The automatic and quantitative analysis of interferometric and optical fringe data

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THE AUTOMATIC AND QUANTITATIVE

ANALYSIS OF INTERFEROMETRIC AND

OPTICAL FRINGE DATA

by

David Peter Towers

Subm itted for the Degree o f

D octor o f Philosophy

in Engineering

at Warwick University

Originally Submitted

7th March 1991,

Revised 28th A pril 1991

Department o f Engineering

Warwick University

A bstract

O ptical interference techniques are used for a wide variety o f industrial mea­ surem ents. U sin g holographic interferom etry or electronic speckle pattern interfer­ om etry, w hole field measurements can be made on diffusely reflecting surfaces to sub-wavelength a ccuracy.

Interference frin ge s are form ed by com paring tw o states of an object. T h e inter­ ference phase co n ta in s information regarding the optical path difference between the tw o object states, a n d is related to the o b je ct deform ation. T h e a utom a tic extraction o f the phase is critica l for optical fringe m etho ds to be applied as a routine to ol. T h e solution to this p ro ble m is the main to p ic o f the thesis.

All stages in th e analysis have been considered : fringe field recording methods, reconstructing th e d a ta into a digital fo rm , and autom atic im age processing algorithm s to solve for th e interference phase.

A new m e th o d fo r reconstructing holographic fringe data has been explored. T h is produced a system w ith considerably reduced sensitivity to environmental changes. A n analysis of th e reconstructed fringe pattern showed th a t m ost errors in the phase measurements are linear. T w o m ethods for error compensation are proposed. T h e op tim u m resolution which can be attained using the m ethod is lambda/90, o r 4 nanometers.

T h e fringe d a ta was digitised using a framestore and solid state C C D camera. T h e image processing followed three distinct stages : filtering the in pu t data, form ing a 'w rapped' phase m a p by either the quasi-heterodyne analysis o r Fourier transform m ethod, and phase unwrapping. T h e prim a ry objective was to form a fully autom atic fringe analysis package, applicable to general fringe data.

A u to m a tic processing has been achieved by making local measurements o f fringe field characteristics. T h e n u m be r of iterations o f an averaging filter is optim ised according to a m easure of the fringe’s signal to noise. In phase unw rapping it has been identified t h a t discontinuities in th e data are m ore likely in regions of high spatial frequency fringes. T h is factor has been incorporated in to a new algorithm where regions o f discontinuous data are isolated according to local variations in the fringe period and d a ta consistency.

The s e m e th o d s have been found to give near o p tim u m results in m any cases. T h e analysis is fully a u tom a ted , and can be performed in a relatively short tim e , «

10

minutes on a S U N 4 processor.

Applications t o static deflections, vib rating objects, axisym m etric flames and tran­ sonic air flows a re presented. S ta tic deflection data from both holographic interfer­ om e try and E S P I is shown.

Contents

1 In troduction 12

1.1 Interference Phase D e t e r m in a t io n ... 14

1.1.1 O p tic a l Heterodyne A n a ly s is ... 19

1.1.2 Q u a si-H e te ro d yn e A n a ly s is ... 20

1.1.3 C a rrie r Fringe M ethods ... 22

1.2 H a rd w a re Requirements for Fringe A n a l y s i s ... 23

1.3 A u to m a te d Evaluation o f General Fringe P a t t e r n s ... 27

1.3.1 P re-P rocessing of Fringe D a t a ... 29

1 .3.2 Frin g e Track in g ... 38

1.3.3 A u to m a tio n of Heterodyne Phase A n a l y s i s ... 46

1 .3.4 A u to m a tic Q u a si-H e te ro d yn e Fringe A n a l y s i s ... 47

1.3.5 A u to m a te d Carrier Fringe Analysis ... 50

1 .3.6 A u to m a tic Phase U n w r a p p i n g ... 55

1.4 S u m m a ry o f Analysis P r o c e d u r e s ...

68

2 In terferom etry, Holography and H olographic Interferom etry 79 2.1 I n t e r f e r o m e t r y ... 79

2 .1.1 L ig h t Sources for Interferom etry and H o lo g r a p h y ... 82

2 .2 H o lo g r a p h y ... - ... 84

2.3 H o lo gra p hic Interferom etry ... 91

2 .3.1 D o u b le Exposure Holographic In te rf e r o m e tr y ... 92

2 .3 .2 T im e -A v e ra g e Holographic In te rfe ro m e try...101

2 .3 .4 S troboscopic Holographic In t e r f e r o m e t r y ... 105

2 .3 .5 C o m b in in g Quasi-H e te ro d yn e Analysis w ith Holographic Inter­ fe ro m e try ...106

2.4 E le ctron ic S peckle Pattern In te rfe ro m e try ... I l l 2.5 A p p lica tio n s o f Holographic Interferom etry and E S P I ...115

2.6 Th e s is F o r m a t ...119

3 Q u asi-H e te rodyn e Analysis o f EYinge P atterns and A u tom atic Im age P rocessin g 124 3.1 D u a l Reference and D ual Reconstruction B eam Holographic Interfer­ o m e try ...124

3.2 Single Reconstruction B eam System for Du a l Reference B eam Holog­ ra p hy ... 132

3 .2 .1 A n a lo g y to M oire Fringe P a t t e r n s ...132

3 .2 .2 M athe m a tica l Description o f the Phase Stepping M echanism . 133 3.3 C a p tu re o f Phase Stepped Fringe Fields from th e Single Reconstruc­ tion B e am S y s t e m ... 143

3.4 P re -P ro ce s s in g O f Fringe D a ta ... 153

3.5 A p p lica tio n o f Phase Ste p pin g A lg o rith m s ... 159

3 .5 .1 T h r e e Image Phase Ste p pin g A n a ly s is ... 161

3 .5 .2 F o u r Image Phase S tepping Analysis ... 161

3.6 Phase U n w r a p p i n g ... 163

4 Accu racy an d Resolution Analysis o f H olographic FYinge D ata 170 4.1 Resolution of the Phase Ste p pin g A l g o r i t h m s ...171

4.1.1 Resolution of Phase Stepping Analysis - Dependence on Fringe M o d u la tio n ... 176

4.1.2 Resolution of Phase Stepping Analysis - Dependence on Image N o i s e ... 176

4 .1 .3 Resolution of Phase Ste p pin g Analysis - Dependence on D e­

te c t o r Lin e a rity... 179

4 .1 .4 Resolution of Phase Stepping Analysis - Dependence on the P h a s e S t e p ...182

4.2 A cc u ra c y Achieved in Phase Stepping A n a l y s i s ... 187

4 .2 .1 P ha s e Calculation w ith Th r e e I m a g e s ... 187

4 .2 .2 P ha s e Calculation w ith Fo u r Im a g e s ... 188

4 .3 A cc u ra cy Lim itations for Single Reconstruction B eam Holographic In­ te rfe ro m e try ... 192

4 .3.1 Holographic Recording and Reconstruction Geometries . . . . 192

4 .3 .2 A n a lysis of the Fringe P attern Formed using a Single Recon­ stru ctio n B e a m ... 200

4 .3 .2 .1 Reference Fringe Com pensation - S chem e 1 . . . . 201

4 .3 .2 .2 Reference Fringe C om pensation - S chem e 2 . . . . 206

4 .3 .3 Lin e a rity Assumptions for the Im aging L e n s ... 207

4 .3 .3 .1 Camera Translation to Phase S tep Linearity . . . . 207

4 .3 .3 .2 C am era Translation t o Pixel Shift L in e a rity -...209

4.4 Analysis o f Objects w ith Substantial D e p t h ...215

4 .4 .1 P ha s e Step A n gle Variation w ith O b je ct D e p t h ...215

4 .4 .2 P ix e l Shift Variation w ith O b je c t D e p t h ... 216

4 .5 S u m m a r y ... 217

5 A pplication s o f Dual Reference B e a m Holographic Interferom e­ try and E S P I 222 5.1 T h e Interferom etric Analysis o f S urface Deform ations ... 225

