phase demodulation from a single fringe pattern based on a correlation technique

Pb

E

1

Rltcepa

atcdrmfiupsetatepc

UBCe

2

hase demodulation from a single fringe patternased on a correlation technique

ric Robin and Valery Valle

We present a method for determining the demodulated phase from a single fringe pattern. This method,based on a correlation technique, searches in a zone of interest for the degree of similarity between a realfringe pattern and a mathematical model. This method, named modulated phase correlation, is testedwith different examples. © 2004 Optical Society of America

OCIS codes: 100.2650, 100.5070, 120.3940.

0ucw

tnp

ustts

fcW

2

Ldrtww

t

i

. Introduction

ecent progress with digital high-speed cameras al-ows us to analyze fast phenomena, such as propaga-ion of cracks, vibration, and impact loading. Theseameras give us a single frame of each step in thexperimental process. With improvements in com-uters, the study of these phenomena by digital im-ge processing has been facilitated.In mechanics, most optical methods based on

nalysis of fringes represent the fringe pattern as arigonometric function and search to extract the me-hanical information contained in the value of theemodulated phase. For example, in shadow moi-e1,2 the demodulated phase represents a displace-ent field and in photoelasticimetry it contains theeld of the principal stresses.3–6 For a better eval-ation of phase, all these optical metrologies usehase-shifting techniques.1,2,7 But for a dynamictudy we cannot use these techniques because sev-ral fringe patterns are necessary. So we must ex-ract the information with only one image. Therere several methods, such as regularized phaseracking8–10 and use of a phase-locked loop,11,12 forxtracting information from only one image. Ahase-locked loop, which is limited to fringes witharrier frequency, gives a measurement precision of

The authors are with the Laboratoire de Mecanique des Solides,nite Mixte de Recherche 6610, Universite de Poitiers, Teleport 2,oulevard Pierre et Marie Curie, B. P. 30179, 86962 Futuroscopehasseneuil Cedex, France. E. Robin’s e-mail address [email protected] 9 December 2003; revised manuscript received 1 April

004; accepted 18 May 2004.0003-6935�04�224355-07$15.00�0© 2004 Optical Society of America

.42 rad for analysis of noisy fringe patterns.12 Reg-larized phase tracking, which is well adapted tolosed fringes, permits determination of the phaseith a precision of 0.5 rad.10

These two methods are based on a local approach tohe problem. In this paper we present a new tech-ique that is based on a pseudoglobal approach to theroblem that uses a correlation technique.In optical metrology, a correlation procedure13–16 is

sually used for determination of displacements andtrain fields. This technique consists of comparingwo digital images, before and after deformation ofhe studied material, and of evaluating the degree ofimilarity between the two images.We aim at determining the demodulated phase

rom a single real fringe pattern. To do so, we useorrelation on the modulated phase �fringe pattern�.e will call our method modulated phase correlation.

. Description of the Correlation Technique

et us describe the principle of the correlation proce-ure for two one-dimensional functions with one pa-ameter. Function h represents the real signal, ghe mathematical model, � the parameter in whiche are interested, and � the parameter of calculationindow.We can find the degree of similarity by minimizing

he relation

C�� D

�h�� g��, ��2d� (1)

f we develop this relation into

�h�� g��, ��2 � h2�� g2��, �� 2�h�� g��, ��.(2)

1 August 2004 � Vol. 43, No. 22 � APPLIED OPTICS 4355

t

I

am

3

Ud

wgIcINe�fifs

waeds

Tmpa

tr

acpr

ced

Fdpwr

C

w

Tws

4

We note that the minimization of C�� correspondso the maximization of

T�� D

h�� g��, ��d�. (3)

n Eq. �3� we see the correlation function.In the literature13–16 both Eqs. �1� and �3� are used,

nd for our technique we have chosen the first for-ulation �Eq. �1��.

