Manuel Servin, J. Antonio Quiroga, and

J. Moises Padilla

Fringe Pattern Analysis for OpticalMetrology

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Manuel Servin, J. Antonio Quiroga, and J. Moises Padilla

Fringe Pattern Analysis for OpticalMetrology

Theory, Algorithms, and Applications

The Authors

Dr. Manuel ServinCentro de Investigacionesen Optica A.C.Leon GuanajuatoMexico

Dr. J. Antonio QuirogaUniversidad Complutense MadridDept. de OpticaMadridSpain

Dr. J. Moises PadillaCentro de Investigacionesen Optica A.C.CampestreMexico

Cover PictureColored ring:The color fringe pattern is the isochromaticpattern of a diametrically loaded ring3D surface:False color 3D unwrapped phase of thewrapped phase shown below.Human eye with circular rings:A human cornea with a Placido ring patternreflected over it.

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V

Contents

Preface XIList of Symbols and Acronyms XV

1 Digital Linear Systems 11.1 Introduction to Digital Phase Demodulation in Optical Metrology 11.1.1 Fringe Pattern Demodulation as an Ill-Posed Inverse Problem 11.1.2 Adding a priori Information to the Fringe Pattern: Carriers 31.1.3 Classification of Phase Demodulation Methods in Digital

Interferometry 71.2 Digital Sampling 91.2.1 Signal Classification 91.2.2 Commonly Used Functions 111.2.3 Impulse Sampling 131.2.4 Nyquist–Shannon Sampling Theorem 141.3 Linear Time-Invariant (LTI) Systems 141.3.1 Definition and Properties 151.3.2 Impulse Response of LTI Systems 151.3.3 Stability Criterion: Bounded-Input Bounded-Output 171.4 Z-Transform Analysis of Digital Linear Systems 181.4.1 Definition and Properties 181.4.2 Region of Convergence (ROC) 191.4.3 Poles and Zeros of a Z-Transform 201.4.4 Inverse Z-Transform 211.4.5 Transfer Function of an LTI System in the Z-Domain 221.4.6 Stability Evaluation by Means of the Z-Transform 231.5 Fourier Analysis of Digital LTI Systems 241.5.1 Definition and Properties of the Fourier Transform 251.5.2 Discrete-Time Fourier Transform (DTFT) 251.5.3 Relation Between the DTFT and the Z-Transform 261.5.4 Spectral Interpretation of the Sampling Theorem 271.5.5 Aliasing: Sub-Nyquist Sampling 291.5.6 Frequency Transfer Function (FTF) of an LTI System 311.5.7 Stability Evaluation in the Fourier Domain 33

VI Contents

1.6 Convolution-Based One-Dimensional (1D) Linear Filters 341.6.1 One-Dimensional Finite Impulse Response (FIR) Filters 341.6.2 One-Dimensional Infinite Impulse Response (IIR) Filters 371.7 Convolution-Based two-dimensional (2D) Linear Filters 391.7.1 Two-Dimensional (2D) Fourier and Z-Transforms 391.7.2 Stability Analysis of 2D Linear Filters 401.8 Regularized Spatial Linear Filtering Techniques 421.8.1 Classical Regularization for Low-Pass Filtering 421.8.2 Spectral Response of 2D Regularized Low-Pass Filters 461.9 Stochastic Processes 481.9.1 Definitions and Basic Concepts 481.9.2 Ergodic Stochastic Processes 511.9.3 LTI System Response to Stochastic Signals 521.9.4 Power Spectral Density (PSD) of a Stochastic Signal 521.10 Summary and Conclusions 54

2 Synchronous Temporal Interferometry 572.1 Introduction 572.1.1 Historical Review of the Theory of Phase-Shifting Algorithms

(PSAs) 572.2 Temporal Carrier Interferometric Signal 602.3 Quadrature Linear Filters for Temporal Phase Estimation 622.3.1 Linear PSAs Using Real-Valued Low-Pass Filtering 642.4 The Minimum Three-Step PSA 682.4.1 Algebraic Derivation of the Minimum Three-Step PSA 682.4.2 Spectral FTF Analysis of the Minimum Three-Step PSA 692.5 Least-Squares PSAs 712.5.1 Temporal-to-Spatial Carrier Conversion: Squeezing

Interferometry 732.6 Detuning Analysis in Phase-Shifting Interferometry (PSI) 742.7 Noise in Temporal PSI 802.7.1 Phase Estimation with Additive Random Noise 822.7.2 Noise Rejection in N-Step Least-Squares (LS) PSAs 852.7.3 Noise Rejection of Linear Tunable PSAs 862.8 Harmonics in Temporal Interferometry 872.8.1 Interferometric Data with Harmonic Distortion and Aliasing 882.8.2 PSA Response to Intensity-Distorted Interferograms 912.9 PSA Design Using First-Order Building Blocks 952.9.1 Minimum Three-Step PSA Design by First-Order FTF Building

Blocks 972.9.2 Tunable Four-Step PSAs with Detuning Robustness at 𝜔 = −𝜔0 1002.9.3 Tunable Four-Step PSAs with Robust Background Illumination

Rejection 1012.9.4 Tunable Four-Step PSA with Fixed Spectral Zero at 𝜔 = π 1022.10 Summary and Conclusions 104

Contents VII

3 Asynchronous Temporal Interferometry 1073.1 Introduction 1073.2 Classification of Temporal PSAs 1083.2.1 Fixed-Coefficients (Linear) PSAs 1083.2.2 Tunable (Linear) PSAs 1083.2.3 Self-Tunable (Nonlinear) PSAs 1093.3 Spectral Analysis of the Carre PSA 1103.3.1 Frequency Transfer Function of the Carre PSA 1123.3.2 Meta-Frequency Response of the Carre PSA 1133.3.3 Harmonic-Rejection Capabilities of the Carre PSA 1143.3.4 Phase-Step Estimation in the Carre PSA 1163.3.5 Improvement of the Phase-Step Estimation in Self-Tunable

