The upper fringe corresponds to an aperture setting of f/10 and the
lower to an aperture of f/64. The damping of the 300 line per inch
of the master is clearly visible.
This method is not always practical in moire work because
exposure times become very long for the small apertures necessary to
filter out coarse grids at one to one magnification. However, since
smaller apertures will always filter some high frequency noise, though
not perhaps the grid lines, they should be used whenever possible.
We next will take advantage of the strong influence of defocus-
sing on the transfer function of a lens which was already evident in
the tests of Section 2.5.
When we move the image plane from the focal plane we leave the
Frauenhofer diffraction region and enter the Fresnel zone where the
diffraction pattern at the output plane will no longer be the Fourier
transform of the aperture. However, Cheatham and Kohlenberg ,
using the approximations of geometrical optics, obtain as the unit
intensity response of the defocussed system
for r < -
h(x,y) = h(r) = for (2.18)
0 for r > --
where I is the intensity of the point source per solid angle, u the
object distance, v the image distance and a the defocal length.
co i-i(w +w )
H(wxWy) = h(x,y)e xx dx dy (2.19)
H(wr) = 2n I 1 v (2.20)
r= x + y wr = V + 2
According to (2.13) an increase in the defocal distance will
close the low-pass filter characteristic of the system. The practical
significance of this fact can be seen in Figure 13 where the micro-
densitometer traces corresponding to the same fringe of a traction test
has been photographed at different focal settings. The upper fringe of
the picture was taken at the correct focus, the lower with the image
plane removed a = 0.0625 in. from the focal plane of the lens.
Further experimentation was conducted to increase the quality of
the image of periodic line patterns, whether they were moire fringes or
grid lines, under incoherent illumination and submerged in strong random
In this connection there exists an extensive literature in the
field of optics dealing with the calculation of the intensity transfer
function HI( x, y) of a lens when aberrations are present. Steel 
and O'Neill have calculated such functions for an annular aperture
and Franyon  reports work done by Sayanagui on a circular aperture
covered by small randomly distributed transparent elements.
The family of modulation transfer functions obtained by the
first two authors, for different ratios of interior to exterior diameter
of the annulus, show how the transfer characteristics of a lens can
be modified to obtain essentially different filtering effects.
As the ratio of the diameters approaches one, the transfer
function develops narrow peaks in the vicinity of w = 0 and w = 28 ,
where Po = kd/2r, k = 2n/X and d is the diameter of the aperture.
Since this effect would be useful to eliminate random noise
from a periodic line pattern which had a spatial frequency equal to 28o,
experiments were conducted with annular apertures of the type described
above. However, the loss of illumination for ratios of internal to
external diameter approaching unity was so severe that photography of
the filtered pattern became impossible.
Turning now to the work done by Sayanagui, this investigator
produced the random dots of vacuum deposition on a flat piece of glass
with parallel faces which he then placed before the lens. By this means
he produced a filter that could eliminate in the image plane P1 of an
incoherent light system, Figure lOc, almost all traces of the screen
mesh of a half-tone photograph introduced in the plane P..
Since we did not have at our disposal the elements to implement
exactly Sayanagui's experiment, we introduced in the air space of an
apochromatic photographic lens a Dirak "comb" consisting of an array of
dark and clear parallel lines and provided the means of rotating this
set of lines with respect to the optical axis of the instrument.
Samples of photographic noise were manufactured by exposing
Kodak Kodalith film through a paper hand towel in an ordinary photo-
graphic contact printer. This artificial noise appeared to be of
a conveniently random nature and could be produced in several densities
by varying the exposure time.
A periodic line pattern was superimposed with the noise to form
the object and was focused on the image plane of a lens containing the
filter described above. The object and image obtained by this procedure
are shown in Figure l1.
The density of lines tested as filters were 50, 100 and 300 lines
per inch. A set of two arrays independently rotatable was also used.
The effect of each filter was to make the lens strongly aberrant
astigmatically. Astigmatism was maximum in the direction perpendicular
to the filter lines and zero in the parallel direction.
A single filter with 300 lines per inch produced the results
shown in Figure 14.
The good results obtained by this method led us to a careful
search of the available literature on the subject of apodization tech-
niques for an analytical explanation of the observed phenomenon but
none seems to have been published to date. We believe that the compu-
tational complexities of such a theoretical development fall beyond the
scope of this research.
A final method of filtering high spatial frequencies of the image
is by averaging the light intensity transmitted through the negative of
a fringe pattern in a photo-reading machine. If the scanning slit of
the instrument is made large enough, the contribution of the individual
lines to the overall intensity is diminished significantly.
We have found that to eliminate completely the influence of
the master grid on the photo-reading machine trace of a fringe produced
with the master in the image plane a slit at least ten times the size
of the individual lines must be used.
For fringes produced with the master in the object plane, slits
three times wider than the lines significantly reduced their influence
on the trace. Figure 15 shows a microdensitometric graph of the same
fringe scanned with a slit opening of 0.084 mm for the top trace and
0.27 mm for the bottom one.
It is interesting to note how all filtered fringes, while having
damped the image of the grid lines, still reproduce faithfully all varia-
tions of the average light intensity. This average is precisely what
carries the information pertaining to the displacement.
NUMERICAL METHODS FOR THE PROCESSING
OF MOIRE DATA
Having produced and recorded a moire fringe pattern with the
photo-optical instrumentation analysed in Chapter II, it remains for us
to obtain the displacements and strains from the continuous light inten-
sity trace of the fringes.
This trace, as we said earlier, contains the useful signal given
by (2.6) plus the spurious variations of amplitude and phase due to
We will call the intensity trace function before filtering I (x)
I(x) = I(x) + N(x) (3.1)
where N(x) is the amplitude and phase noise contained in the signal.
This noise content will depend on the amount of optical filter-
ing performed on the pattern, however, for the purpose of this chapter,
no previous filtering will be assumed to have been made.
We are faced then with the necessity of implementing a numerical
band-pass filter to eliminate the unwanted disturbances of the moire
light intensity trace given by (2.6).
Once this has been achieved, we desire to obtain the displace-
ments given by (1.1) along the direction normal to the master grid lines.
The digitalized version of I (x) can be obtained by the modifi-
cation of a scanning microdensitometer as the Joyce & Loebl Mark III C
shown in Figure 16.
If this instrument is provided with a potentiometer arrangement
that will yield analog voltages proportional to both the density meas-
ured on the pattern and to the coordinate position, these voltages can
then be digitalized on a voltmeter and the numerals printed on tape for
convenient handling in a high speed computer.
The system, consisting of a microdensitometer and a digital
voltmeter, used in the present investigation is shown in Figure 17.
The principle of operation of the microdensitometer was explained
in Section 2.7. It is capable of performing linear density measurements
from 0 D to 6 D, and of linear magnification from 1 X to 1000 X.
The digital data acquisition system is a Dymec 1020 B, that, at
the sampling rate of ten words per minute, has a maximum resolution of
With this system, care must be taken in moire work to use either
a non-linear density wedge in the microdensitometer, as was described in
Section 2.7, or to include operations (2.7) and (2.8) in the digital
program that will handle the numerical operations. Failure to do this
will cause the non-linear response of the photographic film, used to
record the fringe pattern, to produce undesirable waveform distortion.