. Formulation of the Mathematical Model

sually the mathematical representation of a two-imensional fringe pattern is globally expressed by

f � x, y� � A� x, y�cos�� x, y�� B� x, y�, (4)

here A is the amplitude modulation, B is the back-round illumination, and � is the value of the phase.n this global representation �, A, and B are compli-ated functions that depend on coordinates x and y.f we consider only a small zone of interest, denoted�, and centered at coordinates �x, y�, the global

xpression can be reduced to a simpler one in whichdepends on coordinates ��, � and where x and y arexed. Our mathematical model, representing

ringes with constant pitch and orientation, is de-cribed by

f � A, B, , p, �, �, � � A cos��, p, �, �, �� B,(5)

��, p, �, �, � �2�

pcos�� x � ��

�2�

psin�� y � � � �,

(6)

ith N�, centered on coordinates �x, y�. Variables pnd are, respectively, parameters of pitch and ori-ntation of the fringe pattern, and � is the term of theemodulated phase. Equation �6� is used only for itsimplest character.Equation �5� is valid if the zone of interest is small.

he value of orientation can be determined onlyodulo �, and we solve this problem later in this

aper. This is why we identify as the inclinationnd not as the orientation at first.For calculation of the degree of similarity between

he mathematical expression designated f and theeal fringe pattern denoted I, we use the formulation

C� A, B, p, , �� V�,

� f ��, , A, B, p, , ��

� I��, ��2d�d (7)

t coordinates �x, y�. To minimize the C function weannot use a gradient method because we have aeriodic function that depends on and �. For thiseason we must calculate all the values of C that

356 APPLIED OPTICS � Vol. 43, No. 22 � 1 August 2004

orrespond to all the possible values for each param-ter , p, A, B, and �. The minimum of C givesirectly the five parameters that we are looking for:

C� A*, B*, p*, *, �*� � minA,B,p,,�

C� A, B, p, , ��.

(8)

In practice, we use the discret formulation of C:

C� A, B, p, , �� ,��N�

A cos��, p, �, �, ��

� B � I��, ��2. (9)

rom a numerical point of view it is impossible toetermine all the possibilities of the quintuplet �A, B,, , ��, but we can estimate A*, B*, p*, *, and �*ith the help of a specific discrete law for each pa-

ameter. Equations �8� and �9� become

� A{, B{, p{, {, �{� � minA,B,p,,�

( ��,��N�

A cos��, p, �, �, �� B � I��, ��2) ,

(10)

ith

A* � A{ � εA,

B* � B{ � εB,

p* � p{ � εp,

* � { � ε,

�* � �{ � ε�.

o minimize εA, εB, εp, ε, and ε� �i.e., A{ 3 A*, . . .�e use an interpolation of the correlation peak by a

econd-degree polynomial. In Fig. 1 we show the

Fig. 1. Interpolation of the correlation peak.

wet

A

tfcrvtmtp

ccptt

4

F

2am

wp

vt

tmcb

z�ta3f

ay to determine �*interp that we consider the beststimate of the real value of �*. So we have for rank� 1, t, and t � 1 the system

C1 � C�A{, B{, p{, {, ��t � 1��,

C2 � C�A{, B{, p{, {, ��t � 1��,

Cm � C�A{, B{, p{, {, ��t� � �{�. (11)

nd we obtain �* from the relation
interp
Wp

amw

leooWfecs

We can use this interpolation for each parameter ofhe quintuplet. In practice we use this process onlyor �inclination� and � �the demodulated phase�, be-ause these are the only parameters needed. For pa-ameters { and �{ this technique gives the wrappedalues. So the interpolation must take into accounthe discontinuity induced by modulo � and by �odulo 2�. To obtain phase fields we have to apply

he mathematical model to each point of the real fringeattern, that is, to shift N� in the directions x and y.The study is divided into two parts. The first part

oncerns the choice of incremental law and gives aalculated map of each parameter. The second partresents a solution for the determination of orienta-ion from the computation of the inclination and giveshe demodulated phase.

. Building of Incremental Laws

irst we present the boundaries of each parameter.We have chosen to analyze digitalized images with

56 gray levels. As parameters A and B stand formplitude modulation and varying background illu-ination, respectively, they are limited to

B � A � 256,

B � A � 0, (13)

hich lead us to define the progression intervals forarameters A and B:

0 � A � 128,

0 � B � 256. (14)

The terms � and , which are, respectively, thealues of the wrapped phase and of the inclination ofhe fringes, imply

0 � � �,

0 � � � 2�. (15)