PSAs 1183.3.6 Computer Simulations with the Carre PSA with Noisy

Interferograms 1203.4 Spectral Analysis of Other Self-Tunable PSAs 1223.4.1 Self-Tunable Four-Step PSA with Detuning-Error Robustness 1233.4.2 Self-Tunable Five-Step PSA by Stoilov and Dragostinov 1263.4.3 Self-Tunable Five-Step PSA with Detuning-Error

Robustness 1283.4.4 Self-Tunable Five-Step PSA with Double Zeroes at the Origin and the

Tuning Frequency 1303.4.5 Self-Tunable Five-Step PSA with Three Tunable Single Zeros 1313.4.6 Self-Tunable Five-Step PSA with Second-Harmonic Rejection 1333.5 Self-Calibrating PSAs 1363.5.1 Iterative Least-Squares, the Advanced Iterative Algorithm 1373.5.2 Principal Component Analysis 1403.6 Summary and Conclusions 145

4 Spatial Methods with Carrier 1494.1 Introduction 1494.2 Linear Spatial Carrier 1494.2.1 The Linear Carrier Interferogram 1494.2.2 Instantaneous Spatial Frequency 1524.2.3 Synchronous Detection with a Linear Carrier 1554.2.4 Linear and Nonlinear Spatial PSAs 1594.2.5 Fourier Transform Analysis 1644.2.6 Space–Frequency Analysis 1704.3 Circular Spatial Carrier 1734.3.1 The Circular Carrier Interferogram 1734.3.2 Synchronous Detection with a Circular Carrier 1744.4 2D Pixelated Spatial Carrier 1774.4.1 The Pixelated Carrier Interferogram 1774.4.2 Synchronous Detection with a Pixelated Carrier 1804.5 Regularized Quadrature Filters 186

VIII Contents

4.6 Relation Between Temporal and Spatial Analysis 1984.7 Summary and Conclusions 198

5 Spatial Methods Without Carrier 2015.1 Introduction 2015.2 Phase Demodulation of Closed-Fringe Interferograms 2015.3 The Regularized Phase Tracker (RPT) 2045.4 Local Robust Quadrature Filters 2155.5 2D Fringe Direction 2165.5.1 Fringe Orientation in Interferogram Processing 2165.5.2 Fringe Orientation and Fringe Direction 2195.5.3 Orientation Estimation 2225.5.4 Fringe Direction Computation 2255.6 2D Vortex Filter 2295.6.1 The Hilbert Transform in Phase Demodulation 2295.6.2 The Vortex Transform 2305.6.3 Two Applications of the Vortex Transform 2335.7 The General Quadrature Transform 2355.8 Summary and Conclusions 239

6 Phase Unwrapping 2416.1 Introduction 2416.1.1 The Phase Unwrapping Problem 2416.2 Phase Unwrapping by 1D Line Integration 2446.2.1 Line Integration Unwrapping Formula 2446.2.2 Noise Tolerance of the Line Integration Unwrapping Formula 2466.3 Phase Unwrapping with 1D Recursive Dynamic System 2506.4 1D Phase Unwrapping with Linear Prediction 2516.5 2D Phase Unwrapping with Linear Prediction 2556.6 Least-Squares Method for Phase Unwrapping 2576.7 Phase Unwrapping Through Demodulation Using a Phase

Tracker 2586.8 Smooth Unwrapping by Masking out 2D Phase Inconsistencies 2626.9 Summary and Conclusions 266

Appendix A List of Linear Phase-Shifting Algorithms (PSAs) 271A.1 Brief Review of the PSAs Theory 271A.2 Two-Step Linear PSAs 274A.2.1 Two-Step PSA with a First-Order Zero at −𝜔0 (𝜔0 = π∕2) 274A.3 Three-Step Linear PSAs 275A.3.1 Three-Step Least-Squares PSA (𝜔0 = 2π∕3) 275A.3.2 Three-Step PSA with First-Order Zeros at 𝜔 = {0,−𝜔0}

(𝜔0 = π∕2) 276A.4 Four-Step Linear PSAs 277A.4.1 Four-Step Least-Squares PSA (𝜔0 = 2π∕4) 277

Contents IX

A.4.2 Four-Step PSA with a First-Order Zero at 𝜔 = 0 and a Second-OrderZero at −𝜔0 (𝜔0 = π∕2) 278

A.4.3 Four-Step PSA with First-Order Zeros at 𝜔 = {0,−𝜔0∕2,−𝜔0}(𝜔0 = π∕2) 279

A.4.4 Four-Step PSA with a First-Order Zero at −𝜔0 and a Second-OrderZero at 𝜔 = 0 (𝜔0 = π∕2) 280

A.4.5 Four-Step PSA with a First-Order Zero at 𝜔 = 0 and a Second-OrderZero at −𝜔0 (𝜔0 = 2π∕3) 281

A.5 Five-Step Linear PSAs 282A.5.1 Five-Step Least-Squares PSA (𝜔0 = 2π∕5) 282A.5.2 Five-Step PSA with First-Order Zeros at 𝜔 = {0,±2𝜔0} and a

Second-Order Zero at −𝜔0 (𝜔0 = π∕2) 283A.5.3 Five-Step PSA with Second-Order Zeros at 𝜔 = {0,−𝜔0}

(𝜔0 = 2π∕3) 284A.5.4 Five-Step PSA with Second-Order Zeros at 𝜔 = {0,−𝜔0}

(𝜔0 = π∕2) 285A.5.5 Five-Step PSA with a First-Order Zero at 𝜔 = 0 and a Third-Order

Zero at −𝜔0 (𝜔0 = π∕2) 286A.5.6 Five-Step PSA with a First-Order Zero at 𝜔 = 0 and a Third-Order

Zero at −𝜔0 (𝜔0 = 2π∕3) 287A.6 Six-Step Linear PSAs 288A.6.1 Six-Step Least-Squares PSA (𝜔0 = 2π∕6) 288A.6.2 Six-Step PSA with First-Order Zeros at {0,±2𝜔0} and a Third-Order