The nature of the spurious disturbances in the moire light
intensity trace has been discussed in Section 2.6. In order to elimin-
ate these by digital means, a successful band-pass filter must be imple-
In previous work  an in-phase band-pass electronic filter
was used to eliminate spurious variations in the amplitude and phase
of the analog output of the photo-reading machine. The inclusion of
the electronic filter produced transients of long decay time that
lengthened considerably the operating time necessary to process the
To overcome this difficulty, numerical in-phase and in-quadrature
filters were combined in this investigation to separate the amplitude
and phase of the light intensity trace and to yield the phase informa-
tion contained in the signal with minimum error.
The details of these digital filters will be presented in
Upon obtaining the digitalized and filtered version of our
displacement curve (1.1), we still need to implement the process of
derivation by which strains are obtained from displacements.
The definition of the derivative, as the limit of a difference
quotient, has little value if we possess only a discreet number of
experimentally determined values of the function.
The ratio Ay/Ax becomes excessively sensitive to small errors
in the value of y as Ax becomes small . Therefore, we must resort
to a least square approximation for a set of values of the function
in the neighborhood of the point where we wish to obtain the derivative.
The influence of varying the degree of the polynomial approx-
imation, and the use of these techniques of numerical calculus for the
computation of strains, will be seen in Section 3.h.
If the displacement curve is obtained by the use of numerical
in-phase and in-quadrature filters the process of derivation can be
implemented in an analog electronic circuit. The details of this
approach will be presented in Section 3.5.
To evaluate the merit of the different procedures utilized to
retrieve the phase information and to derive the curve of displacements,
two auxiliary tests were performed. These consisted of a bar under
traction and a disk under compression. A brief report on the pertinent
features of these tests is given in Section 3.2 and the comparative
performance of the different methods is discussed in Section 3.6.
3.2 Auxiliary Tests
A bar under axial tension and a disk under diametral compression
were used as auxiliary tests to evaluate the effectiveness of the numer-
ical techniques that will be described in this chapter. Both samples
were made of Hysol 8705 and engraved with a grid of 300 lines per inch.
By means of the tension test a series of parallel equidistant
moire fringes were produced that corresponded to the state of homogen-
eous deformation in the body. The ideal, noiseless, undistorted moire
fringes would therefore give rise to a pure cosine curve of constant
amplitude and frequency as their light intensity trace. Our real
trace, after filtering, should approximate closely the ideal.
The displacement curve along any line parallel to the longi-
tudinal axis of the sample should be a line of constant slope propor-
tional to the strain in the body.
For control purposes the average strain in the body was calcu-
lated over a distance of 7 in. along the longitudinal axis of the
A strain of 34.854 x 10 in./in. was determined. The combined
relative errors of the microdensitometer location of the first and last
fringe and of the hand plot of linear distances was 0.7%.
Turning now to the moire test of the same traction sample, the
pattern was processed in the system of Figure 17. The sampling rate
was ten words per second. The taped output of the digital voltmeter
was translated to a punched card format compatible with an IBM 709
The deck of data cards was then processed by the numerical and
analog methods described in Sections 3.3, 3.h and 3.5 to produce the
results discussed in Section 3.6.
In a second moire test a disk h in. in diameter was loaded
diametrically, Figure 18, and the fringe pattern was secured by the
system of Figure 17. The density trace of the moire fringes across the
horizontal axis of the disk is shown in Figure 18.
In this case a strong frequency modulation exists in the light
intensity variation, therefore, a more general state of deformation
than the one existing in the traction sample is obtained.
The pattern was processed along the x-axis in a manner entirely
similar to that described for the traction sample.
The theoretical solution to the disk problem is well known 
and it affords additional means of evaluating the accuracy of the numer-
ical techniques proposed here. The large deformations imposed here
cause linear theory and the experiment to disagree somewhat due to the
relatively large deformations of the boundaries [24, 25].
However, since it was of great interest to us to determine if
the numerical procedures employed could retrieve the form of the dis-
placement and strain curve from the noise in which they were immersed,
this test was considered suitable.
The density trace of the fringes along the y-axis of the disk
is shown in Figure 19. Under uniform illumination the amplitude of the
trace is seen to diminish as the spatial wavelengths become shorter in
the neighborhood of the applied load.
Though the light intensity trace in this direction was not pro-
cessed to yield the strains, it is shown here as another good example
of the spatial filtering characteristics of the photo-optical system
discussed in Chapter II.
It is clear that when the displacements along the direction
normal to the master grid become large enough to cause the distance
between fringes to approximate the limit of resolution of the optical
system, the contrast in the recorded fringe pattern will be drastically
In the event that this occurs, initial mismatches of contrary
sign to the displacement expected in a given region would increase the
final spatial wavelength of the moire fringes and permit better analysis.
3.3 Displacement Determination by In-Phase
and In-Quadrature Filters
Having obtained, by the use of the electro-optical system, the
digitalized version of the moire light intensity trace of the fringes
plus noise, we possess the numerical values of an empirical function.
We will call this set of numbers I and the continuous function from
which they were obtained I (x).
We make the following assumptions regarding I (x):
a) It is a function that defines a generalized function
b) It is band limited
c) The spectrum of the desired signal I(x) and that of
the noise N(x) are disjoint.
The first assumption holds in all cases. The other two must
be taken as only partially representative of the physical facts.
Assumption (a) is satisfied by all light intensity variations
of moire fringes. In fact, I (x) satisfied Dirichlet's conditions on
its interval of definition, which is the length of the model in the
direction of scanning.
Regarding (b) the moire light intensity trace plus noise has
in general a broad band representation in the frequency plane. The
harmonic content, however, can always be cut short of infinity without
significant loss of information contained in the signal.
Assumption (c) may or may not hold exactly depending on the
noise characteristic of the signal. If it does not hold, the analysis
that follows will be only approximately valid, but sufficiently accurate
for all cases encountered by the writer.
If assumption (c) holds, the frequency composition of the
function to be filtered is contained in a frequency interval fo + Af/2,
where Af is the band width and f the center frequency. All frequencies,
of course, are spatial in moire work.
If we select a sampling frequency f in our electro-optical device
such that f = 2f where f is the highest frequency present in the
data, under assumptions (a) and (b) Shannon's sampling theorem permits
us to write
I E a I (3.2)
k p=-m p k+p (
where Ik indicates a good approximation of I (x) obtained through
However, if the coefficients a are the inverse Fourier trans-
forms of a particular transfer function in the frequency domain, Ik
given by (3.2) will be a filtered version of I (x). We will call the
coefficients obtained in this manner a and for the resulting filtered
version of Ik we will reserve the notation Ik, thus,
I = a (3.3)
k p=-m p k+p
The factors a become, in this case, spatial invariants that
play the role of weight functions of a linear filter. Equation (3.3) can
be thought of as a finite moving average process.
The basic relations of linear filter theory are given in Appendix
A for convenience of the reader.
Since the information on the displacement field is contained in
the phase of the signal I(x), we wish to develop a numerical procedure
capable of separating the phase variations from the amplitude changes
of a signal arriving at a filter.
A simple way of accomplishing this was mentioned by Gabor 
and is applied to our problem in what follows.
Let us assume for simplicity that the light intensity trace of
a moire pattern is a real signal of the form
I(x) = a cos wx + b sin uax (3.4)
This real signal can be replaced by a complex signal of the
H(x) = I(x) + i Iq(x) (3.5)
where I (x) is defined as
I (x) = a sin wx b cos wx (3.6)
(x) = (a ib)eix (3.7)
1 -1 Iq(X)
I(x) = Re (x) = [I(x)2 + I (x)2 ] e tan (3.8)
The function I (x) is a signal in quadrature with I(x) that
transforms the oscillating vector into a rotating vector.