Now we are going to deal with pitch p. In practice,he choice of limits of p is directly linked to the di-ensions of the zone of interest. This zone must

ontain enough information for recognition and muste in accordance with discretization limits. So the

one of interest may have a minimum of one fringefor low-frequency consideration� and must respecthe Nyquist theorem �for high-frequency consider-tion�. For a first approach we used dimensions of3 � 33 pixels for the zone of interest. This impliesor that for p, �, and

2 � p � 33; (16)

�16 � � � 16,�16 � � 16. (17)

e now present the incremental laws used for eacharameter.In practice, the search for all quintuplets would beslow process if we multiplied the number of incre-ents. Instead, we divided the interval associatedith each parameter into 10 increments.Then we built two incremental laws. The first

aw is a linear progression and is used for param-ters A, B, , and �. For pitch p, if we want toptimize the number of increments in the intervalf variation the progression must not be linear.e have chosen a linear progression in terms of

requency �1�p�. For this parameter, we purposelyxceeded the upper limit given in formula �16� be-ause this limit is not clearly fixed. The aim of thistudy is always to demodulate fringes of all dimen-

Table 1. Progressions Used for Parameters A, B, �, �, and p

A B � p

5 20 0 0 4

15 40 �

10�

55

25 60 �

52�

56

35 80 3�

103�

57

45 100 2�

54�

58

55 120 �

2� 10

65 140 3�

56�

513

75 160 7�

107�

517

85 180 4�

58�

523

95 200 9�

109�

531

425880

�*interp �12

��2�t�C2 � �2�t�C1 � �2�t � 1�Cm � �2�t � 1�Cm � �2�t � 1�C1 � �2�t � 1�C2

��t � 1�Cm � ��t � 1��t � 1� � ��t��t � 1� � ��t � 1�Cm � ��t � 1�C1 � ��t�C1. (12)


slrf

fgrtcn

5W

Ir

pTbg

wosuitu3ubT

tuwfyhp

6P

Tu

FT�

Flw

4

ions. All the values of the five parameters areisted in Table 1. In this table we mapped theesult of each parameter �Fig. 2� for a simulatedringe pattern that comprises circles.

In Figs. 2�a�, 2�b�, and 2�c� we show, respectively, theringe pattern and parameters A and B represented inray levels. In Fig. 2�d� the values of pitches are notestricted to the limits given in formula �16�. Indeed,he bigger p calculated in this map is 58 pixels. Wean also remark that the use of the correlation tech-ique reduces the widths of the resultant maps.

. Determination of the Orientation Maps and of therapped Phase Field

f we observe the maps of and � �Figs. 2�e� and 2�f �,espectively� we can see that the discontinuities of


arameter induce discontinuities in parameter �.his is normal because we know not the orientationut only the inclination of fringes, so � is not directlyiven.We saw in Section 4 that the inclination map israpped modulo �. So we must unwrap this map tobtain values of that vary from 0 to 2�, that is toay, the orientation. We can add that even if isnwrapped it is still determined modulo � because it

s impossible, with a single fringe pattern, to knowhe sign of the phase slope. For the unwrapping wesed a classical algorithm,17,18 and we show in Fig.�a� a map of the orientation. Adding � to duringnwrapping must be accompanied, at the same time,y correction of parameter �, which becomes 2� � �.he result of this procedure is shown in Fig. 3�b�.So the phase is completely demodulated. To ob-

ain the continuous phase field �Fig. 3�c�� we can nowse the same unwrapping procedure as that used for. Of course, the unwrapped phase is still knownith an arbitrary constant. The dimensions of the

ringe pattern are 200 � 200 pixels and the full anal-sis requires a computation time of 2 h because of theigh number of iterations induced by the number ofarameters and the number of increments.

. Evaluation of Precision with Simulated Fringeatterns

o evaluate the precision of phase extraction we firstsed the images shown in Fig. 3. For this simulated

ig. 3. Examples of determination of orientation with a simu-ated fringe pattern: We show maps of �a� orientations, �b�rapped phase, and �c� the unwrapped phase.

ig. 2. Example of analysis with a simulated fringe pattern: �a�he initial image gives maps of �b� parameter A, �c� parameter B,

d� pitch p, �e� inclinations , and �f � phase �.

tauap

cwt

o2t

pr

7

NfiM

Fmf

Fw

Fmn

Fcomo

est without noise, we know the phase function ex-ctly and we can easily compare it to the map of thenwrapped phase. From Fig. 4 we can see the goodgreement between the mathematical and the com-uted phases.With a statistical analysis of the global image we

an estimate a precision of measurement of �0.1 radhen p has fewer than 33 pixels. For a pitch greater

han 33 pixels the precision increases to �0.3 rad.The second simulated test concerns the modulation

f the same phase function to which we have added0% amplitude noise and 60% phase noise, yieldinghe results shown in Fig. 5.

ig. 4. Differences between the phase obtained by the MPCethod and values of the mathematical phase: MPC used with a

ringe pattern without noise.

ig. 5. �a� Results from a noisy fringe pattern, �b� map of therapped phase, and �c� map of the the unwrapped phase.