Zero at −𝜔0 (𝜔0 = π∕2) 289A.6.3 Six-Step PSA with a First-Order Zero at 𝜔 = 0 and a Fourth-Order

Zero at −𝜔0 (𝜔0 = π∕2) 290A.6.4 Six-Step PSA with a First-Order Zero at 𝜔 = 0 and Second-Order

Zeros at {−𝜔0,±2𝜔0} (𝜔0 = π∕2) 291A.6.5 Six-Step (5LS + 1) PSA with a Second-Order Zero at −𝜔0

(𝜔0 = 2π∕5) 292A.7 Seven-Step Linear PSAs 293A.7.1 Seven-Step Least-Squares PSA (𝜔0 = 2π∕7) 293A.7.2 Seven-Step PSA with First-Order Zeros at {0,−𝜔0, 2𝜔0,±3𝜔0} and a

Second-Order Zero at −2𝜔0 (𝜔0 = 2π∕6) 294A.7.3 Seven-Step PSA with First-Order Zeros at {0,−𝜔0, 2𝜔0} and a

Second-Order Zero at ±3𝜔0 (𝜔0 = 2π∕6) 295A.7.4 Seven-Step PSA with First-Order Zeros at {0,±2𝜔0} and a

Fourth-Order Zero at −𝜔0 (𝜔0 = π∕2) 296A.7.5 Seven-Step PSA with Second-Order Zeros at {0,−𝜔0,±2𝜔0}

(𝜔0 = π∕2) 297A.7.6 Seven-Step PSA with a First-Order Zero at 𝜔 = 0 and a Fifth-Order

Zero at −𝜔0 (𝜔0 = π∕2) 298A.7.7 Seven-Step (6LS + 1) PSA with a Second-Order Zero at −𝜔0

(𝜔0 = 2π∕6) 299A.8 Eight-Step Linear PSAs 300

X Contents

A.8.1 Eight-Step Least-Squares PSA (𝜔0 = 2π∕8) 300A.8.2 Eight-Step Frequency-Shifted LS-PSA (𝜔0 = 2 × 2π∕8) 301A.8.3 Eight-Step PSA with First-Order Zeros at {0,−𝜔0, ±2𝜔0, π∕10,

−3π∕10, −7π∕10, 9π∕10} 302A.8.4 Eight-Step PSA with Second-Order Zeros at {0,±2𝜔0} and a

Third-Order Zero at −𝜔0 (𝜔0 = π∕2) 303A.8.5 Eight-Step PSA with First-Order Zeros at {0,−π∕6,−5π∕6, ±2𝜔0} and

a Fourth-Order Zero at −𝜔0 (𝜔0 = π∕2) 304A.8.6 Eight-Step PSA with First-Order Zeros at {0,±2𝜔0} and a Fifth-Order

Zero at −𝜔0 (𝜔0 = π∕2) 305A.9 Nine-Step Linear PSAs 306A.9.1 Nine-Step Least-Squares PSA (𝜔0 = 2π∕9) 306A.9.2 Nine-Step PSA with First-Order Zeros at {0,±2𝜔0} and Second-Order

Zeros at {−𝜔0,−π∕4,−3π∕4} (𝜔0 = π∕2) 307A.9.3 Nine-Step (8LS + 1) PSA (𝜔0 = 2π∕8) 308A.10 Ten-Step Linear PSAs 309A.10.1 Ten-Step Least-Squares PSA (𝜔0 = 2π∕10) 309A.10.2 Ten-Step PSA with a First-Order Zero at 𝜔 = 0 and Second-Order

Zeros at {−𝜔0,±2𝜔0,±3𝜔0} (𝜔0 = π∕3) 310A.11 Eleven-Step Linear PSAs 311A.11.1 Eleven-Step Least-Squares PSA (𝜔0 = 2π∕11) 311A.11.2 Eleven-Step PSA with Second-Order Zeros at {0,−𝜔0,±2𝜔0, ±3𝜔0}

(𝜔0 = π∕3) 312A.11.3 Eleven-Step Frequency-Shifted LS-PSA (𝜔0 = 3 × 2π∕11) 313A.12 Twelve-step linear PSAs 314A.12.1 Twelve-step frequency-shifted LS-PSA (𝜔0 = 5 × 2π∕12) 314

References 315

Index 325

XI

Preface

The main objective of this book is to present the basic theoretical principles behindmodern fringe-pattern analysis as applied to optical metrology. In addition to this,for the experimentalist, we present in a ready-to-use form the most commonalgorithms for recovering the modulating phase from single or multiple fringepatterns.

This book deals with phase demodulation of fringe patterns typically encoun-tered in optical metrology techniques such as optical interferometry, shadowmoire, fringe projection, photoelasticity, moire interferometry, moire deflec-tometry, holographic interferometry, shearing interferometry, digital holography,speckle interferometry, and corneal topography.

Compared to previous books in this field, a major novelty of this work is thepresentation of a unified theoretical framework based on the Fourier descriptionof phase-shifting interferometry using the frequency transfer function (FTF). TheFTF, though new in fringe-pattern analysis, has been the standard way of analyzinglinear systems for at least 50 years in the electrical engineering field. The use ofFTF allows the natural and straightforward analysis and synthesis of phase-shiftingalgorithms (PSAs) with desired properties such as spectral response, detuningrobustness, signal-to-noise response, harmonic rejection, and so on.

Another major innovation in this book is the use of stochastic processes to studythe signal-to-noise power ratio of PSAs as a function of the statistical properties ofthe corrupting noise. As recently as 2008, noise analysis in PSAs was done usingerror propagation. This technique assumes that the interferogram is corruptedby a small additive ‘‘noisy’’ variation, and the resulting phase error is found bypropagating this data error all the way down to the PSA arc-tangent function. Theproblem with this approach lies in the fact that the statistical and spectral propertiesof the added noise are not specified, and one cannot tailor the PSA to the particularkind of corrupting noise of the interferometric data. In contrast, by using stochasticprocess theory as applied to linear systems, the signal-to-noise ratio analysis ofPSAs becomes as well defined and as productive as in electrical communicationstheory.