If I(x) is not a simple harmonic the in-quadrature function is
provided by the Hilbert transform of I(x).
The square brackets in (3.8) represent the amplitude of I(x)
while the phase of I(x) is given by
p(x) = tan-1 (3.9)
and the amplitude and phase information contained in I(x) has been
separated to suit our purpose.
This transformation of an oscillating vector function in a
rotating vector function has been widely used in the analysis of narrow-
band wave forms [27, 28, 29]. The combination of this transformation
with numerical filtering has been used by Ormsby  and by Goodman .
We will refer in what follows to I(x) as the in-phase signal
while I (x) will be called the in-quadrature signal. The latter is
obtained from the former by a n/2 phase shift.
Since our analysis involves digital computation with a signal
submerged in noise, we will be working with the sampled version of
I (x) and I (x) which we have called I and (I)k.
q k q k
The purpose of our numerical procedure will be to
a) obtain the filtered version of I^ which we call I
b) produce the 90 degrees out of phase and filtered
(I )k by the expression
(Ik = b 1 (3.10)
4q k p= -m p k(3
where the b have the same interpretation as the a
in (3.3), and are derived from them according to (3.4),
c) find the guotient ofI q)k and Ik and introduce it in
o(x) = tan-1 -n-- (3.11)
d) obtain the displacements given by
U(x) = po(x), and (3.12)
e) derive this displacement curve to produce the strains.
We will apply, to fulfill objectives (a), (b) and (c), the
techniques proposed by Ormsby and Goodman, and upon obtaining a smooth
curve of phases, the change of scale implicit in (3.12) will yield the
The process of derivation (e) will be considered in Section 3.4.
(1) Goodman's linear combination of Tukey filters.
The ideal transfer function of an in-phase and in-quadrature
filter is sown in Figure 20.
Goodman  proposed a linear combination of Tukey filters
that is centered at f is essentially flat from f n/m to f + n/m,
and zero outside the interval f 3n/m to f + 3T/m. The coefficient
m is the number of terms in the trigonometric series of the Tukey
filter transfer function.
The filter function that results from the linear combination is
shown schematically in Figure 21.
The filtering finite moving in-phase and in-quadrature average
processes in the space domain are given by
T n n n 2 -1
m1 8 m- (o.SL + o. 6 cos j -) sin j cos j cos j f
o m m m J f
I +i 1
k n j= j
SI + m (0.5 + 0.6 cos j -) cos j cos j (Ij +Ik
m k m j=1 m m c k+j k-j
+ L (0.54 + O.h6 cos n) cos n cos m f (I + I )}
m o k+m k-m
m-1 (0.54 + 0.46 cos j -) sin 2j sin2 j f -1
)(I = [ m m 0f-
(q k nJ=1
8 n n
Z (0.51 + 0.h6 cos j ) cos j sin j f (I + I)
m j=1 m m o k+j k-j
+ (0.5h + 0.46 cos n) cos n sin m f (Ik + k )}.
m o k+m k-m
The values of the filter transfer function in the interval
If + n/mI to fo T/mI is called the filter roll-off. For Goodman's
filter the roff-off is a straight line inclined with respect to the
vertical axis. In this interval the real and ideal filter differ appre-
ciably. The real filter allows noise of frequency close to that of the
signal to pass onto the output.
Real filters vary as to the monotonic function that is selected
to connect the horizontal portion of the transfer function with the
frequency axis . Tukey, as we said above, selected a straight line.
This produces a sharp discontinuity of the filter function at
f + 3n/m which in turn causes any series approximation used to express
analytically the function to overshoot the value one, due to Gibbs'
effect, at the discontinuity.
Another undesirable feature of this filter transfer function
is the dependence of the band width Af, the roll-off frequency interval,
and the number of terms in the finite moving average process on a single
The selection of one coefficient to suit these three different
purposes'restricts the applicability of this particular filter to prob-
lems where the signal has a narrow-band frequency composition and the
noise and signal spectrum are disjoint.
The only other independent parameter of the filter is the cen-
tral frequency fo
The practical implications of the comments made above will be
explored further in Section 3.5 where comparative quantitative results
(2) Ormsby's filter.
A schematic representation of an Ormsby band-pass filter trans-
fer function is shown in Figure 22.
The roff-off function of this filter is composed of two para-
bolic shoulders at fc and ft connected by a straight line. The length
of the interval of definition of the parabolic transition can be modi-
fied by varying the value of the parameter k.
The independent parameters of the filter are the number of
terms in the trigonometric series approximation of the transfer function
N, the center frequency fo, the cut-off frequency fc, the largest fre-
quency ft, and the value of k explained above which is subject to the
k < c (3.15)
The parabolic transition increases the rate of convergence
of the trigonometric series to the transfer function in the neighbor-
hood of fc, thus permitting better filtering of noise close to the
signal spectrum. The independence of ft and fc insures a narrow roll-
off frequency interval.
The transfer function of this filter is
0 if IftjlI If If t
1 if Ifcl If f I fc I
f fc)2 if fcl < Ifl < Ifc + kAfl|
f f 1)2 if If1l + k+fll < If < Ifll
bf + c if If + kAfll < IfI < Ift kAfl
bf + c if Iftl + kdAf l < Ifl Ifcl kfll
f ft) if Ift kAf 1 fl
f fl) if Iftl < If Iftl + kAfll
where the parameters a,
the condition (3.16).
b, and c are solved in terms of k to satisfy
For the above transfer function the in-phase and in-quadrature
moving average processes that yield the filtered versions of
(I)m can be found to be
I = E h cos 2Tn X I*
m n= -N n o n+m
(I ) = h sin 2nTn X I
m q n=-N n o n+m
h sin 2nn[X +(l-k)X ]+ sin 2nn(X +kX )-sin 2nn(c +X )-sin 2rn X
h = c r c r c r c
f ft f
X = t =t c (3.20)
O s f' c fs r c
The filters (3.13), (3.14), (3.17) and (3.18) were implemented
in an IBM 709 programming code. By several trial runs with different.
value of the independent parameters the optimum performances of both
filters, acting on the data from the two auxiliary tests described in
Section 3.2, were obtained. The quantitative comparison of their per-
formances is presented in Section 3.5.
After optimization (3.11) and (3.12) were implemented to yield
the displacements. The corresponding programs are included in
3.4 Strain Determination by Numerical Differentiation
When the displacement curve has been obtained in the manner
described above, it remains for us to derive numerically this empirical
function to produce the strains.
Previous to this research , the use of an in-phase electronic
band-pass filter did not produce a displacement curve sufficiently smooth
for derivation purposes and a local averaging process
u k 6 u. (3.21)
had to be applied twice before differentiation. The summation extended
over the number of data points in one fringe.
The resulting strains were calculated by a finite difference
k 2 (3.22)
where Ak was the sampling wavelength.
The strains obtained in this manner were again averaged twice
by the formula
k = (3.23)
The analysis of this procedure reveals several undesirable
features which we will now discuss.
The necessity of an amplitude normalization has already been
mentioned as a drawback of the method. Furthermore, since no weights
are associated with the point values of the function in the neighbor-
hood of where the derivative is sought, and the finite difference
approximation is sensitive to small errors in these values, the con-
vergence of the averaging process to the mean is slow.
Though repeated application of (3.21) and (3.23) does smooth
out the displacement and strain curve, it causes us to lose information
at the beginning and end of the moire light intensity trace. In fact,
each average shifts the first point at which the smoothed curve can be
defined a distance 6/2 along the direction of scanning.