5.1.1 Holographic Analysis o f S ta tic Deflections ...225

5 .1 .1 .1 Reference Fringe C om pensation S chem e 1 ... 236

5 .1 .1 .2 Reference Fringe C om pensation S che m e 2 ... 238

5 .1.2 C om parison o f th e Fourier T ran sfo rm and Phase S te p pin g M e th ­ o d s o f Fringe A n a l y s i s ... 240

5 .1.3 E S P I Analysis of Small S tru ctures in the A u to m o tive Industry 247 5 .1 .3 .1 Phase U n w rapping o f the C ylinder Bore Im age . . . 263 5 .1 .3 .2 Phase Un w rap p ing o f th e C ylinder C h a m b e r Image . 264 5 .1.4 A P u lse d Holographic System fo r Q u a n tita tive Vibration Analysis265 5 .2 T h e Analysis o f Phase Objects using Holographic Interferom etry . . . 275

5 .2.1 A n a lysis o f Burner Flames using Dual Reference B eam Holo­ g ra p h ic In te rfe ro m e try ...280 5 .2 .2 Flow Visualisation in a Large Scale Tran son ic W in d Tu n n e l . . 295

6 Conclusions an d Future Prospects 30 8

7 A ckn ow ledgem en ts 3 1 3

I Characteristics o f Laser Speckle 315

I I Solution o f Q u asi-H e te rodyn e Sim ultaneous Equations 3 1 9

I I I Processing o f H olograph ic Plates 322

I V P ublications 326

List of Figures

1.1 Original Phase D istribution to be R e c o rd e d ... 15

1.2 Intensity D is trib u tio n Corresponding to th e Phase Distribution . . . 16

1 .3 Possible In te rp re tation s of Cosinusoidal Fringe P a t t e r n ... 17

1.4 Developm ents in Personal C o m p u te r Processing P o w e r ... 26

1.5 Example o f an Interferom etric Fringe P a tt e r n ... 30

1 .6 Example o f a Holographic Fringe P a t t e r n ... 31

1.7 Example o f an E S P I Fringe P a t t e r n ... 32

1.8 Am plitude S p e c tra of Interferometric, Holographic, and E S P I Fringe D a t a ... 33

1.9 ES P I Fringe Im a g e After 4 Passes o f a 3*3 Averaging Filter’ . . . . 35

1.10 Comparison o f E S P I Am plitude Spectra Before and A fte r Filtering 36 1.11 Possible D ire c tio n s to Track Fringes ... 38

1 .12 5 * 5 W in d o w f o r Implementing Fringe T r a c k i n g ... 40

1 .13 Five Track e d F rin g e s in an Interference Fringe P a t t e r n ... 41

1 .14 Fo ur C o rre c tly Tra ck e d Fringes in a Filtered Holographic Fringe Im age 42 1 .15 Fo ur Fringes Incorrectly Track e d in a Raw Holographic Fringe Image 43 1 .16 W rapped P ha s e m a p Evaluated by Quasi-H e te ro d yn e Analysis . . . 48

1.17 Sym m etrical S p e c tra of Carrier Fringe Im age R a s t e r ... 50

1 .18 Filtered and Tra n s la te d C arrier Fringe S p e c t r a ... 52

1 .19 Unw rapped p ha se o f Holographic Fringes using a M acy type Algo rithm 60 1 .20 Valid Pixel T e s t for Phase U n w r a p p i n g ... 62

1.21 Holographic W ra p p e d Phase M a p w ith Inconsistent Regions H igh­ lighted in W h i t e ... 63

1.22 Examples o f M in im u m W eight S panning T r e e s ...

66

2 .1 Interferometers d u e to Young, Michelson, and M ach-Z e hn d e r . . . 81 2 .2 T h e G abor system for In-Line Holography ... 87 2 .3 Holographic R e co rd in g System w ith O ff-A x is Reference Beam . . . 89 2 .4 Reconstructed w a ve s from an O ff-A x is H o lo g r a m ... 90 2 .5 Fringe Fun ctio ns fo r Different types o f Holographic Interferom etry . 93 2 .6 Double Pulsed Interferogram of a Vibrational M ode of a T ru m p e t

Bell ... 98 2 .7 Definitions to D e te rm in e the Sensitivity V e c t o r ... 99 2 .8 T i m e Average Interferogram of a C ooling To w e r M o d e l ...103 2 .9 Recording S yste m fo r Dual Reference B eam , Double Exposure Holo­

graphic I n te rf e r o m e tr y ... 107 2 .10 Reconstruction S ys te m for Dual Reference B eam , Double Exposure

Holographic I n t e r f e r o m e t r y ... 108 2.11 Reconstruction S ys te m for Phase Ste p pin g, Real T i m e Holographic

I n te rf e r o m e tr y ... 110 2 .12 O p tica l System f o r Electronic S peckle P attern Interferom etry . . . 112 3.1 Du a l Reference B e a m Holographic Interferom etry w ith Reference

Beam Sources C lo se Together or S e p a ra te d ...127 3 .2 Definitions fo r D u a l Reference B eam Reconstruction Analysis . . . 128 3 .3 Phase Sw eeping M o ire F r i n g e s ... 134 3 .4 Single B eam Reconstruction System for D ual Reference Hologram s 135 3 .5 Definitions for D u a l Reference B e am Sensitivity V e ctor Analysis . . 138 3 .6 Single Beam Reconstruction o f D u a l Reference B eam Hologram

-N o O b je ct D e f o r m a t i o n ... 140 3 .7 Deform ation frin ge s O n a C entrally Loaded D i s c ...141 3 .8 Addition o f Reference Fringes A n d Deform ation Fringes in Single

Beam R e c o n s t r u c t i o n ... 142 3 .9 C h i Squared V a lu e s for Image Shift C o rr e la tio n ... 146

3 .10 Definitions t o C alculate Phase Shifting L in e a r i t y ... 147 3.11 Com parison o f C h i Squared distributions for Integer Pixel and 1/4

Pixel Resolution Image T r a n s la t io n ... 151 3 .12 Com parison o f M ethods to Calculate Fringe Field Signal to Noise

R a t i o ...155 3 .13 Com parison o f Measured and Expected Signal to Noise Variations

- Holographic D a t a ...157 3 .14 Com parison o f Measured and Expected Signal to Noise Variations

- E S P I D a t a ...158 3 .15 Sche m atic D ia g ra m of a W rapped Phase M ap S how ing Fringe W ra p ove r

L i n e s ...164 4 .1 Input D a ta t o Phase Stepping A lgo rith m S im u l a t i o n ...172 4 .2 Phase M e a s u re m en t Resolution in Phase Stepping Analysis . . . . 173 4 .3 Phase M e a s u re m en t Errors resulting from a Reduction in Fringe

M odulatio n ...177 4 .4 Phase M ea s u re m e n t Error Variation vs Image N o i s e ... 178 4 .5 Distortion Effe cts o f N on-Linear C am e ra Transfer Function . . . . 180

4.6

Unw rap p e d P hase Distributions S im ulating the Effect of D etector

N o n -Lin e a rity ...181 4 .7 Effect o f P ha s e Step Errors on the Unw rapped Phase D istribution . 182