Finally, we offer a comprehensive description of the most common spatial andtemporal interferometric techniques such as Fourier transform interferometry,spatial phase shifting, phase unwrapping, self-calibrating algorithms, regularized

XII Preface

phase demodulation algorithms, and the regularized phase tracker and asyn-chronous self-tuning algorithms, among others.

This book is organized as follows:Chapter 1 reviews the theory behind linear systems theory in the Z and the

frequency domain, and briefly introduces its application to quadrature filters forphase estimation. This chapter covers most of the digital signal analysis backgroundused as theoretical basis for the rest of the book. Also, in this chapter an introductionto the theory of stochastic process as applied to linear systems is given to be usedin the rest of the book. One important application of stochastic processes to linearsystems is to find the signal-to-noise power ratio of any PSA when interferogramsare corrupted by additive white Gaussian noise.

Chapter 2 describes the main PSAs when the fringe patterns are recorded ata regular phase-sampling rate. We analyze some classical PSAs using the FTFframework presented in Chapter 1. We also discuss the synthesis of phase-shiftingfilters based on first-order building blocks, analyzing their spectral FTF response,detuning errors, signal-to-noise ratio, and harmonic response.

Chapter 3 includes linear and nonlinear PSAs for the phase estimation oftemporal fringe patterns with unknown phase steps, including tunable linear PSAsand self-tunable nonlinear PSAs. We close this chapter with the presentationof two self-calibrating PSAs, namely the iterative least-squares and the principalcomponent analysis methods.

Chapter 4 presents techniques for analyzing single-image fringe patterns withspatial carrier, using the classical Fourier transform, spatial synchronous detection,and the windowed Fourier transform. Also, we discuss the demodulation ofpixelated spatial-carrier interferograms along with their harmonic response. Thischapter ends with an introduction to the regularized quadrature filters.

Chapter 5 discusses the case when a single-image fringe pattern with closedfringes (without spatial and/or temporal carrier) is analyzed. In particular, wediscuss the regularized phase tracker and the local robust quadrature filters.Finally, we present the vortex transform and the general quadrature transform,which provide very good examples of the role of fringe direction in fringe-patternprocessing. This chapter ends with explaining how the direction information canbe used to transform any 1D PSA to a general n-dimensional PSA.

Chapter 6 deals with phase unwrapping, starting with elementary unwrappingtechniques, followed by the use of the phase tracker as phase unwrapper. Also,we introduce a set of linear recursive filters that use the phase predictor–correctorparadigm to obtain fast and robust phase unwrappers. Finally, we include thedetection of noise-generated phase inconsistencies to improve the noise robustnessof the phase-unwrapping process widely used in optical metrology, radar, andmedical images.

Appendix A includes a list of 40 PSAs and covers many published PSAs aswell as some new ones introduced in this book, all of them using the FTF PSAdesign paradigm. The 40 ready-to-use PSAs show the large variety of PSA designrequirements and constraints. Another intention of this appendix is to familiarizethe readers with a large number of PSA examples, which in the end would permit

Preface XIII

them to design their own PSAs tailored to their specific interferometric demands.As we already know, there is nothing like ‘‘the perfect PSA,’’ and the best PSA willalways depend on the kind of optical metrology application at hand.

Finally, Manuel Servin and Moises Padilla wish to acknowledge the financialsupport from the Centro de Investigaciones en Optica A. C. (CIO) and the MexicanConsejo Nacional de Ciencia y Tecnologia (CONACYT). J Antonio Quiroga wishesto acknowledge the financial support from the Universidad Complutense Madridand the Spanish Ministry of Science.

XV

List of Symbols and Acronyms

L{⋅} generic linear system𝛿(t) dirac delta functionIII(t) Comb functionh(t) impulse response functionr = (x, y) position vector (x, y) in the interferogram plane𝜌 =

√x2 + y2 radial position in the interferogram plane

q0 = (u0, v0) spatial carrierq = (u, v) spatial frequencies position vector𝜔 temporal frequency𝜔0 radial carrier, temporal carrier or in general a 1D carrierI(x, y, t) = a + b cos[𝜑(x, y, t)] general expression for a spatiotemporal interferogramI(r) = a + b cos[𝜑(r)] general expression for a spatial interferogramI(t) = a + b cos[𝜑(t)] general expression for a temporal interferogramI𝜑 = A0 exp[i𝜑(x, y, t)] analytic signal associated with a general interferogramI(𝜔) = [I(t)] temporal Fourier transform of a temporal interferogramI(u, v) = [I(x, y)] = [I(r)] spatial Fourier transform of a spatial interferogram{f (n)} = F(z) Z-transform of a discrete-time signalX(t), Y(t) temporal realization of the random process X and YE(X) ensemble average of the random variable XRX (t1, t2) ensemble autocorrelation functionfX (x) probability density function (pdf) of the random variable Xpsd power spectral density of a (random) stochastic processangle(f ) angle or phase of the complex number fRe{z} real part of complexIm{z} imaginary part of complex z𝜑 interferogram modulating phase�� interferogram demodulated or estimated phase𝜑W wrapped version of phase 𝜑

W[⋅] wrapping operator𝜔0 estimated 𝜔0 calculated from average intensity values I(x, y)c(x, y) general spatial carrier phaseH(𝜔) frequency response or FTF of a temporal digital filterH(u, v) = H(q) spatial frequency response or spatial FTF of a spatial digital filter

XVI List of Symbols and Acronyms

𝛻𝜑=(𝜑x, 𝜑y) interferogram instantaneous spatial frequencies𝜑t =

∂𝜑∂t

interferogram instantaneous temporal frequency1D One dimensional space2D Two dimensional spaceAIA Advanced iterative algorithmAWGN Additive white-Gaussian noiseBIBO Bounded-input bounded-output stability criterionC1 Space of continuous functions up to the first derivativeC2 Space of continuous functions up to the second derivativeDTFT Discrete time Fourier transformFTF Frequency transfer function or frequency responseFIR Finite impulse responseIIR Infinite impulse responseLTI Linear time-invariantPCA Principal component analysisPSA Phase shifting algorithmLS−PSA Least-squares phase-shifting algorithmPSI Phase shifting interferometryROC Region of convergenceRPT Regularized Phase TrackerRQF Robust quadrature filterLRQF Local robust quadrature filterS/N Signal-to-noise power ratioGS∕N Signal-to-noise power ratio gain