A series of two averaging processes for the displacements and
two for the strains would erase all information on the strain curve
contained in the two moire fringes closest to every boundary.
In the thermal tests of Chapter IV, the strain determination at
the boundaries of the model was critical since the largest circumfer-
ential stresses occurred at these points. To overcome this difficulty
and to obtain increased accuracy in the derivation procedure, we em-
ployed a method suggested by Lanczos . It provides for the approx-
imation, in the least square sense, of a polynomial of arbitrary degree
n, to a set of experimentally obtained values of a function in the
neighborhood of the point where we wish to define the derivative. This
polynomial is then derived.
The expression for this derivative is
E op (x+uh)
dp(x) g=-s (3.2h)
where h is the sampling wavelength of the data and s the degree of
Since the least square approximation is performed in the small,
no significant error is committed. The variation of the degree of the
polynomial can be conveniently performed by the inclusion of s as an
independent parameter in the programming code.
To obtain the derivative at the beginning and end of the dis-
placement curve, (3.2h) can be modified , or a reflection of the
experimental curve can be performed with respect to the first and last
data point. Both methods have been used in the numerical examples
of Section 3.5.
The fitting of a polynomial to the entire displacement point
function by the method of least squares did not offer a satisfactory
solution because it implied making a decision on the degree of the poly-
nomial approximation that endangered the objectivity of the procedure.
The techniques of differentiation that rely on integration were
considered [22, 33, 34]. The one proposed by Lanczos, as the limit of
(3.24) when the sampling wavelength becomes very short, yielded the
best results with less computing time.
At the limit (3.24) becomes
dP(x) -= t f(x+t) dt (3.25)
where dx is the sampling wavelength and I defines the limits of the
The strain curve obtained by (3.24) or (3.25) may or may not
be smooth enough for our purposes depending on the amount and the kind
of noise present in our signal. Since the derivative process tends to
magnify small errors in the displacement function that is being derived,
we may wish to apply a final smoothing process to the strain curve
obtained. For this purpose we select a filter of the form 
e = E w e(n+r) (3.26)
n r= -i r
where en are the raw values of the derivative, wr the weights of the
filter and e' the smooth derivative values.
If r = for all r,(3.26) reduces to a local average process
identical to (3.23), however, (3.26) affords the possibility of select-
ing the war's to produce a more rapid convergence of the noise ridden
function to the smooth for smaller values of i. The choice of the best
w for a given filtering application depends on the type of noise in
the curve of derivative. As will be seen in Section 3.6, several trials
are often necessary to obtain the optimum filter weights.
The programming codes corresponding to this section are pre-
sented in Appendix C.
3.5 Strain Determination by a Hybrid Computer System
The previous two sections were concerned with purely numerical
methods of displacement and strain determination. We will now describe
a process that combines the digital procedure of in-phase and in-
quadrature filtering that produces the displacement curve with a filter-
ing and derivative circuit obtained on an analog computer. For this
purpose we combine a Ramo Wooldridge RW 300 digital computer with an
Applied Dynamics AD2 digital system to produce a hybrid computer.
Operations (3.3), (3.10), (3.11) and (3.12) are implemented and
performed on the digital section of the hybrid while the equivalent of
Equations (3.2h) or (3.25) are done on the analog section of the system.
The displacement curve produced by digital methods is smoothed
by an analog exponential filter circuit. This filtered version of the
displacement curve is derived by another circuit to produce the strain
distribution. The details of these circuits are given in Section 3.6.
The advantage of this alternate process of differentiation is
the speed with which the curve of strains can be obtained from the
analog and the ease with which the parameters that control the amount
of filtering on the displacement and strain curve can be varied.
We see that, though the digital methods of Sections 3.3 and 3.h
permit us to take advantage of the continuous law (1.1), reduce the
real time of pattern analysis and increase the objectivity of moire
data processing, the versatility of the procedure is somewhat less than
ideal since several runs must be made to optimize the independent para-
meters involved. This can make the method costly in computer time.
As far as we have been able to determine, there exists to date
no electronic system that can perform the orthogonalizations (3.6) or
(3.10), so we have had to settle for the combination of numerical and
analog data processing described above.
3.6 Comparative Performance of the Proposed Techniques
In this section we will present the results obtained by the use
of the numerical and analog techniques discussed in Sections 3.3, 3.h
and 3.5 acting on the digitalized version of the light intensity trace
corresponding to the tests of Section 3.2.
For the sake of brevity we will agree to call this section
Filter 1 to Goodman's linear combination of Tukey filters, and Filter 2
to Ormsby's filter.
a) Displacement determination
As was said in Section 3.3, (3.13) and (3.1h) that correspond
to Filter 1 were implemented in an IBM 709 programming code.
The independent parameters of this filter are the number of terms in
the trigonometric approximation of the transfer function m, and the
central frequency of the signal f
In another program (3.17), (3.18) and (3.19) were implemented
for Filter 2. This filter has as independent parameters the number
of terms in the trigonometric expansion N, the center frequency fo, the
cut-off frequency fc, the termination frequency ft and k a factor that
defines the extent of the parabolic transition of the roll-off. Both
programs contained (3.11) and (3.12) to produce the displacements.
Traction test results
The homogeneous state of deformation of the model affords the
possibility of determining quite accurately the band width of the
numerical filter. In fact, the transfer function of the filter reduces
1 for f = f
0 for f / f
However, the actual intensity trace of the moire fringes of the trac-
tion sample showed that at a constant sampling speed the number of
data points per fringe varied somewhat due to noise. Therefore, to
determine the parameter f the average value of the wavelengths was
chosen. This f was the same for both Filter 1 and Filter 2.
The criterion for the selection of the number of terms in the
trigonometric expansion, m and N for Filters 1 and 2, is different for
Since for Filter 1, m defines also the band width of the
transfer function, it is necessary for it to be large enough so that
the quotient n/m be very small. This increases the number of terms in
(3.13) and (3.14), making the use of the filter costly in computer time
to retrieve constant phase cosine waves immersed in noise.
For Filter 2 the situation is entirely different. The exist-
ence of independent parameters to define the number of terms in (3.17)
and (3.18) and the extent of the band width make it very practical for
the purposes of this test.
The roll-off frequency interval can also be narrowed conven-
iently by the use of the parameters fc, ft and k.
The displacement curve obtained with Filter 2 is shown in
Figure 23. The computer values appear as discrete points while the
theoretical curve is drawn in full line.
The displacements were obtained over base lengths of measure-
ment of 6, 6/h, 6/5, 6/8, 6/10 and 6/00. The standard deviations at
the different levels of fractional fringe interpolation varied between
an upper bound of + 0.001 p or 21' 36" of arc and a lower bound of
+ 0.0003 p or 6' 29" of arc. The maximum variance between two means
at different levels of interpolation was found to be 0.00025 p.
The consistency of these results, and the fact that no cyclic
variation in the errors of the displacement determination present in
earlier work  was found to exist in this method, demonstrate that
a substantial improvement has been achieved.
The results reported above indicate that the minimum instantan-
eous angle that can be meaningfully measured by this procedure is 10
This'implies that we have effectively divided the grid pitch into 360
equal parts, that is, a five-fold increase over previous methods of
fractional moire fringe interpolation.
Disk under compression test results
In this case the light intensity trace along the x-axis of the
ring presents strong spatial frequency modulation, Figure 18. This
indicates that the frequency composition of the signal is contained in
an interval f + Af/2, where Af is not negligibly small as in the case
of the traction sample. This fact allows Filter 1 to perform more
effectively here than in the previous test and we will report the com-
parative performance of both filters.