When the noise level is high, we obtain �Fig. 6� arecision of �0.4 rad for small values of p and of �0.8ad for the other values �large p�.

. Examples of Real Fringe Patterns

ow we present three kinds of fringe pattern. Therst one �440 � 510 pixels; Fig. 7�a��, obtained from aichelson interferometer, represents the out-of-

ig. 6. Differences between the phase obtained by the MPCethod and values of the mathematical phase: MPC used with aoisy fringe pattern.

ig. 7. �a� Fringe pattern obtained from an interferometer andorresponding to an elastic cracked plate under loading. �b� Therientation and �c� the wrapped phase were calculated by the MPCethod. �d� The unwrapped phase represents the field of the

ut-of-plane displacement.


plhtia7

rftppoG

mfm

tpfhoIv�

8

WdgmiittA

Fmepdifference of principal stresses.

Fgwpears when the phase is unwrapped.

4

lane displacement of an elastic cracked plate underoading.19 From this interferogram we can see aigh level of speckle noise that causes the quality ofhe fringe pattern to deteriorate. Here it is interest-ng to note that the discontinuity of orientationslong the crack tip is well calculated �Figs. 7�a� and�b��.The second fringe pattern �156 � 247 pixels� rep-

esents subtraction of the principal stresses obtainedrom three-dimensional photoelasticimetry.3 A pho-oelastic bar under torsion loading gives us an exam-le of noisy fringes �Fig. 8�. We can see that theroposed method is highly efficient. The result wasbtained in 2 h with a personal computer of at least 2Hz.The last example analyzed concerns a shadowoire fringe pattern �428 � 645 pixels�. We can see

rom Fig. 9 discontinuities of 2� on the orientationap, but they are not a problem because the orien-


ation is given modulo 2�. The demodulation of thehase is well performed even if the contrast of theringe pattern is weak. We can also see that theigh-frequency modulation that is inherent in the usef shadow moire does not affect the phase extraction.ndeed, frequencies of this modulation correspond toalues of pitches under the limit defined in Table 1p � 4 pixels�.

. Conclusions

e have presented in this paper a new method foremodulation of a two-dimensional phase from a sin-le fringe pattern �open or closed fringes�. Thisethod, based on the correlation technique, consists

n searching, in a zone of interest, for the mathemat-cal solution that best resembles real fringes. Whenhe best models were found, we saw how to extracthe phase from the orientation unwrapping process.s we chose a simple mathematical model to repre-

ig. 8. �a� Fringe pattern from three-dimensional photoelastici-etry obtained from a bar in a torsion loading test. �b� The ori-

ntation and �c� the wrapped phase were calculated by theroposed method. �d� The unwrapped phase gives the field of the

ig. 9. The shadow moire method applied to part of a telephoneives �a� a fringe pattern with which �b� the orientation and �c� therapped phase are calculated. �d� A relief of the telephone ap-

sbttWptatwspaTrcaoad

R

1

1

1

1

1

1

1

1

1

1

ent the real fringe pattern, the zone of interest muste adjusted to this condition. For this reason we sethe dimensions of this zone to 33 � 33 pixels and usedhe parameters’ progression referenced in Table 1.

e have seen that determination of the modulatedhase is made with the help of the fringes’ orienta-ion, so this requires the use of a good unwrappinglgorithm to obtain the orientation from the inclina-ion.17 However, this algorithm is also necessary ife want to unwrap the phase. With the help of

imulated fringes, we estimated the precision ofhase retrieval at 0.4 rad for extremely noisy fringesnd at 0.1 rad for a fringe pattern without noise.he demodulation of the proposed method is highlyobust because it can operate even if the noise level isonsiderable. This is so because of the pseudoglobalspect of the calculus procedure. The disadvantagef the method lies in its long computing time, but ancceleration procedure to solve this problem can beeveloped.

eferences1. G. Mauvoisin, F. Bremand, and A. Lagarde, “Three-

dimensional shape reconstruction by phase-shifting shadowmoire,” Appl. Opt. 33, 2163–2169 �1994�.