1

1Digital Linear Systems

1.1Introduction to Digital Phase Demodulation in Optical Metrology

In this chapter, we review the theory behind digital signals and their temporalprocessing using linear time-invariant (LTI) systems. The analysis of digital LTIsystems is based on their impulse response h(t), their Z-transform H(z), theirfrequency transfer function (FTF) H(𝜔), their harmonic response, and their stabilitycriteria. We then briefly discuss the equivalence between phase-shifting algorithms(PSAs) and quadrature linear filters tuned at the temporal phase-sampling rateof 𝜔0 radians per sample. Also, we analyze the aliasing phenomena produced byhigh-order harmonic distortion of the continuous interferogram being sampled.

In this chapter, we also discuss regularized low-pass filtering and its applicationto fringe-pattern denoising. Convolution spatial filters (such as the 3 × 3 averagingfilter) mix up valid fringe data inside the interferogram boundaries with outsidebackground where no fringe data is defined. This linear mixing of fringes andbackground distorts the modulating phase near the interferogram boundaries. Incontrast, regularized linear filters optimally decouple the fringe data inside theinterferogram from the outside background.

Finally, we discuss the theory behind stochastic processes to analyze the responseof LTI systems to stochastic input signals X(t). We define and analyze theirprobability density function (PDF), or fX (x), their ensemble average E{X}, andtheir stationary autocorrelation function RX (𝜏). We then continue by defining thepower spectral density (PSD) SX (𝜔) for X(t). This result is then used to show thatthe input PSD, SX (𝜔) of X(t), changes to |H(𝜔)|2SX (𝜔) when processed by an LTIsystem whose FTF is given by H(𝜔).

1.1.1Fringe Pattern Demodulation as an Ill-Posed Inverse Problem

A fringe pattern is defined as a sinusoidal signal where a continuous map, analogousof the physical quantity being measured, is phase-modulated by an interferometer,Moire system, and so on. An ideal stationary fringe pattern is usually modeled by

I(x, y) = a(x, y) + b(x, y) cos[𝜑(x, y)], (1.1)

Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications, First Edition.Manuel Servin, J. Antonio Quiroga, and J. Moises Padilla.c© 2014 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2014 by Wiley-VCH Verlag GmbH & Co. KGaA.

2 1 Digital Linear Systems

where {x, y} ∈ ℝ2; a(x, y) and b(x, y) are the background and local contrast func-tions, respectively; and 𝜑(x, y) is the searched phase function.

In physics and mathematics, an inverse problem is a general framework thatis used to convert the observed measurements into information about a physicalobject or system under study [1]. Clearly, Eq. (1.1) represents an inverse problem,where the fringe pattern I(x, y) is our measurement and the searched informationis given by the phase 𝜑(x, y). An inverse problem is said to be well posed ifthe mathematical model of a given physical phenomenon fulfills the followingconditions:

• A solution exists,• The solution is unique, and• The solution depends continuously on the data.

On analyzing Eq. (1.1), one can see that the phase function 𝜑(x, y) cannot bedirectly estimated since it is screened by two other unknown functions, namelya(x, y) and b(x, y). Additionally, 𝜑(x, y) can only be determined modulo 2π becausethe sinusoidal fringe pattern I(x, y) depends periodically on the phase (2π phaseambiguity); and its sign cannot be extracted from a single measurement withouta priori knowledge (sign ambiguity) because of the even character of the cosinefunction [cos(𝜑) = cos(−𝜑)]. Finally, in all practical cases, some noise n(x, y) isintroduced in an additive and/or multiplicative manner, and the fringe pattern maysuffer from a number of distortions, degrading its quality and further screeningthe phase information [2, 3].

It must be noted that, even if careful experimental setups could preventthe screening of 𝜑(x, y) due to the unknown signals a(x, y), b(x, y), andn(x, y), one would still have to deal with the sign ambiguity and the 2πphase ambiguity. Because of these ambiguities, the solution for this inverseproblem is not unique; this is illustrated in Figure 1.1, where severalphases (from an infinite number of possibilities) produce exactly the samesinusoidal signal.

In short, the phase demodulation of a fringe pattern, as the one modeled inEq. (1.1), can be viewed as an ill-posed inverse problem where some sort ofregularization process is required in order to obtain a proper phase estimation.However, despite its intrinsic difficulties, it is rather easy to visualize a possiblesolution for this inverse problem. First, let us rewrite Eq. (1.1) by means of thecomplex representation of the cosine function

I(x, y) = a(x, y) + 12

b(x, y){exp[i𝜑(x, y)] + exp[−i𝜑(x, y)]}. (1.2)

Now, if somehow one is able to isolate one of the analytic signals in Eq. (1.2), say,(1∕2)b(x, y) exp[i𝜑(x, y)], we have

tan ��(x, y) =Im{(1∕2)b(x, y) exp[i𝜑(x, y)]}Re{(1∕2)b(x, y) exp[i𝜑(x, y)]}

, (1.3)

where b(x, y) ≠ 0. Computing the arc-tangent of the above formula, one obtains awrapped estimation of the phase under study, that is, 𝜑(x, y) mod 2π. Thus, the

1.1 Introduction to Digital Phase Demodulation in Optical Metrology 3

(a)

(b)

(c)

(d)

−10

0

10

−10

0

10

−10

0

10

−4 −2 0 2 4

−1

0

1

Figure 1.1 Numerical simulation of several phases (a–c) producing exactly the same sinu-soidal signal (d). For ease of observation, only a horizontal slice is shown.

final step of this fringe pattern demodulation process usually involves an additionalphase unwrapping process. Nevertheless, when working with good-quality data,this last step is straightforward. Next, we will illustrate the easiest way to obtainthese analytic signals.