4.8

Absolute P h a s e Measurement Errors fo r Inaccurate Phase Steps . . 184

4.9

Relative P ha s e Measurement Erro rs for Inaccurate Phase S teps . . 185 4 .1 0 Phase M e a s u re m en t Resolution against Phase S tep M agnitude . . 186 4 .11 C alculated P ha s e Step Distribution - Expected V a lue 120 Degrees . 190 4 .12 C alculated P ha s e Step Distribution - Expected V a lue 80 Degrees . 191 4 .13 R e con struction Arrangem ent in tro d ucin g N otation for Sensitivity

V e ctor E r r o r ... 192 4 .1 4 G eo m e try fo r Spherical Reference B e am Source P o s i t i o n s ... 197 4 .15 G e o m e try fo r Collimated Reference B e a m s ... 199

4 .16 Phase D is trib u tio n Caused by Dual Reference Beam s in th e Holo­ gram Plane ...203 4 .17 Phase Erro r R em aining after Linear C o m p e n s a tio n ... 204 4 .18 Phase S tep V a ria tio n with Trave rse D i s t a n c e ...210 4 .19 Experim ental Arra n ge m en t to Assess Linearity of Pixel Shifting . . 211 4 .20 Pixel Shift V a ria tio n with Trave rse D i s t a n c e ... 214 5.1 Holographic Record in g Arra n ge m en t to Examine th e Deformation

o f a M etal D i s c ... 226 5 .2 G eom etry o f D is c O bject Used for S ta tic Deform ation Tests . . . . 227 5 .3 Filtered and Tra n s la te d Cosinusoidal Fringe Pattern o f the Deformed

D i s c ...230 5.4 W rapped P ha s e M ap of the Deformed D i s c ...231 5.5 Unw rapped P hase Map of the Deformed Disc - Normalised G rey

Scale I m a g e ... 232 5.6 U n w rapped P h a s e Map of the Deformed Disc - Mesh P l o t ... 233 5 .7 Filtered and Tra n s la te d Cosinusoidal Fringe P attern o f the Reference

Beam F r i n g e s ... 234 5 .8 U n w rapped P ha s e Map of th e Reference Beam Fringes - Mesh P lo t 235 5 .9 C om parison o f Reference Fringe Com pensation Schemes Applied to

Holographic Deform ation D a t a ...237 5.10 Cross S ection o f Reference Frin ge Intensity S how ing Distorted Cos­

inusoidal F r i n g e s ... 238 5.11 Cosinusoidal Frin g e Image Suitable for B oth Fourier Tran sfo rm and

Phase S te p p in g A n a lysis ... 241 5.12 W ra p p ed P h a s e M ap formed by F F T M etho d ... 243 5.13 W ra p p e d P h a s e M ap formed by the Phase Stepping M etho d . . . 244 5.14 Unw rap p e d Pha se M ap from the Fourier Tran sfo rm M etho d - N or­

malised G re y Scale I m a g e ... 245 5.15 Un w rap p e d P ha s e M ap from th e Phase S tepping M etho d - Mesh P lot246

5.16 Phase D is trib u tio n Showing the Difference between the Results from

the Phase S te p p in g and Fourier Tran sfo rm T e c h n iq u e s ... 248

5 .17 Fibre O p tic B a se d ES P I S y s t e m ...250

5.18 O p tica l S ystem U se d to Examine C a r Engine Block ...251

5.19 Filtered Cosinusoidal Fringe Image o f the C ylinder B o r e ... 253

5 .20 W ra p p ed Phase M a p for the Cylin d er B o r e ... 254

5.21 U nw rapped P h a s e M ap for the Cylin d er Bore - Normalised G rey Scale I m a g e ...255

5.22 Unw rapped P h a s e M ap for the C ylinder Bore - Mesh P l o t ...256

5 .23 Filtered C osinusoidal Fringe Image o f th e Cylinder Head C ha m be r . 257 5.24 W ra p p ed P hase M a p for the C ylinder Head C ha m be r ... 258

5.25 Unw rapped P ha s e M ap for the C ylin d er Head C ha m b e r - Normalised G re y Scale I m a g e ...259

5.26 U n w rapped P ha s e M ap for the Cylin d er Head C h a m b e r - Mesh P lo t 260 5.27 Progression o f t h e M inim um W e ig h t Spanning Tre e in the Bore Image261 5 .28 Progression o f t h e M inim um W e igh t Spanning T re e in the C ha m be r I m a g e ...262

5.29 High Speed Reference Beam S w itching C o m p o n e n t s ...266

5.30 Holographic A rra n g e m e n t for Vibration A n a l y s i s ...269

5.31 Filtered C osinusoidal Fringe Image o f th e V ib ra tin g S h e e t ...270

5.32 W ra p p ed P hase M a p for the Vib ra tin g S h e e t ... 271

5.33 Unw rapped P h a s e M ap for th e V ib ra tin g Sheet - Normalised G re y Scale I m a g e ...272

5.34 U nw rapped P h a s e M ap for th e V ib ra tin g Sheet - M esh Plot . . . . 273

5.35 Progression o f t h e M inim um W e ig h t Spanning Tre e in the V ib ra tin g Sheet I m a g e ...274

5 .36 Mesh P lo t o f R a m p Corrected Unw rap p e d Phase M a p ...276

5.37 Cross Section o f Original and R am p Corrected U n w rapped Phase M a p s ...277

5.38 O ptical A rra n g e m e n t to Record a Holographic Optical Elem ent from a Diffusing S c r e e n ...282 5.39 Dual Reference Ho lo graphic Recording System for Sm all A xisym

-m etric F l a -m e s ...284 5.40 Filtered Cosinusoidal Fringe Image of th e Lam inar Flam e ...285 5.41 W rapped Phase M a p for the Lam inar F l a m e ...286 5.42 Unw rapped Phase M a p for the Lam inar Flam e - Normalised G rey

Scale I m a g e ...287 5.43 Unw rapped Phase M a p for the Lam inar Flam e - Mesh P lo t . . . . 288 5.44 Mesh Plot o f a La m in a r Flame after Linear Ram p C om pensation . 289 5.45 Use of a M ultich a n n e l H O E to form M ulti-D irectio nal D a ta of a

Phase O b je c t ... 291 5.46 O p tica l S ystem t o A p p ly a 180 degree Phase Change to a Reference

Beam ...292 5.47 C onventional D o u b le Exposure Interferogram of a F l a m e ...293 5.48 Double Exposure Interferogram o f a Flam e with Reference Beam

Phase M o d u l a t i o n ...294 5.49 Schem atic D ra w in g o f the Transonic W in d Tu n n e l at th e Aircraft

Research Associa tio n ...296 5.50 Holographic R e co rd in g System for Tran s on ic Flow in a Te s t C ell . . 297 5.51 Reconstructed Im a g e from an Interferogram of a Sta tic W in g and

E n g i n e ...298 5.52 Holographic R e co rd in g System for Tran son ic External Flow s - W in g

R oot C am era P o s i t i o n ...300 5.53 Holographic R e cord in g System for Tran son ic External Flow s - Side

W a ll C am era P o s i t i o n ... 301 5 .54 Reconstruction fr o m an Interferogram o f Transonic External Flows 303 1.1 S ubjective S pe ck le formed in the Image Plane o f a L e n s ... 316

Chapter 1

Introduction

Interferometric m ethods were established over 100 years ago w ith the w o rk o f N e w ­ ton, Yo un g, Michelson, M ach, an d Zehnder. Standard interferometric m e th o d s were limited to examining specularly reflecting surfaces. T h is restriction was rem oved ap­ proximately 25 years ago by the discovery of holographic interferom etry by Powell and Stetson [1]. Since then m any developm ents have occurred in the techniques used to record and interpret the in form ation contained in an interferogram.