The central frequency f was determined in terms of data point
numbers by inspection of a standard IBM 80 x 80 listing of the data.
For Filter 1 several trial runs had to be made to obtain the best combi-
nation of the only two parameters available, m and fo
Turning now to Filter 2, it again proved to be easier to tune
in this test than Filter 1. The reasons for this were explained
earlier. The displacement curve obtained is shown in Figure 24. Both
filters proved to be able to retrieve the form of the displacement
curve precisely. The good approximation between the numerical values
of the strains derived below and those obtained by independent means,
confirms quantitatively this accuracy.
b) Strain determination
To compute the strains from the displacement curve produced by
the filters, (3.24), (3.25) and (3.26) were programmed and the compara-
tive performance of (3.24) and (3.25) analysed. Due to the wealth of
data points in our possession, (3.25) could be used with some advantage
To obtain the derivatives at the beginning and end of the sample
it was found useful to reflect the displacement curve of the disk under
compression with respect to the first and last data point locations.
However, for the displacement curve of the traction sample the reflec-
tion technique produced a sharp discontinuity with no gain in the accu-
racy of the derivative. In this case the weights of the polynomial
approximation implicit in (3.2L) were varied at the ends of the trace
as recommended by Lanczos .
For the strain curve obtained from the traction test no smooth-
ing with (3.25) was performed, so the values reported below correspond
to the straightforward application of (3.24).
For the ring under compression test (3.25) was applied since
some scatter of values was noticeable after derivation.
The strains obtained in the manner described above between two
successive fringes are shown in Figure 25a. The small scale of the
plot is intended to indicate the variations about the mean and the
average strain determined in Section 3.2.
A histogram was prepared to show the relative frequency of
occurrence of the random variable. This variable was subdivided into
discrete intervals spanning 10 x 10-6 in./in. The resulting graph is
shown in Figure 25b.
The characteristics of this distribution are of interest.
A saddle-shaped function replaces the more familiar bell-shaped
Gausian distribution we might have expected to obtain. However, the
saddle shape insures that 98% of the values of the random variable fall
within the interval 4 + 50 x 10-6 which is quite sufficient for most
The individual strains plotted in Figure 25a and included in the
listing, correspond to a base length of measurement of 6/00. If the
98% certainty that any individual measurement is contained in the inter-
val 4 + 50 x 106 in./in.is taken as a figure of merit of the proposed
techniques, we obtain an accuracy of 50 x 10-6 in./in. in the strain
determination. This is well in excess of previous results in moire work.
Disk under compression test
The numerical differentiation of the displacement curve of
Filters 1 and 2 were performed with (3.24) and subsequently filtered
with (3.26). Several trial runs were necessary to obtain the best
values of the parameters s, i and w The resulting curve of strains
can be seen in Figure 26, where one-half of the strain curve corre-
sponds to Filter 1 and the other to Filter 2, for the same values of
s, i and w.
The full line curve corresponds to the moire hand analysis of
the problem done by Sciammarella , while the dashed curve represents
the results obtained by Durelli and Mulzet .
The latter investigators reported natural strains defined as
e = n(l + o
where Z and Zf are the initial and final lengths, respectively.
Since moire yields Eulerian strains defined as
the results of  were transformed accordingly to permit comparison.
The value of Poisson's ratio for both  and our tests was
0.h65 while in [2h] this value was between 0.47 and 0.48. Since the
strain distribution is a function of this ratio some small inaccuracy
has occurred .
The modulus of elasticity in  and here is Eulerian while in
[2U] it is natural; however, for strains less than 0.03 in./in. both
almost coincide .
The percentage of errorscommitted by Filters 1 and 2 with respect
to the strains obtained in  and  is presented in the following
As can be seen from the table and Figure 26, Filter 2 produces
a smoother curve that more nearly approximates those of  and [2h].
Both filters do not perform well near the edges of the disk where the
strains are approaching zero. This is believed to be due to the fact
that long wavelengths have not been properly filtered in the quadrature
RELATIVE ERRORS IN THE STRAINS OF THE DISK
Filter 1 Filter 2
Location Relative to Location Relative to
x/r   x/r  
-1.0 0 0 0.0 4.3 5.5
-0.9 44.5 44.5 0.1 0.0 2.7
-0.8 20.8 20.8 0.2 1.6 1.1
-0.7 22.0 17.0 0.3 7.5 0.0
-0.6 0.0 0.0 0.4 9.3 0.0
-0.5 10.7 4.0 0.5 5.8 3.4
-0.4 9.4 0.1 0.6 16.0 16.0
-0.3 16.4 9.1 0.7 11.0 16.0
-0.2 3.7 0.0 0.8 21.0 21.0
-0.1 1.1 3.8 0.9 60.0 60.0
0.0 3.2 4.2 1.0 0.0 0.0
Now we wish to present the results obtained with the hybrid
system of Section 3.5.
The phases obtained with Filter 2 were smoothed with the cir-
cuit of Figure 27a; the resulting displacement curve is shown in Fig-
ure 27b. These displacements were derived by the circuit of Figure
27c to yield the strains of Figure 27d. The disymmetry in the strain
curve is due to the fact that the digital and analog sections of the
hybrid system are time locked; thus, there is a progressive delay in
the rate at which the data stored in the RW 300 is relayed to the AD2.
The resulting strain curve compares with that obtained in 
as shown in Figure 28. Again the maximum strain at the center of the
ring is 3.6% lower than that of . Several sections of the curve
coincide to within + 0.5%. Other errors calculated are 8.3% at
x/r = 0.3, and 8.6% at x/r = 0.50.
From the consideration of Figures 26 and 28 we conclude that
the hybrid system yields a strain curve with less error than that pro-
duced by purely numerical techniques for the amount of filtering used
in the comparison. However, further numerical filtering could con-
ceivably diminish the errors.
The strains produced by analog means also present larger errors
at the edges of the disk for the same reasons explained above.
It might appear paradoxical that at this final stage of the
development of a completely computerized system of strain analysis
we should check our results against those produced by hand analysis.
However, it must be remembered that the ability of the human eye to
detect trends in a curve submerged in noise and to select the proper
tangent is often difficult to replace by another system.
In a case such as this of a ring under compression where an
intelligent analyst has a good idea of what results to expect, the
fact that any piece of myopic hardware can compete with him is quite
gratifying since total objectivity and tremendous savings in labor and
time are welcome bonuses.
THE THERMAL TEST
4.1 The Experimental Set-Up
A general view of the experimental set-up can be seen in
Figure 29. Basically it consists of four units: the model, the photo-
graphic equipment, the heating and cooling system and the temperature
a) The models
The models were disks made of AISI 304 stainless steel of
nominal composition 0.06% C, 16% Cr, and 8% Ni. The disks were stress
relieved by heat treating. The outer diameter of all models were
5.5 in., the interior diameter varied from 0.5 to 2.5 in. in steps of
0.5 in. The thickness of all rings was 1 in.
We will report here only the results corresponding to what we
will call Ring No.1 with OD = 5.5 in. at 1120F and ID = 2.5 in. at
9200F, and Ring No.2 with OD = 5.5 in. at 15800F and ID = 0.5 in. at
The models were engraved with a chemically etched cross-grid
of 300 lines per inch on one face. A description of the engravure
technique, together with other information regarding the preparation
of models for high temperature testing can be found in Appendix A.
b) The photographic equipment
The photographic equipment consisted of a Saltzman enlarger
converted to a photographic camera. The details of the transformation
were reported earlier by Ross . The interesting features of this
camera are its rigidity, the precision gearing that permits accurate
positioning of its back and front, and the microinch counters that allow
faithful reproduction of relative positions.