1.1.2Adding a priori Information to the Fringe Pattern: Carriers

A fringe pattern obtained as the output of a measuring system may be modifiedby the optoelectronic/mechanical hardware (sensors and actuators) and software(virtual sensors and actuators) of the system [4]. With these modifications, one isable to introduce known changes in the argument of the sinusoidal signal

I(x, y, t) = a(x, y) + b(x, y) cos[𝜑(x, y) + c(x, y, t)], (1.4)

where c(x, y, t) is a known function (typically a reference plane) and is calledthe spatiotemporal carrier of the interferogram. By design, a carrier must be ahigh-frequency signal in comparison with the searched phase 𝜑(x, y). That is‖‖𝛻c(x, y, t)‖‖ > ‖‖𝛻𝜑(x, y, t)‖‖max , (1.5)

4 1 Digital Linear Systems

where we define (locally) this nabla operator as

𝛻 =(

∂∂x

,∂∂y

,∂∂t

). (1.6)

For instance, for a stationary phase (which shows no explicit time dependence)given by in 𝜑(x, y), and a spatial carrier c(x, y), the following condition must befulfilled:√( ∂c

∂x

)2+(∂c∂y

)2

>

√(∂𝜑∂x

)2

+(∂𝜑∂y

)2

. (1.7)

The spatial and/or temporal carriers are of extreme importance in modern inter-ferometry: first of all, their presence allows us to solve the sign ambiguity since ingeneral cos(𝜑 + c) ≠ cos(−𝜑 + c). They also allow us to isolate the analytic signal(1∕2)b(x, y) exp[i𝜑(x, y)] which practically solves the phase demodulation process(the proof of this last point will be postponed until we review some basic conceptsof Fourier analysis). Some typical examples of the carrier functions are as follows:

• linear temporal carrier [5, 6]

c1(t) = 𝜔0t; (1.8)

• tilted (spatial) carrier [7, 8]

c2(x, y) = u0x + v0y; (1.9)

• conic carrier [9]

c3(𝜌) = 𝜔0𝜌; 𝜌(x, y) =√

x2 + y2; (1.10)

• 2 × 2 pixelated carrier [10–12]

exp[i c4(x, y)] = exp

[i𝜔0

(1 23 4

)]∗∗

∞∑m=0

∞∑n=0

𝛿(x − 2m, y − 2n), (1.11)

where 𝜔0 = π∕2, and ∗∗ is the two-dimensional convolution operation;• 3 × 3 pixelated carrier [13]

exp[i c5(x, y)] = exp⎡⎢⎢⎣i𝜔0

⎛⎜⎜⎝1 2 38 9 47 6 5

⎞⎟⎟⎠⎤⎥⎥⎦ ∗∗

∞∑m=0

∞∑n=0

𝛿(x − 3m, y − 3n), (1.12)

where 𝜔0 = 2π∕9.

Since digital interferometry is a research area under continuous development, itis impossible to list all useful spatiotemporal carriers; again, these are just somecommonly used examples. For illustrative purposes, in Figures 1.2–1.5 we showhow these carriers modify the fringe pattern.

The temporal linear carrier approach (shown in Figure 1.2) allows us to demod-ulate closed-fringe interferograms [5, 6]. However, this method is not useful (inprinciple) to study fast-varying phenomena since it requires a(x, y), b(x, y), and𝜑(x, y) to remain stationary during the phase-step acquisition.

1.1 Introduction to Digital Phase Demodulation in Optical Metrology 5

(a) (b) (c)

Figure 1.2 Numerical simulation of a closed-fringe interferogram, phase-modulated with alinear temporal carrier 𝜔0t. The piston-like phase step between successive samples is 𝜔0 =2π∕3 rad.

(a) (b)

Figure 1.3 Simulation of a closed-fringe interferogram (previously shown in Figure 1.2a)phase-modulated with a linear spatial carrier (a), producing an open-fringe interfero-gram (b).

(a) (b)

Figure 1.4 (a) Circular pattern with binaryamplitude projected over an eye usinga Placido mire and (b) its spectrum asobtained by the FFT2 algorithm. The larger

spectral flares in the spectrum are due tothe binary profile of the projected pattern,and these lead to harmonic distortion.

6 1 Digital Linear Systems

(a) (b)

Figure 1.5 Simulation of a closed-fringe interferogram (previously shown in Figure 1.2a)phase-modulated with a four-step pixelated carrier (a), producing a 2D pixelated carrierinterferogram (b).

The spatial linear carrier approach (shown in Figure 1.3) allows us to demodulateopen fringe patterns from a single image, making this technique particularly usefulto study fast dynamic phenomena [7, 8].

The conic carrier (shown in Figure 1.4) has been used to measure the topographyirregularities of the human cornea since 1880 [14]. Traditionally, these irregularitieswere analyzed by means of a sparse set of estimated slope points, integrated alongmeridian lines to obtain the topography of the testing cornea [15]. However,recently it has been proved that these patterns of periodic concentric rings can bephase-demodulated by means of synchronous interferometric methods, providingholographic phase estimation at every point of the region under study. A detailedreview of this topic is available in [9].

The 2D pixelated carrier (shown in Figure 1.5) was originally proposed as a spatialtechnique for the simultaneous acquisition of four phase-shifted interferograms, tobe demodulated using a ‘‘temporal’’ PSA, but recently it has been shown that spatialsynchronous demodulation allows higher quality measurements [10–12]. The nine-step pixelated carrier was proposed as a logical extension of this technique to allowfor the analysis of nonsinusoidal signals in fast dynamic phenomena [13]. Wechoose to include only one illustrative example for both cases because the four-stepand nine-step pixelated carrier interferograms are visually indistinguishable.