C on cu rre n tly the range o f applications for which th e m ethods offer a solution has grow n. Th e s e applications include :

i) non-destructive testing ( N D T ) [2, 3],

ii) strain analysis due to m echanical or thermal lo ading (4 , 5,

6

], iii) modal analysis o f vibrations [7 ,

8

, 9],

iv ) profile measurement [

10

,

1 1

,

12

],

v ) phase object analysis in clu d ing compressible flows [1 3 , 7 ,1 4 ], te m p e ra tu re measurement [15, 16], a n d plasma diagnostics [1 7 ].

In every case, the required m easurem ent parameter is encoded as a phase cha n ge of the illum ination. W he n tw o (o r m o r e ) light waves are superimposed a frin ge p attern (o r inte rfe rogram ) is produced.

T h e results of an interfe rom e tric experiment are often required by n on-optical specialists. Hence there is a need for methods to analyse a ‘general’ frin ge pattern in order to provide solutions to th e measurement problems listed above. T h e result of any analysis should take th e form of a contiguous phase d istribution. T h i s data may be processed fu rthe r to give the required m easurem ent parameter field assuming some knowledge of the optical system and the object under test. Fo r ro utin e usage, the analysis should be achieved in a fully autom ated m anner and be cost effective. Autom a te d fringe analysis also implies th at an optical system may be used by engineers of different disciplines w ith o u t re-training in a new field.

T h e qu an tity of data co ntained in a fringe field is vast, of the order o f \ G B yte . Furtherm ore, the phase data required is encoded as an intensity m od u lation , usually a cosinusoidal or Bessel fu n ctio n. Im age sensors, such as photodiodes o r cha rg e coupled devices, detect the intensity o f t h e incident light. Hence some means o f data decoding is required to extract the phase data.

Several stages of processing are required to overcom e the problem s o f a utom atic analysis and extraction o f th e phase data.

i) Firstly, optical techniques are required to form th e fringe fields so th a t th e phase data can be decoded.

ii) A sensor is needed to d e te c t the intensity in the fringe field. T h is requires interfacing to an analyser in order to extract the phase inform ation . T h is data m ust then be stored.

iii) T h e complete data analysis system requires automation to ensure cost effectiveness.

T h e current solutions t o these three points are described in sections 1 .1 1.2 1.3. M o s t of the examples in th is work are o f fringe fields w ith a cosinusoidal inten­ sity profile. T h is covers a w id e range of applications : static deform ations, vibration analysis, compressible flows, a n d the measurement of tem perature fields. Vibration analysis, compressible flows, an d the measurement of tem perature fields were

ied using a pulsed laser, w hich also facilitates th e study of transient e ve n ts . T h e occurrence o f Bessel function fringes arises from the study o f vibrations w h e n the exposure tim e for the interferogram is much longer than the tem poral period of the motion [18].

1.1 Interference Phase Determination

Optical techniques have been developed since the first use o f optical interferom eters [19]. Subsequently moire system s, holographic interferom eters, electronic speckle pattern interferometers ( E S P I ) , and shearing systems have been used. In th e case of a static deformation applied to an ob je ct, the fringes formed are cosinusoidal. Th is can be represented by a spatially va ryin g intensity, i ( x , y ) , given by :

• ( * . » ) = • * ( * . » ) • ( 1 + • » (* > » )• CO« ( ¿ ( * . v ) ) ) . ( 1 1 )

where a ( x ,y ) is the local b a ckg rou n d intensity, m ( x , y ) is th e m odulation o f the fringes, and <f> is the phase change t o be measured.

T h e initial analysis o f these frin ge s was by 'fringe co un tin g' (also called th e allo­ cation o f fringe order n u m be rs); e ith e r directly from the fringe field o r a photog ra ph ic record. T h e a m o u n t o f information contained in th e field was reduced by considering only the centres o f the bright (o r d a rk ) bands m aking up the fringe p a tte rn . As a result phase contours were p ro du ce d , and not a contiguous phase map.

A single cosinusoidal fringe m a p contains insufficient data for a unique solution of the phase to be determ ined. T h i s is due to th e ambiguous nature o f th e fringes. T h e point m ay be illustrated w ith reference to figures 1.1 1.2 1.3. Figu re 1 .1 shows an original phase function to be re corded. A n interferometer produces th e cos o f this function (see figure 1 .2 ). T h is g ra p h may be interpreted in m any different w ays as the phase gradient is not encoded in to the cosinusoidal fringe pattern. H e n c e a t the fringe m axim a, it is unknown w h e th e r to add or subtract 2 xfrom the p hase. T h e possible interpretations of figure 1 .2 are shown in figure 1.3.

Figure 1 - Original Phase Distribution

Figure 1.1: Original Phase Distribution to be Recorded

[image:18.362.27.330.16.429.2]

C

os(

Pha

[image:19.363.37.342.12.424.2]

se)

Figure 2 - Ideal Cos(Phase), or Intensity, Distribution

Figure 1.2: Intensity Distribution Corresponding to the Phase Distribution

[image:20.360.21.325.14.412.2]

Figure 3 - Possible Solutions from Sinusoidal Fringe Pattern

Figure 1.3: Possible Interpretations o f Cosinusoidal Fringe Pattern

Fringe counting relies on human interpretation of the fringe p a tte rn . In this way the sign o f each fringe may be designated from one fringe ce n tre t o the next. T h is process requires a priori information a b o u t the experiment and th e expected result. Even w ith this knowledge of the field, th e re is no guarantee th a t th e correct phase distribution will be achieved. For the successful use o f fringe co u n tin g , the user requires detailed knowledge o f both the o p tic s and the item u n de r test.

A higher measurement resolution could be achieved by interpolation between fringe centres. T h is technique allows m easurem ents to

¿5

of a fringe t o be made in a few cases [20, section 2 .3 ] [19].

1.1.1

Optical Heterodyne Analysis

T h e limitations described 'above were largely rem oved using th e he terodyne and quasi- heterodyne techniques [21, 22, 2 3]. B y o p tic a l heterodyning th e phase information could be directly extracted from the field. T h i s was achieved b y reconstructing the fringe data w ith a beat frequency superposed on the optical frequency. In practice this is achieved by form ing the fringe field w ith tw o waves o f slightly different optical frequency [21]. The s e waves wx and ui

2

m a y be represented by :

» , ( * . » ) = (1 .2 )

and

i » , ( x , » ) = (1 .3 )

where a,-,* =

1 , 2

is th e amplitude o f each w a ve , u » j,i =

1 , 2

is th e optical frequency, = 1 ,2 is the phase o f the wave, and 3? denotes the real p a rt. A photodetector will measure th e tim e-dependent intensity o f th e superposition o f these tw o waves [

2 1

]

■ ( * , » , < ) = |w , + Wil* (1 .4 )

• ( * . » . < ) = » ) + +

2 » i ( i > » ) « > ( * . » ) « * [ ( < « i — » * ) • + * i ( i , l r ) - (1 .5 ) * ( * , » , 0 =

0

|J ( i , l l ) + o ,

1

( l , » ) + 2 o , ( x , K ) a , ( x , v ) c o « [ n i + ^ ( X , » ) ] (1 .6 ) where Q = u>x — u

>2

and ^ ( x , y ) = <t>i(x,y) — f a ( x ,y ) . P ro vid in g th at i l may be resolved by the photodetector, th e o p tic a l interference phase can be measured electronically as th e phase of the beat frequency. O n ce the phase is available rather than a function o f th e phase, the a m b ig u ity in assigning fringe orders is removed. T h u s heterodyning allows a further stage o f th e analysis process t o be automated.