Perfect parallelism of the image and object planes can be
obtained and preserved for a wide range of object and image distances.
The back of the camera can be rotated 3600
The film was placed against the emulsion side of a high resolu-
tion photographic plate which served as the master. Through this plate
the film was exposed. The moire fringe pattern was formed therefore in
the image plane of the camera between the image of the model's lines
and the master plate.
To make adjustments of focus and to obtain the desired mismatches,
the image on the focal plane of the camera could be inspected with the
aid of a removable ground glass. Once the adjustments were made the
ground glass would be removed, the film loaded in its place, and the
The convenient mismatches could be obtained by simply changing
the distance between the front and back of the camera, while the object
was kept focused. The rotation of the back of the cameras permitted the
elimination of all rotation mismatch by minimizing the number of fringes
in the image. A further verification of the absence of rotation mismatch
could be performed by varying the size of the image. If the number of
fringes increased or diminished without changing direction no such mis-
match was present.
The apochromatic photographic lens with a time shutter tested
in Chapter II was used to form the image.
A variety of films were evaluated prior to running the test.
The films found to record better the fringes at different temperatures
are given below.
FILM PERFORMANCE AS A FUNCTION OF TEMPERATURE
Temperature Film Speed Developer
200 Super Panchro 250 DK 50
300 Royal Pan 00 DK 50
500 Contrast 50 DK 11
800 Super Speed 125 DK 50
1600 Royal Ortho 400 DK 50
As we can see, for low temperatures, panchromatic films record
well the moire patterns. As temperatures increase, the fringe contrast
is lost with this type of films and orthochromatic films should be used.
The reason for this was given in .
An important factor in the successful photography of fringes at
high temperatures has been found to be the type of light sources used
. For all our photography an actinic cold light source with pre-
dominant radiation in the wavelength X = 0.5 x 10 cm was used.
Good contrast was obtained in the temperature range between
3000F and 16000F with several orthochromatic films of varying speeds.
However, the fastest should be preferred due to the convective air
currents that may exist in the space between the model and the lens.
These changes in the air density caused by the heat gradient may seri-
ously hinder photography by producing what appears to be a movement of
To keep them at a minimum, the model was surrounded by a chamber
and evacuated by a vacuum pump. A polished plate of fused silica
separated about 1/4 in. from the model surface closed the chamber towards
the camera providing an excellent window. Finally, about 5 in. above
the model surface, a polished glass plate was placed parallel to the
With these precautions and fast photographic films the convec-
tion current problem was overcome completely.
c) The heating and cooling system
The induction heating technique was selected because it allowed
an accurate control of the heat input and a perfect simulation of the
boundary conditions of the problem.
The induction heating coils were wrapped around the outer circum-
ference of the rings. The diameter of the coils exceeded that of the
rings. The diameter of the coils exceeded that of the ring by 1/8 in.
which permitted the thermal insulation of the coil with a uniform layer
of a fibrous potassium titanate produced by Dupont under the trade name
The coils functioned as the primary of a transformer and the
model as a loaded secondary. This heated the outer perimeter of the
rings to a depth of 0.030 in. The power source was a 5 KW radio fre-
quency generator with an operating frequency of 0.375 MC.
The cooling fixture consisted of a hollow aluminum hub, inter-
nally cooled by water, to which the interior circumference of the rings
were attached. The flow of water through the hub could be controlled.
Preliminary tests, with volumes of water that varied between 0.2 and 2.0
gal/min, showed that the amount of flow had little influence on the
temperature gradient that could be maintained. This we believe was due
to the fact that the hub together with its stand were made of aluminum,
thus providing a large area for effective heat dissipation to the air.
Figure 30 is a close-up view of the model, induction heating coils,
insulator and cooling system.
d) The control system
Iron-constantan thermocouples were fixed by crimping to the
lower face of the rings. The cold joint was kept at 320F. All thermo-
couple wires were connected through a radio frequency filter and a
selection switch to a Leeds and Northroup bridge balance.
In every ring five thermocouples were attached along a radius.
One was placed at the outer radius, one at the interior radius, and the
remaining three at regular intervals between the first two.
Errors involved in the determination of the temperature distri-
bution were less than 1%. The steady state temperature distribution of
Ring No. 1 and Ring No. 2 were compared with the theoretical distribution,
under the assumption that the coefficient of thermal conductivity is
independent of the temperature. Good correlations were obtained as is
shown in Figures 31 and 32.
To control the uniformity of the heat input around the outer
circumference and the output around the interior circumference, some
rings were prepared with thermocouples fixed on the external and inter-
nal perimeter separated ninety degrees of arc one from the other.
After steady state heat conduction had been achieved the thermocouple
readings indicated that the temperature at the outer circumference was
uniform to within + 3.5% and the interior circumference to within + 4.0%.
4.2 The Operation of the Test
For each ring, the radio frequency generator was set at 2 KW to
begin heating. The temperatures were read off the thermocouples and
sufficient time was given for the gradient to stabilize. Once the
temperatures remained stationary over a period of fifteen minutes,
steady state conditions were assumed to have been obtained. A picture
of the fringe pattern was then taken and the temperatures recorded.
The plate voltage of the generator was then increased by 0.5 KW.
Again time was given for the thermal circuit to reach a steady state
condition and another picture was taken, recording the corresponding
4.3 Theoretical Solutions to the
Thermal Stress Problem
The interest of theoreticians in finding a solution to this
thermal boundary value problem, considering the material nonhomogeneous
due to the temperature and therefore radial dependence of E, v and a,
has been considerable. However, experimental data to judge the relative
merit of the necessary simplifying assumptions has been lacking.
Hilton  has obtained a closed form solution for a hollow
cylinder subjected to a steady state temperature difference between the
interior and exterior wall under the assumption that E and C are arbi-
trary functions of the temperature and v = 0.5. He also presented
a series solution for the corresponding plane stress problem.
Chang and Chu  studied independently essentially the same
problem with both mechanical as well as thermal loading. They did not
restrict Poisson's ratio to any particular value. However, the tempera-
ture distribution they assumed across the thickness of the cylinder wall
does not appear to be physically meaningful if steady state heat trans-
fer is supposed to exist.
Trostel [o0] solved the problem of thermal stresses in a thick-
walled pipe for which v = 0.5, the coefficient of thermal conductivity
and thermal expansion are linear functions of the temperature, and
Young's modulus is a quadratic function of the temperature.
Nowacki [h1] gives further references in the Polish and Russian
Our temperature measurements, Figures 31 and 32, agree reason-
ably well with the solution to the heat conduction equation under the
assumption that the coefficient of thermal conductivity is independent
of the temperature in the range considered. This physical fact was
assumed correctly by Hilton and therefore our experimental values were
compared with those predicted by his theory.
This comparison however is of a qualitative nature, since
Hilton's assumption that the material is incompressible does not permit
a quantitative comparison between the theoretical values of the stresses
and those measured in a real material.
