systematic errors of phase-shifting speckle interferometry

m

Systematic errors of phase-shifting speckle interferometry

Pascal Picart, Jean Claude Pascal, and Jean Marc Breteau

A theoretical and numerical investigation of the systematic phase errors in phase-shifting speckleinterferometry is presented. The theoretical investigation analyzes the behavior of some systematicerror induced by intensity variations in two cases of data-computing techniques. The first case dealswith the technique in which the phase change is computed, unwrapped, and then linearly filtered; thesecond case deals with the technique in which the data are linearly filtered before the arctangentcalculation and then unwrapped. With the first filtering technique it is shown that it is preferable whenthe phase change is of relatively low spatial frequency, leading to a particularly accurate method. Withthe second case it is demonstrated that an important parameter of speckle interferometry is the modu-lation factor of the interference frame that induces phase errors when the data are filtered before thearctangent calculation. We show that this technique is better than the first when the phase change iscomposed of high-spatial-frequency variations. The theoretical investigation of the two techniques iscompared with numerical simulations, considering the frequency characteristics of the phase change, andthis shows a good match between theory and simulations. © 2001 Optical Society of America

OCIS codes: 120.0120, 120.2650, 120.3180, 120.3940, 120.4630, 120.5050.

1. Introduction

Electronic speckle-pattern interferometry ~ESPI! hasproved in the past few years to be a useful industrialtool for the measurement of full-field surface defor-mation of naturally rough objects.1 For example,ESPI has been successfully applied to the inspectionof several composite structures,2 to the analysis of

echanical characteristics of masonry,3 and to theinspection of aluminum honeycomb panels.4 More-over, ESPI allows the displacement field in the threeorthogonal axes to be examined independently.5Consequently, industries are showing an increasinginterest in optical inspection techniques because oftheir advantages over traditional techniques: Rela-tively large areas can be inspected in one shot with-out the need to paste a finite number of strain gaugerosettes on the surface to be studied, no disposablerecording medium such as the holographic technique

The authors are with the Ecole Nationale Superieured’Ingenieurs du Mans, rue Aristote, 72085 Le Mans Cedex 9,France. P. Picart [email protected]! and J. C. [email protected]! are also with the Institutd’Acoustique et de Mecanique, Avenue Olivier Messiaen, 72085 LeMans Cedex 9, France. J. M. Breteau [email protected]!is also with the Laboratoire de Physique de l’Etat Condense, Av-enue Olivier Messiaen, 72085 Le Mans Cedex 9, France.

Received 18 July 2000; revised manuscript received 22 January2001.

0003-6935y01y132107-10$15.00y0© 2001 Optical Society of America

is needed, there is no requirement for surface prep-aration with a grating as with moire interferometry,and so on. However, the accuracy of phase-shiftingspeckle interferometry has not been clearly estab-lished. The accuracy of the apparatus is closely re-lated to two different errors: systematic error,which gives the distance between the mean experi-mental value and the exact value, and random error,which shows the variation of the measurement re-garding the mean experimental value. Random er-ror is a part of the repeatability of the apparatus,whereas systematic error characterizes the trueness.In this paper we focus on the systematic error of themeasurement of the phase change, which is to bemeasured by means of phase-shifting speckle inter-ferometry. We present both a theoretical and a nu-merical study of the behavior of the systematic errorsof such a method. We study the systematic error bytaking into account some filtering procedures thatcan be used. Indeed, we consider both filtering ofthe raw unwrapped phase map and filtering of theraw data just before the arctangent calculation. Weexamine the phase error versus the iteration number,considering two filtering processes and two kinds ofphase change, a low- and a high-frequency phasechange. We deal here with linear filtering of thedata. It should be noted that nonlinear filtering canbe used with, for example, median filtering. How-ever, we will compare the behavior of the systematicerror with a same linear filtering scheme. Indeed,the theory of linear filtering is systematic and well

1 May 2001 y Vol. 40, No. 13 y APPLIED OPTICS 2107

sat

w

tfT

TtD

t

2

S

B

t

p

F

T3bsc

2

known, whereas the few nonlinear processes, whichwe call filtering, do not have a very theoretical foun-dation and must often be studied by use of abun-dantly numerical simulations. This point needsfurther analysis, which will not be discussed here.Moreover, the use of linear filtering allows for esti-mation of the random error of the measurement.6 InSection 2 we first give some background on the phase-shifting speckle technique. Section 3 presents thetheoretical analysis of the systematic errors, consid-ering the filtering process of the data. A numericalexperimental investigation is presented in Section 4by considering the systematic error introduced by thecalibration parameter of a speckle interferometer.In Section 5 some conclusions are drawn about theability of the filtering process to provide highly accu-rate measurement in phase-shifting speckle inter-ferometry.