T w o disadvantages remain for he terodyne analysis. Firstly, th e data may only be read o u t in a point-w ise m anner. T h is in h e re n tly limits the processing speed for a

com plete interferogram . Secondly, due to th e trigon o m etric nature o f th e fringes, a given phase difference 6 cannot be distinguished between 6 + m 2ic, m integer. T o overcom e this, a co un t o f the multiples of

2

x is also made as th e data is scanned out. T h i s count can be m ade unambiguously, as opposed to fringe counting, as the interference phase itself is known at all points.

1.1.2

Quasi-Heterodyne Analysis

T h e quasi-heterodyne technique allows the interference phase to be calculated unam ­ biguously as in heterodyne analysis. In this case, the intensity o f a complete fringe field is measured sim ultaneously with a d e te c to r array. Several fringe images are acquired where the phase o f the fringe field is varied either in discrete steps [

22

] or continuously [24]. W h e n discrete steps are added to the frin ge field phase, the m ethod is usually referred to as ‘phase s te p p in g ’ . A num ber o f intensity values are then available for each point in the fringe field. F ro m this data th e interference phase can be calculated in a point-w ise manner.

Conceptually, the m in im u m number of im ages required for quasi-heter.odyne anal­ ysis is tw o [25]. In this case, if a positive phase shift of ^ is applied the direction in which th e fringe maxima m ove indicates w h e th e r th e fringe order n u m b e r is increasing or decreasing. In practice three, four, or five im ages are norm ally used [23,

8

, 26]. W he n three images are used w ith a phase s te p o f 90 degrees, th e th re e images take the fo rm [23] :

where i#(x, y ) is th e intensity at the position (x , y) when the phase step is 0. T h is can be considered as a set o f simultaneous equations w ith un kn o w n s a(x,y), and m (x ,y ). B y algebraically eliminating a (x ,y ) and m (x,y). th e phase may be

* o (* .») = « ( * , » ) . ( 1 + " • ( * ,» ) . CM ( # * , » ) ) ) ,

• »(*.») = •(*.*)•(•-">(*.»)• •“ >(*(*,»))),

•

mo

(* .» ) - o(i,ll).(l -m (x ,* ).co .(* < j:,y ))),

(1 .7 )

(1.8)

(1 .9 )

calculated by :

<t>(x,y) = tan- i f

2

i» o (x , y ) - i

0

( x , y ) - i n o ( x , y )

1

l * o (* » v ) ~ * ie o (* »y) J

(

1

.

10

)

T h e same expression can also be derived as a special case of the analysis by Grievenkamp (27), w here the least squares fit of a sinusoid to th e phase stepped intensity values is con­ sidered.

T h e phase values are evaluated using an a rc tangent fu n ctio n [23]. Hence the results are ‘w ra p p e d ’ into the range — ir to ir. T h e map produced by calculating the phase across th e whole field is called a w ra p p e d phase map. F o r a total variation in optical path difference across an image o f > 2ir, the w ra p p e d phase map will contain discontinuities. T h is makes the w h o le field difficult t o interpret and can cause errors when further com putation m u st b e performed on th e data, for example th e determ ination of strain. T o overcom e th is problem, the phase discontinuities m u st be offset by ‘phase unwrapping’ [22]. In th is process 2 x is added or subtracted from the w rapped phase values every tim e a w ra p o v e r point is reached. W hence, the unwrapped phase map is produced, conta in in g a continuous phase distribution.

T h e variation of background illumination a n d modulation d e p th may also be cal­ culated using :

< o ( * . » ) - N o ( * . y )

<»• (# (* ,»))- « n (# * ,»))

(

1

.

11

)

and

a ( x ,y ) = t

0

( x , y ) — < » ( x , y ) m ( x , y ) c o s ( ^ ( x , y ) ) . ( M

2

) Variations in these tw o parameters allow th e d a ta quality to b e assessed across an image. T h e fringe contrast, o r modulation d e p t h , m ( x , y ) also allows object discon­ tinuities such as holes to be detected a uto m a tica lly [28, 29].

T h e use o f fo ur images in the phase s te p p in g algorithm allow s the phase step to be determ ined w itho ut calibrating the phase shifting m echanism . Alternatively, an increase in th e n u m be r o f images used can in tro d u ce a degree o f redundancy in the

solution w hich in tu rn minimises the errors in the calculated phase (if there exists errors in the in pu t param eters). For exam p le, Hariharan proposed using five images w ith a phase step of 90 degrees to m inim ise measurement erro rs due to inaccurate phase steps [2 6 ].

W ith th e curre n t com m ercial availability o f detector arrays and image digitisers, this m ethod offers rapid data acquisition a n d d ire ct co m pu te r analysis.

1.1.3 Carrier Fringe M ethods

Fringe order am biguities can also be re m o ve d by the addition o f ‘carrier fringes'. In this technique, a linear phase variation is a d d e d to the fringe field as a function of image co -o rdinate. T h is can be achieved in p ra ctice by tiltin g on e o f the mirrors in an interferom eter, thereby givin g a ‘tilted w a v e fr o n t'. T h e intensity in the fringe pattern m ay then be represented by :

»(*,

y) =

a ( x ,y ) .( l + m ( * ,y ) . cos (2w/0* + ¿ ( x ,y ) ) ) , (1 1 3 )

w here /„ is the spatial carrier frequency a ssum ed to be in th e x direction. T h e carrier frequency m u st be sufficiently high so as t o p ro du ce m onotonically increasing fringe order num bers. W h e n this is achieved, th e fie ld can also be described as a finite fringe field, and is characterised by th e absence o f a n y closed lo op fringes. An exam ple of such a field is given in figure 1.5 (contained in section 1 .3 .1 ), w hich was produced by a classical specular interferom eter.

A finite fringe field can be processed b y fringe co un tin g w ith o u t am biguity as th e fringe orders are known to increase m on o to n ica lly in th e direction of th e tilted wavefront. T h e phase data obtained re presents the required measurement param eter added to th e phase of th e tilted w avefront. T h e phase change due to the tilt m u st therefore be calculated and subtracted fr o m t h e original phase information obtained. T h e required param eter m a y then be calcu late d from the resulting phase distribution. T h e usefulness of carrier fringe m e tho d s in interferom etry w a s not fully realised until electronic imaging hardw are and m ore pow erful co m pu te r systems became

able 1.3.5.

1.2 Hardware Requirements for Fringe Analysis

T h e most obvious means to handle large q u an titie s o f information whilst allowing flexi­ ble means o f data processing is to utilise c o m p u te r technology. C om p u te r technology has developed rapidly over the past

20

years and w ith ever-increasing processing power it has enabled the practical im p le m e nta tion of m any o f the optical techniques described above. T h e general requ ire m en ts o f a com puter system to perform fringe analysis are listed below.

i ) C om putational power. T h is is lim ite d by the processing speed o f the microprocessor.

ii) T h e facility to run high level c o m p u te r languages hence allowing flexible user programing.

iii) Large data storage space.

iv ) T h e facility to interface image digitisers, and other devices, to computers thereby form ing an integrated s yste m fo r industrial use.

v ) Rapid data transfer from the s to ra ge devices and co m pu te r interfaces.

v i) A suitable means to o u tpu t th e d a ta .

v ii) Portability, for on-site industrial inspection and analysis.

T h e requirements for each o f these ite m s is continually g row in g due to th e industrial need to evaluate more complex problem s.

T o take advantage o f flexible c o m p u te r analysis, the data m ust first be interfaced to the co m pu te r system. T h is involves t w o components : a sensor to detect the intensity in the fringe field, and a digitiser t o convert the o u tp u t o f the sensor in to a digital fo rm .