All investigators agree that the effect of a temperature depend-
ent G and a is to produce different values of the stresses across the
ring than those the elastic homogeneous theory with constant G and a
Whether the stresses increase or decrease will depend on the
relative variations of G and a with temperature. No definite statement
can be made, one way or the other, unless particular numerical values
For both rings Hilton's plain-strain solution was computed,
assuming a linear variation for G and a, and v = 0.5. Although in this
case the plane-strain solution and the plane-stress solution will give
different results, numerical computation of the plane-stress solution
was not attempted in view of the lengthy calculations involved and the
poor rate of convergence that Hilton has disclosed in his paper.
These values were compared with the elastic solution 
assuming G = G equal to a constant. G is taken as the shear
modulus corresponding to the average temperature in the cylinder.
The coefficient of thermal expansion was also taken as 01 = a
using for the average value the same definition as for the shear
modulus. v was taken equal to 0.5.
For Ring No. 1 both approaches yielded almost identical results
with a maximum departure of 4.3%. The results corresponding to
Ring No. 2 have been plotted in Figure 4O.
The value of the yield stress as a function of temperature, for
304 stainless steel, is as follows:
YIELD STRESS OF 304 STAINLESS STEEL AS A
FUNCTION OF TEMPERATURE
T (OF) a (psi)
The values in this table must be kept in mind when considering
Figures 38, 39, and 40. Wherever the thermal stresses exceed the
value of the yield stress at that temperature, the numerical value of
the stress has no significance beyond showing the comparative effect
of the temperature dependence of E, v and a .
We must also realize that the values reported in Table 3 have
been obtained under physical conditions which are different from those
prevailing in the ring, since the stresses were produced by mechanical
loads. Furthermore, the reported yield stresses correspond to an
arbitrary definition which does not reflect a real discontinuous transi-
tion in the behavior of the material.
h.h Experimental Determination of
Thermal Strains and Stresses
The moire patterns at the corresponding temperatures can be
seen in Figures 33 and 34. Degradation of the image due to the process
of reproduction is evident.
From the moire fringe patterns, the strains shown in Figures 35
and 36 were computed, using the continuous light intensity-displacement
law presented in  and [h]. The details of the automatic processing
of the information were presented in .
For Ring No. 1 the strains measured experimentally, Figure 35,
show very small departures from the corresponding elastic strains.
This is interesting in view of the fact that the value of the stresses
would indicate that plasticity has occurred in the vicinity of the inner
and outer circumference. Thus, no such elastic theory is rigorously
For Ring No. 2 the measured strains, Figure 36, show a more
definite departure from those predicted by the elastic theory with
constant coefficients. However, the elastic theory, while clearly not
applicable, would again seem to yield similar values for the strains.
The variation of the mechanical properties of the stainless
steel AISI type 304 (18Cr 8 Ni) used in our tests, are shown in Fig-
ure 37. These values were obtained from references  and [h4].
Figures 38 and 39 show the comparison of theoretical elastic
stresses with constant coefficients with stresses determined from experi-
mentally measured strains by the Duhamel-Neumann law with variables
E, v and CY.
For Ring No. 1 the differences are small. This coincides with
Hilton's theory. For Ring No. 2, the experimental circumferential
stresses a are larger than those of the homogeneous elastic theory as
predicted by Hilton's theory. The radial stresses however do not agree.
Hilton's solution shows an increase in a while our experiment indicates
a decrease in this stress.
In interpreting the above results, the reservations made in
Section 5.3 concerning the meaning of the stresses when the yield limit
has been exceeded, should be kept in mind. In Hilton's numerical
examples the stresses are also of such magnitude as to exceed locally
the yield stress of the material.
The tests reported in this chapter seem to indicate that for
304 stainless steel up to 1600 F of maximum temperature and for gra-
dients of 680 F or less the value of the thermal strains do not differ
greatly from those calculated by the elastic theory using E avr, avr
4.5 Residual Strain and Stress Determination
Due to the large circumferential stresses present at both the
interior and exterior perimeter of Ring No. 2, the final moire pattern
at room temperature after heating was analysed.
The resulting strains have been plotted in Figure 41. The
values of these total residual circumferential and radial strains in
the vicinity of the hole and the outer boundary are different from
zero and coincide in sign with the strains produced by the thermal load.
The strains measured in this manner, due to the geometrical
character of the moire phenomenon, are the final resulting strains in
the body that was initially strain free before the loading. As such,
they are the sum of both plastic and residual elastic strains in the
In order to separate both components of strain the retrieval
of the elastic strain was undertaken by means of a destructive test
proposed by Sachs .
This test consists of boring out the central portion of the ring
around the hole and measuring the variation of the exterior diameter
for a given bore size.
To accomplish this, the ring was placed on a vertical milling
machine, Figure 42, and secured at the outer perimeter. The boring
was made with sharp carbide tool bits with slow machine feeds and abun-
dant cooling. Two jets of water mixed with compressed air were directed
towards the interior circumference of the ring for heat removal.
After each cut the sample was allowed to cool to room tempera-
ture before performing a measurement. The ambient temperature was
recorded to within 1/2 F and the measured values corrected accordingly.
The variations in the external diameter were measured by means
of a Brown and Sharpe transistorized electronic gage capable of reading
0.000010 in. The instrument was connected so that the measurements
would not be affected by rigid body motions of the test sample as
The diametral readings were taken every 300 around the outer
circumference of the ring and averaged. The stresses were then calcu-
lated in a straight forward manner [h4]. The plot of circumferential
and radial stresses is shown in Figure h3. The experimental values of
the circumferential stress satisfied well the equilibrium condition.
From the stresses the residual elastic strains were computed, Figure U4.
Having obtained the total and the elastic residual strains, the
residual plastic strains were computed according to the expression
eT = eE + eP
The residual plastic strains are shown in Figure 45.
The analysis of Figures4l, 43, 4h and 45 show that the total and
plastic strains, where they are different from zero, are of the same
sign as those produced by the thermal gradient applied. The elastic
strains and stresses are of opposite sign to the ones produced by the
The ratio of plastic circumferential to plastic radial strain
at the edges of the disk, where a state of uniaxial loading can be
assumed to exist, is 0.150 inside and 0.475 outside. These ratios
differ from the value 0.500 we would have expected to obtain under the
assumption of incompressible plastic flow.
The reasons for this, we suspect, are the accumulation of
experimental errors in the rather indirect process by which the plastic
strains are found and, the extrapolation necessary to obtain the resid-
ual, total and elastic circumferential strains at the outer edge of the
CONCLUSIONS AND RECOMMENDATIONS
In this research the analysis, based on the MTF of the photo-
optical system used to produce and record the moire fringe pattern,
has been introduced as a first step in the application of moire to
experimental stress analysis.
It has resulted in a better understanding of the optical filter-
ing capabilities inherent to the process, and it provides a quantitative
measure of the limit of resolution of the lens, the film and the lens-
film combination. Furthermore, it permits a priori information on the
amount of contrast that can be expected in the image of grids or fringes
of a given spatial wavelength.
The strong dependence of the MTF on the proper focus, the paral-
lelism of object and image plane, the rigidity of the system and the
aperture, were demonstrated numerically.
The comparison of the performances of the ideal and real lens-
film combinations reveals large differences that amply justify the extra
effort of optical testing prior to a moire application.
This is particularly important in problems of stress concentra-
tion where the displacement gradient around notches or cracks can become
large. Under these conditions, the distance between moire fringes may
be so small that their contrast in the image plane can be drastically
Though initial mismatches of adequate sign could correct this
problem, it is clear that for a given pitch and no mismatch, there
exists for every photo-optical system a limit to the displacement that
can be measured dependent on the MTF.