Example: Synchronous Demodulation of Open FringesFor illustrative purposes, let us assume a vertical open-fringe interferogram phase-modulated by a linear spatial carrier in the x direction, given by

I(x, y) = a(x, y) + b(x, y) cos[𝜑(x, y) + u0x],= a + (b∕2) exp[i(𝜑 + u0x)] + (b∕2) exp[−i(𝜑 + u0x)], (1.13)

where we have omitted the spatial dependency in a, b, and 𝜑 for simplicity.Applying the spatial synchronous demodulation method, the so-called the Fouriermethod [7, 8], first we multiply our input signal with a complex reference signal

1.1 Introduction to Digital Phase Demodulation in Optical Metrology 7

(a) (b) (c)

Figure 1.6 Several steps of the spatialsynchronous demodulation of an open-fringe pattern interferogram. The input sig-nal is shown in panel (a). Panel (b) shows

the real part of the synchronous productexp(−iu0x)I(x, y). The estimated phase ��(x, y)modulo 2π, as obtained from Eq. (1.16), isshown in panel (c).

(which is a value stored in the digital computer) oscillating at the same frequencyas our lineal carrier:

f (x, y) = exp(−iu0x)I(x, y), (1.14)

= a exp(−iu0x) + (b∕2) exp(i𝜑) + (b∕2) exp[−i(𝜑 + 2u0x)].

In general, the spatial variations of the phase are small in comparison to the carrier(Eq. 1.5), |𝛻𝜑|max ≪ u0, so the only low-frequency term in the above equation is theanalytic signal (b∕2) exp(i𝜑). Thus, applying a low-pass filter to Eq. (1.14), we have

LP{ f (x, y)} = (1∕2)b(x, y) exp[i𝜑(x, y)], (1.15)

where the low-pass filter LP{⋅} is preferentially applied in the Fourier domain formore control in the filtering process. Taking the ratio between the imaginary andreal part of this complex-valued analytic signal, we have

tan ��(x, y) =Im{(1∕2)b(x, y) exp[i𝜑(x, y)]}Re{(1∕2)b(x, y) exp[i𝜑(x, y)]}

, (1.16)

where b(x, y) ≠ 0. Computing the arc-tangent of the above equation, the estimatedphase ��(x, y) is wrapped within the principal branch (−π, π]; so there is a 2πphase ambiguity as illustrated in Figure 1.6. Usually, a priori knowledge of thephenomenon indicates that ��(x, y) should be continuous so the final step in thedemodulation process is to apply a regularization condition that removes this2π ambiguity.

1.1.3Classification of Phase Demodulation Methods in Digital Interferometry

To summarize our previous discussion, the main objective of fringe patternanalysis is to estimate a usually continuous phase map 𝜑(x, y) from the inputintensity values I(x, y, t). This means solving an ill-posed inverse problem wherethe signal of interest is masked by unknown functions, plus the sign ambiguity and

8 1 Digital Linear Systems

the 2π phase ambiguity problems. The simplest way of action is to actively modifythe fringe pattern in order to provide additional information, that is, introducingspatial or temporal carriers.

The inclusion of phase carriers not only solves the sign ambiguity problem but italso provides spectral isolation between the unknown signals in the interferogram(this will be discussed in detail in Chapters 2 and 4). On the other hand, the2π phase ambiguity is intrinsic to fringe-pattern analysis, so some unwrappingmethod is usually required as the last step of a phase-demodulation process[16, 17]. Nevertheless, there are notable exceptions that estimate nearly directly theabsolute phase without 2π phase ambiguity, such as the temporal heterodyningtechnique [18], as well as phase demodulation methods that directly estimatethe unwrapped phase, such as the linear phase-locked loop [19], temporal phaseunwrapping [20], hierarchical absolute phase measurement [21], and the regularizedphase tracking [22].

According to the above, a possible classification for the phase demodulationmethods in fringe pattern analysis is as follows: whether a phase carrier isrequired; whether this carrier is a spatial and/or temporal one; and whetherthe estimated phase is wrapped within a single branch (requiring an additionalunwrapping processing) or without 2π ambiguity. In Figure 1.7, we present aschematic representation of this proposed classification for some commonly used

Spatialcarrier

Temporalcarrier

Withoutcarrier

Squeezing

Conic

Linear

Pixelated

Phase-shifting

Temporal phaseunwrapping

Regularizedphase

tracking

Unwrapping

Linear PLL

Multiple images

2-D Hilbert

a+bcos(φ+c)

φ(x,y)^

Single image

A0exp(iφ)

Figure 1.7 Schematic classification of somecommonly used phase estimation methodsin modern fringe pattern analysis. Here wetry to illustrate that the intermediate tar-get in most methods is to isolate the ana-lytic signal A0 exp[i𝜑(x, y)], from where one

can straightforwardly compute the wrappedphase ��(x, y) modulo 2π. On the other hand,some methods combine both the fringedemodulation and the phase unwrappingprocesses, obtaining directly the estimatedphase ��(x, y) without the 2π ambiguity.

1.2 Digital Sampling 9

phase estimation methods. We want to stress that this scheme is illustrative andby no means exhaustive.

In the following chapters, we will analyze several methods to estimate the analyticsignal (1∕2)b(x, y) exp[i𝜑(x, y)] highlighting their positive features and drawbacks.However, in order to do this, first we need to review some basic mathematical tools.For a beginner to this topic, this will serve as a quick reference guide for linearsystems theory. Advanced readers can skip the rest of this chapter and return toit only in specific cases that we will refer to whenever we are unable to keep thediscussion self-contained in the following chapters.

1.2Digital Sampling

Despite the fact that (analog) macroscopic phenomena are properly modeled ascontinuous functions, nowadays virtually any processing required is done on digitalcomputers. Thus, typically one of the very first steps in fringe pattern analysis isto perform some analog-to-digital (A/D) conversion, the so-called digital samplingprocess. In this section, we analyze some mathematical functions commonlyused to model digital signals and systems. This will allow us to understand andcope with many problematic phenomena (e.g., spectral overlap with high-orderdistorting harmonics) that arise in fringe pattern analysis as consequence of thedigital sampling process.

