Title: Information retrieval from moiré fringe patterns in thermal strain fields
Full Citation
Permanent Link: http://ufdc.ufl.edu/UF00097883/00001
 Material Information
Title: Information retrieval from moiré fringe patterns in thermal strain fields
Alternate Title: Moiré fringe patterns in thermal strain fields
Physical Description: xvii, 169 leaves. : illus. ; 28 cm.
Language: English
Creator: Sturgeon, Donald L. G., 1937-
Publication Date: 1966
Copyright Date: 1966
Subject: Thermal stresses   ( lcsh )
Engineering Mechanics thesis Ph. D
Dissertations, Academic -- Engineering Mechanics -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
Thesis: Thesis--University of Florida, 1966.
Bibliography: Bibliography: leaves 165-168.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00097883
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000559114
oclc - 13439414
notis - ACY4560


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April, 1966


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.





LIST OF TABLES . . . . . .

LIST OF FIGURES ............

LIST OF SYMBOLS . . . . . .

ABSTRACT . . . . . . . .



2.1 Introduction .
2.2 Optical Analysis



Modulation Transf
Modulation Transf

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 . . . .


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





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


5.1 Conclusions . . . . .
5.2 Recommendations . . . .

APPENDIX A . . . . . . . . .


A.1 Linear Systems . . . .
A.2 Definition of a Digital Filter
APPENDIX B . . . . . . . . .


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 . . . . . . . . .


APPENDIX D . . . . . . . . .

FIGURES . . . . . . . .

APPENDIX E . . . . . . . . .


LIST OF REFERENCES . . . . . . .
























Table Page

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


Figure Page

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


Figure Page

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


Figure Page

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




a ,a
P' n


(w xw y


in x y




Ci (w)

Co (w)

Co, Cl,






Defocal length or interior radius of a ring or a constant

Weight functions of a linear in-phase filter

Shannon's coefficients

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


E (x)







f (a,B)






HA(U xu )

h (x,y)


Sampling wavelength

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


Modulus of elasticity or film exposure

Modulus of elasticity for average temperature

Negative exposure variation along the x direction

Spatial frequency

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

Band width

Roll-off frequency interval

Complex amplitude distribution over the surface of
the aperture

Complex conjugate of f(a,B)

Shear modulus

Shear modulus for average temperature








1in( x' y)


o ( x' wy)




(q) k
(1 )k

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

Background intensity

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
q k


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
point number

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
or ring

s Summation limit


t Time

T Transmittance or temperature

T Transmittance of a negative
T Transmittance of a positive
T (uw) Maximum of a sinusoidal transmittance function at
frequency w

T (w) Minimum of a sinusoidal transmittance function at
frequency w

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








e n

k n







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

Natural strain

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
desired signal

Mean value of a random variable

Poisson's ratio

Poisson's ratio for the average temperature










x y

Argument function of the coordinate x

Normal stress

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

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



Donald L. G. Sturgeon

April, 1966

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

numerical methods.

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

[1, 2].

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 [3] and tested by Ross, Sciammarella and Sturgeon [4].

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

master grid.

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 [5]. 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 [6].

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 [6]. By means of an initial mismatch the frequency of the moire

signal before and after deformation was made to fall within the range of

the instrument.

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 [6]. The errors

involved in this step were found to be considerable and are discussed

fully in [6].

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

law (1.1).

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

stress analysis.

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.



2.1 Introduction

The photo-optical system used to produce and record moire

patterns, has the ability to perform as a linear low-pass filter of

spatial frequencies.

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

moire information.

We will call system synthesis the processes by which we act on

the system to change its frequency response characteristics to suit

our purposes.

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

luminance, then

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

expressed as

M(w) = C (2.2)

where C (w) and Co(u) are the contrast of the image and the object at

frequency w.

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 [7] 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

fringe photography.

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 [8] 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 [9].

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 [10].

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 [11] 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 [6].

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 [12]

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 [10].

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

object plane.

2.6 Light Intensity Trace of Moire Fringes

The light intensity variation along a line in a moire pattern

given by

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 [4] 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)
n n

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

discussed here.

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

filter requirements.

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 [13].