Initially, com puters were used in c o n ju n c tio n w ith a digitising tablet to allow th e e n try o f fringe m axima or minim a from a p hotog ra ph . In this case, the sensor was the hum an eye and discrete x , y c o -o rd in a te pairs were in pu t to the com puter. T h e data was detected and processed in a seq u e n tia l manner, thereby limiting the processing speed. S om e increase in the p ro cessing rate was achieved by replacing the hum an eye w ith a photodiode and co m pu te r in te rfa c e [15]. T h e co m p u te r utilised the detected intensity to scan th e diode along frin g e m axim a or m in im a . The s e m ethods o f acquiring digital fringe information were t im e consum ing and o n ly provided phase data at discrete points along the fringe centres.

Electronic cameras and image digitisers w e re needed to fo rm a more efficient interface between th e optical and co m pu te r s ys te m s . In this way, a grid of intensity values is detected sim ultaneously and som e o f th e problems described above are removed. A detector array inherently possesses a parallel detection o f the intensity a t a set o f regularly spaced grid points. T h e lim itin g factors are th en the size of the array and the speed at w hich the data can be re a d out from th e individual sensors. O n e o f the first reports o f such a device b e in g used in fringe analysis was made in 1974 by B runing et al [22]. T h e o u tp u t f r o m a two dimensional array o f 32 by 32 photodiodes was digitised electronically a n d fed directly in to a com puter. T h e co m pu te r controlled th e image digitisation p ro ces s and the operation o f a piezo­ electric ( P Z T ) phase stepping device. T h u s f u ll synchronisation o f phase stepped fringe field capture w as possible. T h e quasi- he te ro d yn e phase calculation algorithms w ere im plemented in software to produce th e p h a se map. T h is w ork formed the basis fo r m any quasi-heterodyne analysis systems.

T h e main lim itations evident from th e w o rk b y B runing et al are described below.

i ) T h e resolution o f the detector array (a n d d igifiser) limited th e range of fringe fields w hich could be analysed. T h i s is determined b y th e Nyquist sam pling theorem which states th a t o n e frin g e must be im aged over at least tw o detector elements [30, p .6 7 -6 8 ].

ii) A n execution tim e o f 1 m inute was achieved f r o m the Bruning system. Considerably greater co m pu te r power w ould b e required to solve higher resolution data o f general objects.

iii) T h e fringe fields analysed had a low noise c o n t e n t, hence sim plifying the com putational analysis. T h is resulted from t h e use of a Tw ym a n -G re e n interferom eter w ith the fringe field being p ro je c te d directly onto th e face o f the detector array. Hence the problem s o f speckle noise were not present.

iv ) T h e objects examined were o f simple form , a g a in simplifying the analysis. Points one and tw o are inter-related. W ith the in cre a se in resolution of electronic sensors and image digitisers the data space is a u g m e n te d . Hence faster processors are required.

Progress on the a utom a tic analysis o f fringe fie ld s was inhibited by th e lack of com m ercial d etector arrays o f sufficient resolution a n d th e shortage o f sufficient com­ puting power to perform the analysis. Advances in im a ging hardware occurred with the emergence o f video technology. T h is took th e fo rm o f ~ 600 x 525 pixel solid state charge coupled device ( C C D ) cameras. T h e developm ent of real tim e video digitisers is technologically linked to th a t of c o m p u te r hardware. In th e mid 1980s, both th e Intel 80286 range o f personal co m puters ( P . C . ) and 512 x 512 x

8

bit frame grabbers became available. T h is scale o f image re solution proved sufficient to analyse m any industrial problems and the com putational p o w e r produced an acceptable exe­ cution speed. A t this tim e , it became practical t o a tt e m p t fully autom a tic processing systems for a w ide variety o f fringe patterns.

N e w advances in V L S I (v e ry large scale in te g ra te d circuits) technology can be expected in the fu tu re as a consequence o f th e ra p id grow th th at has occurred in personal co m pu te r pow er. Figure 1.4 shows th e d e ve lo p m e n t of the microprocessors used in IB M and com patible P .C .'s in term s o f th e processor power in M IP S (M illions of Instructions Per S econ d ) against the in tro d u c tio n date of the chip . T h is graph shows an increasing rate o f grow th especially w ith t h e introduction o f th e i860

cessor. T h e increased processing power facilitates e ith e r quicker execution times or the implementation of more complex software analysis schemes. A n example of this is the application of artificial neural networks ( A N N ’s ) w h ic h have the capacity to learn from previous experience. T h is type o f 'intelligent' softw a re may be applicable to the analysis o f general fringe patterns in the future. S im ila r advances in C C D resolution are currently being achieved w ith commercial syste m s fo r 2048 x 2048 pixels reaching the marketplace (although at an inhibitive price fo r frin g e analysis system s). T h e in­ creased image resolution means th at fringe fields w ith higher spatial frequency fringes will be adequately sampled and processed. T h e co m bin a tio n of these tw o technolo­ gies w ill provide greater flexibility in the range o f applications for the computerised, quantitative analysis o f interferometric fringe data.

Figure 1.4: Developments in Personal C om p u ter Processing Power

1995

[image:29.363.35.321.13.410.2]

1.3 Automated Evaluation of General Fringe

Pat-terns

W it h th e availability o f co m pu te r hardware a n d high level program m ing languages, the theoretical ideas and algorithms for frin ge analysis could be im plem ented. T h is allows flexible system development and the incorporation o f new ideas in software. T h e com plexity o f th e analysis is then lim ited b y the algorithm used. Furtherm ore, th e autom ation of a fringe analysis system fr o m the operation o f peripheral devices to im age capture, data processing and sto ra ge m ay all be controlled by means of software.

T h e objective o f software analysis o f frin ge patterns is then to perform fully au­ to m a tic processing o f th e fringe data resulting in a contiguous phase distribution. A t this stage, it is w orth sum m arising the pro ble m s facing an autom a tic fringe analysis package. T h is list has been compiled w ith reference to work by Trolinger [31) and oth e r workers [32, 33).

A - i ) speckle noise, A - i i ) diffraction noise. A - i i i ) uneven fringe background, A - i v ) varying fringe contrast,

B - i ) unknown fringe sign,

C - i ) discontinuous and split fringes,

C - i i ) closely spaced fringes, C - i i i ) broad cloudlike fringes, C - i v ) extraneous fringes,

D —ii) lack o f a known reference position,

E - i ) the data reduction process should be fa s t and provide sufficient resolution of phase and the spatial detection o f th e phase.

The s e points m ay be grouped in to several categories. Item s A - l to A - 4 are caused by th e scattering characteristics o f th e surface under examination and any non-uniform ity in the interfering w ave fron ts. Th e s e problems are normally approached by pre-processing the fringe fields to a ttain b e tter and more uniform signal to noise ratio ( S N R ) over the entire field [32, 3 4 ].

T h e unknown fringe sign (ite m B - l ) m a y be reconciled by th e optical m ethod used (h eterodyne, quasi-heterodyne, carrier frin g e e tc .) [35], or by prior knowledge o f the experiment and hum an interaction (frin ge co u n tin g ). T h e latter is a non-autom atic process.

Items C - l to C - 4 specify the degree o f complexity o f th e phase data in the field. Problems m ay arise in the correct determ ination o f the interference phase for several reasons : when the phase changes rapidly and the fringe density approaches the N yquist sampling lim it o f the d e te cto r [33], or the fringes are very broad and some noise is present in th e data. O t h e r ambiguities m ay also be present such as discontinuous fringes. In this case different answers will be obtained depending on the route taken to relate th e phase at on e poin t to th at a t another [36, 37, 3 8, 39]. Extraneous fringes are due to extra co h e re n t waves being detected which add faint fringes to th e desired interference p attern.

T h e tw o points, D - l and D - 2 , are p a rtly caused by the experimental arrangement and the ‘view* o f th e object w hich is ob ta in e d . A point in the field which is known to have zero phase change, or equivalently n o m ovem ent, allows absolute measurements to be made. T h is is not always a necessary requirement, for example in N O T it is often the relative phase change across a specim en that is o f interest.