2. J. Degrieck, W. Van Paepegem, and P. Boone, “Application ofdigital phase-shift shadow moire to micro deformation mea-surements of curved surfaces,” Opt. Lasers Eng. 36, 29–40�2001�.

3. J. C. Dupre and A. Lagarde, “Photoelastic analysis of a three-dimensional specimen by optical slicing and digital image pro-cessing,” Exp. Mech. 37, 393–397 �1997�.

4. J. Villa, J. A. Quiroga, and J. A. Gomez-Pedrero, “Measure-ment of retardation in digital photoelasticity by load steppingusing a sinusoidal least-squares fitting,” Opt. Lasers Eng. 41,127–137 �2004�.

5. H. Aben and L. Ainola, “Isochromatic fringes in photoelastic-ity,” J. Opt. Soc. Am. A 13, 750–755 �2000�.

6. R. A. Tomlinson and E. A. Patterson, “The use of phase-stepping for the measurement of characteristic parameters inintegrated photoelasticity,” Exp. Mech. 42, 43–50 �2002�.

7. Y. Morimoto and M. Fujisaa, “Fringe pattern analysis by a

phase shifting method using Fourier transform,” Opt. Eng. 33,3709–3714 �1994�.

8. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulationof a single interferogram by use of a two dimensional regular-ized phase tracking technique,” Appl. Opt. 36, 4540–4548�1997�.

9. M. Servin, J. L. Marroquin, and J. A. Quiroga, “Regularizedquadrature and phase tracking �RQPT� from a single closed-fringe interferogram,” Reportes Tecnicos del Cimat 2001,Guanajuto, GTO, Mexico, 22 August, 2003�.

0. R. Legarda-Saenz, W. Osten, and W. Juptner, “Improvement ofthe regularized phase tracking technique for the processing ofnonnormalized fringe patterns,” Appl. Opt. 41, 5519–5526�2002�.

1. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Fringe patterndemodulation with a two-dimensional digital phase locked loopalgorithm,” Appl. Opt. 41, 5479–5487 �2002�.

2. M. Servin, R. Rodriguez-Vera, and D. Malacara, “Noisy fringepattern demodulation by an iterative phase locked loop,” Opt.Lasers Eng. 23, 355–365 �1995�.

3. H. A. Bruck, S. R. McNeill, M. A. Sutton, and W. H. Peters,“Digital image correlation using Newtown–Raphson method ofpartial differential correction,” Exp. Mech. 29, 261–267 �1989�.

4. M. A. Peng Cheng, M. A. Sutton, H. W. Schreier, and S. R.Mcneill, “Full-field speckle pattern image correlation withB-spline deformation function,” Exp. Mech. 42, 344–352�2002�.

5. H. W. Schreier and M. A. Sutton, “Systematic errors in digitalimage correlation due to undermatched subset shape func-tion,” Exp. Mech. 42, 303–310 �2000�.

6. H. Weber and R. Lichtenberger, “The combination of specklecorrelation and fringe projection for the measurement of dy-namic 3-D deformation of airbag caps,” in International Unionof Theoretical and Applied Mechanics Symposium on Ad-vanced Optical Methods and Application in Solid Mechanics,A. Lagarde, ed., �Kluwer Academic, Dordrecht, The Nether-lands, 1998�, pp. 619–626.

7. F. Bremand, “A phase unwrapping technique for object reliefdetermination,” Opt. Lasers Eng. 21, 49–60 �1994�.

8. B. Gutmann and H. Weber, “Phase unwrapping with thebranch cut method: role of phase field direction,” Appl. Opt.39, 4802–4816 �2000�.

9. L. Humbert, V. Valle, and M. Cottron, “Experimental deter-mination and empirical representation of out-of-plane dis-placements in a cracked elastic plate loaded in mode I,” Int. J.Solids Struct. 37, 5493–5504 �2000�.


phase demodulation from a single fringe pattern based on a correlation technique

Documents