It is noteworthy that we will use t for the independent variable when working withunidimensional (1D) signals and systems; thus we will refer to continuous-time anddiscrete-time functions. Nevertheless, this is just a convention and the followingtheory also applies for 1D spatial processing.

1.2.1Signal Classification

By definition, a signal is everything that contains information. Signals in engineer-ing systems are typically classified in five different groups:

1) Continuous-time or discrete-time2) Complex or real3) Periodic or aperiodic4) Energy or power5) Deterministic or random.

Continuous-time and discrete-time signals. A signal is defined to be a continuous-time signal if the domain of the function defining the signal contains intervals ofthe real line f (t) where t ∈ ℝ. A signal is defined to be a discrete-time signal if thedomain of the signal is a countable subset of the real line { f (n)} or f [n], wheren ∈ ℤ.

10 1 Digital Linear Systems

In most cases, discrete signals arise from uniform sampling of continuous-timesignals. However, these sampled signals can also be represented by continuousfunctions (as we will see in Section 1.2.3). Thus, the following definitions andconventions apply to both continuous and sampled signals:

Real and complex signals. In optics, we often work with complex (analytic) signalsof real arguments. In general, a complex signal is given by

f (t) = Re[ f (t)] + iIm[ f (t)], (1.17)

where i =√−1. Or, in polar form, the modulo of the signal is defined by

|f (t)| = √f (t) f ∗(t) =

√{Re[ f (t)]}2 + {Im[ f (t)]}2, (1.18)

and its phase (modulo 2π) is given by

angle [ f (t)] = arctanIm{ f (t)}Re{ f (t)}

. (1.19)

A word of caution: in modern programming languages, this operation is calledatan2(⋅) and it uses two arguments. Unlike the single argument arc-tangentfunction, atan2(⋅) is able to retrieve the searched angle without sign ambiguitywithin (0, 2π).

Periodic and aperiodic signals. A signal is said to be periodic if repeats itself intime. The function f (t) represent a periodic signal when

f (t) = f (t + kT), ∀ k ∈ ℤ (1.20)

and the fundamental frequency of a periodic signal is given by 1∕T .

Energy and power of signals. The energy of a signal f (t) is a real and nonnegativequantity given by

U{ f (t)} = ∫∞

−∞||f (t)||2 dt. (1.21)

If U{ f (t)} exceeds every bound, we say that f (t) is a signal of infinite energy. Forsuch cases, it is useful to calculate the power of the signal, which represents theenergy per unit time. It is defined by

• aperiodic signals

P{ f (t)} = limT→∞

12T ∫

T

−T

||f (t)||2 dt; (1.22)

• periodic signals

P{ f (t)} = 1T ∫

t+T

t

||f (𝜏)||2 d𝜏. (1.23)

1.2 Digital Sampling 11

Deterministic signal Random noise Observed signal

+ =

Figure 1.8 Signals observed in nature, which are usually composed by deterministicsignals distorted by some degree of random noise.

Deterministic and random signals. Most of the time, we deal with deterministicsignals distorted in some degree by random noise (Figure 1.8). The kind of noisetypically observed in fringe pattern analysis can be modeled as a well-knownstochastic process; however, the theory of stochastic processes is so vast that adetailed review is beyond the scope of this book. In Section 1.9, we will brieflyreview some basic aspects of this theory, but for now we will assume that we aredealing with purely deterministic signals.

1.2.2Commonly Used Functions

Dirac delta function. Also called the unit-impulse function, the Dirac delta is(informally) a generalized function on the real number line that is zero everywhereexcept at zero where its value tends to infinity. However, quite often is better todefine the Dirac delta function by its properties as

∫∞

−∞f (t)𝛿(t − t0)dt = f (t0), (1.24)

which is also constrained to satisfy the identity

∫∞

−∞𝛿(t) dt = 1. (1.25)

For convenience, some algebraic properties of the Dirac delta function are listed inTable 1.1.

The Dirac delta function is graphically represented as a vertical line with anarrow at the top. The height of the arrow is usually used to specify the value ofany multiplicative constant, which will give the area under the function; anotherconvention is to write the area next to the arrowhead (Figure 1.9).

Unit step function. Also called Heaviside’s step function, it may be defined by meansof the Dirac delta as

u(t) = ∫t

−∞𝛿 (𝜏) d𝜏 =

{0 for t < 0,

1 for t > 0.(1.26)

12 1 Digital Linear Systems

Table 1.1 Properties of the Dirac delta function.

Properties Observations

𝛿(t − t0) = 0 For all t ≠ t0𝛿(−t) = 𝛿(t) Dirac delta is an even function𝛿(at) = (1∕|a|)𝛿(t) Scaling property∫ ∞−∞ f (t)𝛿(t − t0)dt = f (t0) Definition as a measure𝛿(g(x)

)=∑

i 𝛿(x − xi)∕|g′(xi)| Where xi are the roots of g(x)f (t)𝛿(t − t0) = f (t0)𝛿(t − t0) Valid under the integration symbolf (t) ∗ 𝛿(t − t0) = f (t − t0) Shifting property𝛿(x, y, z,…) = 𝛿(x)𝛿(y)𝛿(z)… n-dimension generalization

t0 0

1

(a) (b)t

a0

t0

Figure 1.9 (a) Usual representations of the impulse function 𝛿 (t) and (b) the shifted (andescalated) impulse function a0𝛿

(t − t0

).

Rectangle function. The rectangle function of unit height and base is defined by

II (t) =

{0 if |t| > 1∕2,

1 if |t| < 1∕2.(1.27)

This function can also be represented by means of the unit step function as

II (t) = u(

t + 12

)− u

(t − 1

2

). (1.28)

The step function and the rectangle function are illustrated in Figure 1.10.

Dirac comb. The so-called Dirac comb is a periodic distribution of Dirac deltafunctions that plays an important role in the sampling process:

III(t) =∞∑

n=−∞𝛿(t − n). (1.29)

This generalized function is illustrated in Figure 1.11.

t0

1 1

−0.5(b)(a)

0 0.5 t

Figure 1.10 Unit step function (a) and rectangle function (b).