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)
n n2

where C2 is a constant. If we print this negative on a positive film

with a v such that

n p =l (2.11)

we obtain

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-

ful techniques.

We will describe in what follows several experiments in which

the quality of moire fringes has been substantially increased by

optical filtering.

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
re Aspectively.

HA is therefore the amplitude transfer function of the optical

system that operates on the object amplitude spectrum to give the image

amplitude spectrum.

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

this plane.

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 [17].

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 [18] 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 [18],

using the approximations of geometrical optics, obtain as the unit

intensity response of the defocussed system

for r < -
a- v
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)
y CO

H(wr) = 2n I 1 v (2.20)
u aLwr


r= x + y wr = V + 2
x y

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 [19]

and O'Neill[20] have calculated such functions for an annular aperture

and Franyon [21] 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.



3.1 Introduction

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)

so that

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

0.0001 V.

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-

mented numerically.

In previous work [6] 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

fringe patterns.

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

Section 3.3.

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 [22]. 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 [23]

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
s aa
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,
k k
Sm I
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 [26]

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)

and then

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

I (x)
p(x) = tan-1 (3.9)
I(X) (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 [30] and by Goodman [31].

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
k k'
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
p p
in (3.3), and are derived from them according to (3.4),

and (3.6),

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 [31] 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
o o
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 [32]. 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

parameter m.

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

are presented.

(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

ft f
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).


I and

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
q m

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



1 a(

1 a(



H(f) =


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
Lk(l-k)X (nn)3

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

Appendix C.

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 [37], 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


Uk+1- ~k-I
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 [22]. 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)
dx s
2 a2h

where h is the sampling wavelength of the data and s the degree of

the polynomial.

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 [22], 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

integration interval.

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 [35]

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
n r
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

H(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

each filter.

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 [6] 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

over (3.24).

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 [22].

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.

Traction test

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

engineering applications.

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 [1], while the dashed curve represents

the results obtained by Durelli and Mulzet [24].

The latter investigators reported natural strains defined as
Zf o
e = n(l + o

where Z and Zf are the initial and final lengths, respectively.

Since moire yields Eulerian strains defined as

Zf 0o

the results of [24] were transformed accordingly to permit comparison.

The value of Poisson's ratio for both [1] 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 [24].

The modulus of elasticity in [1] and here is Eulerian while in

[2U] it is natural; however, for strains less than 0.03 in./in. both

almost coincide [24].

The percentage of errorscommitted by Filters 1 and 2 with respect

to the strains obtained in [1] and [24] 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 [1] 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




Filter 1 Filter 2
Location Relative to Location Relative to
x/r [1] [24] x/r [1] [24]

-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 [1]

as shown in Figure 28. Again the maximum strain at the center of the

ring is 3.6% lower than that of [1]. 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.



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 [36]. 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

camera closed.

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.



Maximum ASA
Temperature Film Speed Developer

200 Super Panchro 250 DK 50

300 Royal Pan 00 DK 50

500 Contrast 50 DK 11
Process Ortho

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 [37].

An important factor in the successful photography of fringes at

high temperatures has been found to be the type of light sources used

[37]. 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

the object.

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

fused silica.

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

of Typersul.

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 [38] 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 [39] 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

would predict.

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

are considered.

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 [22]

assuming G = G equal to a constant. G is taken as the shear
Savr avr
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:



T (OF) a (psi)

75 33,000
500 31,500
700 30,000
900 24,000
1100 13,000
1300 11,500
1500 10,000

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 [38].

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 [3] and [h]. The details of the automatic processing

of the information were presented in [6].

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 [43] 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

and ac

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 [51].

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

a whole.

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

thermal load.

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

ring [h6].



5.1 Conclusions

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 [6] 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.

5.2 Recommendations

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 [1]. 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.



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

o 0
L[*(x xo)] = p(x xo) (A .3)

that is, L is invariant under a space transformation.

Then, if

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

l(f) is

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
ca s

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

series, then

ca s
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)

becomes f
N 2nT-ir
(f) = E a e s (A.18)
Sn= -N n

and (A-17)

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 [33] and is sometimes referred to as a finite moving average

process [32].

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