Finally, fast data processing and a specified resolution is required for the measure­ m ent m ethod to be industrially satisfactory.

of th e problems are solved by th e use o f the appropriate optical methods, e.g. the determination o f fringe order n u m b e rs. T h e extent to which the data quality is im proved and the phase determ ined is dependent on the com plexity of th e algorithm and th e initial quality o f th e d a ta . A fu rth e r consideration is the analysis speed which is especially relevant to th e industrial application o f the system.

1.3.1 Pre-Processing o f Fringe Data

T h e extent to which pre-processing m u st be applied is dependent on the optical m ethod used to generate th e d a ta . F o r fringe fields generated by classical specular interferom etry and some holographic fringe patterns th e data may be o f sufficient quality for direct phase calculation. However this cannot be guaranteed, and the case o f speckle interferom etry and m o ire systems need to be considered. Examples of interferometric, holographic, and E S P I fringe fields are given in figures 1.5 1.6 1.7. In each case the fringes are cosinusoidal and have been captured using a C C D camera and 512 x 512 x 8 bit fram e g ra b b e r. It is clear from th e form of these images th at th e noise content in each case is v e ry different. T h e noise may be assessed further by calculating the fast Fourier tra n s fo rm ( F F T ) of a row o f pixels in th e image. T h e resulting amplitude spectra are s how n in figure 1.8. T h e signal power in the high frequency range can be used as an estim ate of the noise co nte n t. T h e proportion of noise is seen to increase from interfe rom e tric to holographic to ES P I fringe images.

T h e methods of pre-processing frin ge data have been reviewed as part of a paper by Reid on a utom atic fringe analysis[32]. Further details on the m ethods are contained in references [33, 6 ]. It is assumed t h a t the fringe data has been digitised by a suitable cam era and fram e grabber.

T h e pre-processing technique t o be applied depends o n th e type o f noise present. W h e n the spatial frequency o f th e noise is higher than th a t o f the fringe data then a lo w pass filter m ay be applied. T h i s normally takes th e form of a window (o r kernel), w ith either the average o r median of the intensities contained within the w ind o w being determ ined. T h is va lu e is then used to supplant the intensity at the

R ow Spectra of Interferometric, Holographic and ESPI Data

Figure 1.8: A m p litu d e Spectra of Interferometric, Holographic, and E S P I Fringe

Data

[image:36.364.12.325.12.412.2]

centre of the w in d o w . T h e window shape is normally a square a rra y o f pixels or a cross. M ore sophisticated approaches have also been implem ented, f o r example the spin filter a lg o rith m s o f Y u [40]. T h is technique uses the average g r e y scale over a line in a specific d irection. T h e line is rotated, spun, around th e p o in t of interest until it lies as close as possible to a constant phase line. T h e ave ra ge value over the line is then assigned to the point in question. In this way, a more refined filtering of the data is p ro du ce d which reduces th e blurring effect of conventional square window filters. Square s h a p e d windows remain the most com m only used b e cau s e of the rapid execution tim es, approxim ately 30 tim es faster than the spin filter a lg o rith m .

A n effective re d u ction in speckle and electronic noise is produced b y all the m eth­ ods described. T h i s can be illustrated on the E S P I image in figure 1 .7 . B y applying fo ur successive ite ra tio n s of an averaging filter over a 3 x 3 w in d o w , th e image in figure 1.9 is p ro d u ce d . T h e spectra o f the same ro w in the original a n d the filtered images are shown in figure 1.10. T h e noise reduction achieved is a p p a re n t.

Similarly, in t h e case o f uneven illumination o r fringe co ntra st, th e fringe data m ay be separated fro m the background by a high pass filter. T h i s approach was implemented by B e c k e r [41] using regional averages. These values a re sm oothed over the w hole image a n d the result subtracted from th e interferom etric d a ta to form a m ore uniform b a ckg ro u n d intensity.

T h e alg orithm s above may also be implemented in frequency s p a ce rather than in th e spatial d om a in [42, 6 ]. T h is is achieved by evaluating the F F T o f a raster in the fringe data. A m a s k m ay be placed in th e fourier plane and then th e in ve rse transform taken to yield m o d ifie d intensity d a ta . B y appropriate choice o f t h e mask cut-off frequencies, th e desired noise reduction, high frequency or b a ckg rou n d variation, can be achieved on th e fringes [42].

O n e disad van tag e o f the frequency domain approach is the e x e cu tio n tim e which can be ~ 20 tim e s greater when com pared with the corresponding s p a tia l processing m ethod. T h is c a n be explained by com paring th e actual filtering processes being performed in each case. T h e F F T m e tho d produces the convolution o f the window function w ith th e im a ge , whereas the spatial domain approach on ly considers a small

Row Spectra of ESPI Data Before and A fter Filtering

region of the field ( 3 x 3 pixels) w ith unity am plitude for each p o in t in the window. T h e tim e p enalty in using frequency domain filtering has recently been overcome by using dedicated hardware [43].

T h e noise reduction algorithms as described above have a lim ite d usefulness when th e noise source has approximately th e same frequency as th e fringes. In this case it is very difficult to improve the data effectively. W h e n th e noise is stationary, i.e. not related to th e position o f the fringes, it is possible to s u b tra c t th e background variation by using tw o fringe images w ith a phase difference o f * radians between th em [31, 4 4 ]. T h e tw o images then represent th e same fringe field but a bright fringe in one im a ge becomes a dark fringe in the oth e r. B y p o in t-w is e subtraction of th e tw o fields, th e noise will be cancelled out. T h e process pro du ce s higher contrast fringes a t the expense of a more com plex optical arrangem ent.

In all the references cited in this section, the noise reduction algorithm has been suited to the d a ta concerned. Hence the analysis is a utom a tic o n ly if fringe patterns o f a sim ilar qu a lity are to be processed. A topic o f curre n t research is the automatic detection o f th e necessary degree o f smoothing.

1.3.2 Fringe Tracking

Some o f th e first codes to really tackle autom atic fringe processing were based on the idea o f ‘fringe tra ckin g'. T h e aim o f this technique is t o use an algorithm to a utom atically track the fringe maxima (o r m in im a ), see for exam p le [45, 4 6, 4 7, 48, 4 9, 3 2 ]. T h e fundamental concept is to examine a w indo w o f pixels around a given point. T h e average intensity is computed for all possible d ire ctio n s leaving the point in question, see figure 1.11. T o prevent the track from co ntinu a lly circling, only th ree of these directions from those possible are normally considered : th e original direction to reach th e point in question, and the directions either side o f t h e original direction [45]. If a fringe minima is to be traced, the m inim um o f the a verage intensities along these three directions is taken to reach a new point. T h e process is repeated until some criteria is satisfied or the edge of the image is m et.

3 * 3 Pixel Grid

Pixel to be Considered

Figure 1.11: Possible Direct

Previous Direction to Pixel in Question

ons to Track Fringes

3 Possible Directions to move

Forward

O th e r m ethods based on skeletonising the fringe data before fringe tra ckin g have been used [50, 51, 15]. Skeletonising normally results in a binary image achieved by the application of a threshold operator. T h e 'w h ite ' fringes in the image can then be thinned and tracked to form continuous lines representing frin ge maxim a. T h e analysis of a com bustion flame interferogram by Brya n sto n-C ross e t al [15] is interesting as a spatial resolution of at least 4000 x 4000 pixels w as required to adequately sample the fringe field. T h is w ork demonstrates th at fringe tra c k in g m ay be performed over a m u ch larger data space than the standard video resolution o f 512 x 512 pixels.