In this work the first steps towards effective optical filter-
ing of a moire signal have been taken. Again the MTF appears to be the
logical tool for the evaluation and comparison of different techniques.
The remarkable properties of a lens in a coherent light field,
as an analog device that performs a Fourier transform and displays the
Fourier spectrum of a signal, has been brought to the attention of
experimental stress analysts.
This idea has been applied with success to the filtering of the
grid lines in a moire fringe pattern.
An incoherent light apodization filter has been implemented
empirically to retrieve periodic line patterns immersed in strong
random noise. The potential of this filtering technique was thus
brought to bear on a problem of particular interest in moire work.
Methods of numerical in-phase and in-quadrature filtering were
applied to overcome certain shortcomings of previously used electronic
in-phase filters. With them the phase information contained in the
moire light intensity trace has been obtained without regard to
As a result, the minimum phase angle that can be meaningfully
measured has been reduced from 50 in previous work  to 10. This
implies that we have effectively divided the grid pitch into 360 equal
The accuracy of the displacement determined at this level of
interpolation for a carefully tuned filter, has been found to be no
more than 30 x 10- p and no less than 100 x 10- p.
The process of differentiation by which strains are obtained
from the experimental curve of displacements, was critically reviewed.
The solutions presented offer a degree of accuracy and reproducibility
in excess of previous results in moire work. An average error of
50 x 10-6 in./in. was determined for the strains in an accurately tuned
The thermal strain problem of a ring under axisymmetrical steady
state temperature gradient was solved completely by the moire method.
Interesting results on the similarity in the magnitude of the strains
predicted by the linear, homogeneous and isotropic theory of elasticity
and their experimentally determined values, have been reported.
Since the stresses in the ring during the application of the
thermal load exceeded at some points the value of the yield stress at
the corresponding temperature, the total residual strains were obtained
from the final moire pattern. The residual elastic strains were
obtained by the boring-out method and the residual plastic strains were
then obtained by superposition.
This experiment shows that when elastic and plastic components
of deformation coexist in a body, moire, due to its geometrical charac-
ter, will yield the algebraic sum of the strains. The separation into
their respective components can be accomplished only after one of the
two is measured by means other than the nondestructive analysis of the
final moire pattern.
The techniques of model grid engravure able to withstand high
temperatures, have been presented in detail. The photoengraving tech-
niques, the composition of etchant baths, the engravure times and the
current densities reported have produced consistently good results.
We hope they will provide useful guidelines for workers in this field.
The ability of moire to measuredisplacementsis well known [L7,
48, h9], however, its application to strain analysis implies a process
of differentiation which is prone to error.
The methods of hand and numerical derivation are costly in real
and computer time, respectively, therefore, efforts should be made to
implement optically this procedure.
One such technique was proposed by Dantu [$0] and another by
Parks and Durelli . Both, however, involve a rigid body motion of
the moire pattern or the deformed grid with respect to itself to pro-
duce fringes that are the loci of points of equal strain. This shift
results in a slight indetermination as to what point in the body is sub-
jected to a given strain.
An optical method of derivation that does not require this dis-
placement would be ideal. In this connection, the optical properties of
coherent light systems may yield the basis for an alternate solution.
The minimum photographic, optical, electronic and digital
instrumentation necessary to produce, record and analyse moire patterns
along the lines proposed in this research should be determined. The
apparatus used here is considered to exceed in many cases what is neces-
sary to obtain results of engineering accuracy.
In the numerical retrieval of phases from the moire light inten-
sity trace it is conceivable that the decomposition of the signal in
other than a trigonometric series could yield faster convergence and
more accurate results. This matter would merit further analysis.
Electronic in-phase and in-quadrature filters should be applied
for the processing of moire data in order to increase the versatility
and speed of the analysis.
If the statistical properties of the noise in the moire light
intensity trace were known, the requirement that the noise and signal
spectrum should be disjoint could be relaxed. This would permit the
implementation of a filtering technique more powerful than the ones
presented here. However, the task of characterizing the noise content
of moire fringe patterns obtained under diverse conditions of practical
interest has not been done to date.
Based on the complete strain determination made for the stain-
less steel ring, a thermoplastic analysis in terms of stresses can be
attempted. This will require the knowledge of the stress-strain rela-
tion in the elastic and plastic region of behavior of the metal.
Some of these ideas are being pursued in the continuing research
effort of which this dissertation is a part; others remain as avenues
possibly worth exploring by anyone attracted to this matter of large
field strain analysis by a method of such direct visual appeal as moire.
ON LINEAR FILTER THEORY
A.1 Linear Systems
For the purpose of this appendix a linear system is a linear
operator L, such that for all functions y(x) and *(x) in the space
domain and all scalars a, b, the following relation holds
L[a c(x) + b*(x)] = aL[p(x)] = + bL[*(t)] (A.1)
and that, if
L[*(x)] = cp(x) (A.2)
and x is a real constant, then
L[*(x xo)] = p(x xo) (A .3)
that is, L is invariant under a space transformation.
L[6(x)] = h(x) (A.h)
L[6(x xo)] = h(x xo) (A.5)
it follows that
p(x) = *(e)h(x ) de (A.6)
If we call c(x) the output of the linear system L and *(x)
the input, (A.6) indicates that the output of L is given in terms
of the input and a unique function h(x) called the weight function
or impulse response of L.
The Fourier transform of h(x) is called the transfer function
of L and is given by
H(f) = L h(x)e2nifx dx (A.7)
If we recognize (A.6) as the convolution product we can write
cp(x) = *(x) h(x) (A.8)
(f) = Y(f) H(f) (A.9)
where (f), V(f) and H(f) are the Fourier transforms of cp(x),
*(x) and h(x) in that order.
From (A.9) we obtain immediately the transfer function that
will produce a desired output whose transform is (f) from an input
H(f) = f) (A.10)
The expression for H(f) is in general complex and of the form
H(f) = A(f)eie(f) (A.11)
We will restrict our attention to linear transformations
involving no phase shift, thus
e(f) = 0 (A.12)
We define a low-pass filter as a filter for which A(f) is
small in some sense for Ifl > fc, where f is called the cut-off
frequency. A band-pass filter is a filter where A(f) is small outside
an interval If + "fI, Figure 18.
The ideal band-pass filter would have the following transfer
function, Figure 3-la
i IA-fI f If +'I
0 2 0 2
H(f) = A(f) = (A.13)
O f I > If > If + Il
0 2 0 2
A.2 Definition of a Digital Filter
If the function *(x) defines a generalized function, is band
limited and we possess its sampled values obtained at a sampling fre-
quency fs which is at least twice as large as the largest frequency fa
present in the function, Shannon's sampling theorem guarantees that t(x)
can be uniquely determined from its sampled values
= *(4.n) n
by the infinite series
t(x) = E a n (A.14)
n= n n
If H(f) is such that R(f) can be written as a trigonometric
H(f) = E a e
n= -a n
The transform of (A .15) will be
i(x) = E a 6(x +
n= -cc n f
If we define the convolution h(x) n' where *n is the space
sampled version of *(x), then the sampled version of the desired output
C = c(-) will be given by
cp() = Z an *(m) (A.17)
s n= s
For computational purposes the infinite series (A.15), (A.16)
and (A.17) have to be replaced by finite summations, then (A.15)
(f) = E a e s (A.18)
Sn= -N n
cn = E a # (A .19)
n n= -N n m+n
where nc is a good approximation of Y(-!) through (A.19).
Equation (A.19) is the fundamental formula of digital filter-
ing  and is sometimes referred to as a finite moving average