A n implementation of th e algorithm by B u tto n [4 5 ] has been made w ith a 5 x 5 pixel w indo w . T h e algorithm has been extended by considering the 16 directions shown in figure 1.12 and the a utom atic detection of a s ta rtin g point and initial direction for tra ckin g. T h e result for five fringes of the in te rfe rogram in figure 1.5 is shown in figure 1.13. A similar result can be obtained for th e holographic fringe data in figure 1.6, four tracked fringes are shown in figure 1 .14 . In this case the fringe data required filtering by a single pass of a 3 x 3 averaging w in d o w prior to fringe tracking. T h e sta rtin g points of th e tracked fringes were set a t j u s t below the sta rt o f the fringe data in th e image. T h e need for filtering can be dem onstrated by examining figure 1.15 w hich shows th e result when the same tra c k in g algorithm is executed on th e raw fringe data. T h is implies th a t a certain im age q u a lity is required for successful fringe tracking.

D u e to these problems, the development of fringe tra ck in g algorithms have fol­ lowed tw o separate paths : either the invention o f m ore sophisticated algorithm s for both im age enhancement and fringe tracking, or im p le m e ntin g semi- a utom a tic sys­ te m s requiring operator intervention. Both these approaches have been followed by different groups.

Yatagai et al [46] implemented a more critical te st fo r determ ining fringe maxima points by testing in several directions for each poin t. Based on the statistics o f speckle noise [2 0 , p.30 -3 6], it is more likely th at a dark speckle w ill occur in a bright fringe than a bright speckle in a dark fringe. Hence it is m ore d ifficult to track fringe maxima than m inim a in the presence of speckle noise and th e m ore sophisticated tests are

5 * 5 Pixel box for Determining Tracking Direction

Directions over which average of Pixel Grey Scales is Taken

Figure 1.12: 5 * 5 W indow for Im plem enting Fringe Tracking

necessary.

T h e problem of broken fringe tracks has been addressed by Liu et al [5 1 ]. In this work the ends o f fringe tracks which are not at the edge o f the image are identified. A sector is then defined using one such fringe end p o in t as the centre. If a second fringe end poin t is found within the sector then the tw o e n d points are joined.

In contrast Funnell [4 7 ], Yatagai et al [46] and P a rth ib a n [52] have developed interactive fringe tracking systems. T h e ir philosophy co nce d e s th at a fully a utom atic algorithm is not possible and consequently a hum an in te rfa ce is required to correct for mistakes.

Assum ing th at the fringe extrema can be tracked co rre ctly, there are still tw o disadvantages which are apparent for all fringe tra ckin g m e th o d s. Firstly, the fringe order num bers (p o in t B - l ) must be assigned m anually unless some prior knowledge can be utilised, o r carrier fringes have been added to th e o rig in a l fringe field. Secondly, phase data is only available on the fringe maxima and m in im a thereby restricting the use of the data especially for strain analysis w here spa tia l derivatives are required. T h is may be overcome by fitting a curve to the known frin g e maxima and/or minim a points, see for example [53, 54]. However this analysis is n o t usually employed due to the execution tim e required [32].

Fringe tra ckin g cannot therefore form a full solution t o th e autom atic analysis of interferograms, see section 1.3. However, the m e tho d is still used in classical lens testing interferometers because of the simplicity o f th e o p tic a l system required, and the high performance of fringe tracking algorithms on spe cu la r interferometric data (see figure 1 .1 3 ). A fringe tracking analysis system has so m e advantages in certain other cases.

i) W h e re an infrequent analysis is to be performed.

ii) T h e com plexity o f heterodyning optical systems is n o t practical. iii) In the analysis o f tim e average fringe fields (c o n ta in in g Bessel function

T h e first autom atic holographic heterodyne system was constru cte d by Dandliker and can produce data at a rate of approximately 1 second per point in the interferogram [28]. B y scanning the detector using a motorised x , y stage, a contiguous phase distribution can be obtained for the whole field at any required spatial resolution. T h e data collection and detector position are co m p u te r controlled. T h e lim it o f 1 second per point arises due to the re- positioning o f th e scanned detector and the tim e to log the detected phase data. It may be possible to increase the data collection rate by continuously scanning a detector across th e fringe field and reading the phase data a t intervals providing th at the stability o f th e optical system can be m aintained. A different approach was adopted by Massie [5 5 ] w here an image dissector camera was used to select w hich part of the interference field is sampled at any poin t in tim e. T h e system constructed allowed data collection at 5/j s per point. The refore , to cover th e n u m be r o f points available in a 512 x 512 fram estore, i.e. 262144, requires approximately 1.3 seconds.

T h e advantages of heterodyne phase m easurem ent are th e very small phase differ­ ences w hich may be resolved unambiguously. U p to o f a fringe can be achieved. T h e spatial resolution is high and only lim ited by the motorised traverse stage (o r image dissector ca m e ra ) and the characteristic speckle size in the image. Som e noise im m un ity is also gained as the phase to be m easured is independent of extraneous fringes (ite m C - 4 ) w hich are not modulated a t th e beat frequency, and the interfering wavefront amplitudes (item s A - 3 and A - 4, see equation 1 .6 ).

Despite these benefits, heterodyne systems are confined to a stabilised laboratory bench [56]. T h is is due to the stability requirem ents o f th e reconstruction optics over the duration o f the data read-out and the need for a ccu ra te hologram repositioning after development. C learly this is not appropriate for perfo rm in g rapid measurements in an industrial environm ent. Th e re is no facility to pre-process the fringe data (p o in ts A - l to A - 4 ) and hence successful analysis is dependent o n achieving sufficient signal in th e detected intensity at the recording stage o f the interferogram [57].

1.3.3 Autom ation o f H eterodyne Phase Analysis

T h e autom ation of quasi-heterodyne analysis systems was a p p a re n t after the work by B ru n in g et al. In general a com puter system is linked to : a C C D video camera via an image frame grabber, and a device fo r varying the phase of the fringe field. Synchronisation o f the image capture system an d fringe field phase is achieved through software. It has been shown by subsequent w orkers th at th e set o f phase stepped images (typically three, four, or five, see section 1 .1 .2 ) can be ca p tu re d in consecutive video frames (see for example [8 ]). T h is is d on e by moving th e phase shifting device in the read-out tim e of the C C D camera. H e n ce digital image ca p tu re requires ^ of a second, where n is the number of images.

O n ce the data is in digital form, the data processing m ay be achieved by a sequence of software functions. T h is allows flexibility in th e analysis a n d independent fringe pre-processing (section 1 .3 .1 ) followed by phase co m putatio n. M a n y phase calculation algorithms are available and have been reviewed by Creath a n d W yk e s [58, 56]. As described in section 1.1.2, the resultant phase is evaluated m o d u lo 2 x . T h e image formed is called a 'w rapped' phase map. T h e w ra p pe d phase m a p corresponding to figure 1.6 is shown in figure 1.16 where the phase values b etw een —* and x have been encoded as grey scales from 0, black, t o 2 5 5, white. T h e cosinusoidal images were smoothed using a 3 x 3 averaging filter prior to calculation o f the wrapped phase values.

T h e a utom atic processing of a wrapped phase m ap, i.e. ‘ phase unwrapping’, will be discussed after the following section.

T h e advantages of quasi-heterodyne analysis are :

i) th e rapid, a utom atic data acquisition w h ich directly yields th e fringe data in a digital form,

ii) and the facility for software analysis.

T h e phase measurement resolution is also high, up to o f a frin ge [58], whilst the spatial resolution is fixed by : the num ber o f pixels in the d ig ita l frame grabber (o r

1.3.4 Automatic Quasi-Heterodyne Fringe Analysis