2. Basic Fundamentals of the ElectronicSpeckle-Pattern Interferometry Technique

The task of ESPI is to measure a displacement ~or atrain!, seen as an optical phase change, induced byn object of study submitted to loading ~mechanical,hermal, pneumatical!. The computation of the

phase-change term Dw is achieved by means of reg-istering a set of N interferometrical phase-shiftedframes before the object is loaded. These phase-shifted frames, referenced Ei, are written as

Ei 5 a 1 b cos@c 1 ~i 2 1!f#,

i 5 1, 2, 3, . . . , N, (1)

here a, b, and c are high-spatial-frequency randomparameters and f is the phase shift introduced be-ween each frame. Then N new phase-shiftedrames are acquired after the object has been loaded.hese frames are referenced Fi:

Fi 5 a 1 b cos@c 1 Dw 1 ~i 2 1!f#,

i 5 1, 2, 3, . . . , N, (2)

he spatial dependence of Eqs. ~1! and ~2! is volun-arily omitted. An estimation of the phase changew is computed mod 2p with an arctangent formula,7

Dw 5 atanFS~Ei, Fi!

C~Ei, Fi!G , (3)

where S~Ei, Fi! and C~Ei, Fi! can be shown to be equalo kb sin Dw 1 dS~Ei, Fi! and kb cos Dw 1 dC~Ei, Fi!,

respectively, and the combination of the frames ~Ei,Fi! in dS~Ei, Fi! and dC~Ei, Fi! depends on the chosenalgorithm8,9; k is a coefficient depending on thephase-shifted frames used for the computation, and bis closely related to the parameter b. For example,for a 2 3 3 algorithm derived from Frantz et al.10 wehave k 5 4 and b 5 b2; for the popular 2 3 4 algo-rithm11 we have k 5 4 and b 5 b2; for the 2 3 4algorithm derived from Carre12 we have k 5 4 and

108 APPLIED OPTICS y Vol. 40, No. 13 y 1 May 2001

b 5 b ; for a 2 3 6 algorithm derived from Zhao andurrel13 we have k 5 16 and b 5 b2; and for the

recent 2 3 2 phase-step technique proposed by vanrug and Somers14 we get k 5 1 and b 5 b, the

computed phase change being Dwy2 1 py4. Sincehe terms dS~Ei, Fi! and dC~Ei, Fi! can be high-

frequency noises that can be caused, for example, byspeckle decorrelation, it is common practice to ap-ply a low-pass filtering to the numerator and thedenominator of Eq. ~3!. Thus we now consider the

hase-change computation to be

Dw 5 atanFSf~Ei, Fi!

Cf~Ei, Fi!G , (4)

where Sf ~Ei, Fi! 5 f @kb sin Dw# 1 f @dS~Ei, Fi!#, Cf ~Ei,i! 5 f @kb cos Dw# 1 f @dC~Ei, Fi!#, with f @. . .# being

the applied linear filtering procedure. Usually, thelinear filtering is based on a two-dimensionalfinite-IR filter represented by a ~2M 1 1! 3 ~2N 1 1!matrix that contains the coefficients of the impulseresponse h ~M is related to the vertical length of thefilter, and N is related to the horizontal length; M andN are expressed in pixel units!. Usually, the two-dimensional linear filter is described as follows:

h 5 3h~2M, 2N! . . . . . . . . . h~2M, N!

u u uh~0, 2N! . . . h~0, 0! . . . h~0, N!

u u uh~M, N! . . . . . . . . . h~M, N!

4with (

m52M

m51M

(l52N

l51N

h~m, l ! 5 1. (5)

he convolution kernel can be an average filter of size3 3 or greater, or with M Þ N. The filter size can

e adjusted when we take into account the speckleize and the spatial characteristics of the phasehange.

3. Theoretical Analysis of Systematic Error

The accuracy of an interferometrical device is deter-mined by considering two different error types: sys-tematic error and random error. In practice,systematic and random error are mixed. Generally,it is rather difficult to detect systematic errors andeven more so to identify their origin. Thus it is eas-ier to estimate systematic errors by an analyticalstudy of the phase sensitivity. Inversely, randomerrors can be directly estimated by means of the ex-perimental results. The behavior of systematic er-rors is strongly influenced by the filtering that iscarried out in Eq. ~4!. In this section we first exam-ine the behavior of systematic errors when the phasechange is directly computed with the raw data andthen smoothed, and, finally, we consider the influenceof the filtering process on the systematic error of themeasurement when filtering before the arctangentcalculation.

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tt

A. Phase Calculation with Raw Data

A systematic error in the phase measurement gener-ated by a variation on a parameter p can be describedy the following equation15:

dDw 5]Dw

]pdp 1

12

]2Dw

]p2 ~dp!2 1 · · · 11k!

]kDw

]pk ~dp!k

1 . . . . (6)

Considering Eqs. ~3! and ~6! and a variation on theparameter of the denominator and the numerator ofthe arctangent function, this can be rewritten as

dDw 5]Dw

]SdS~Ei, Fi! 1

]Dw

]CdC~Ei, Fi!

112

]2Dw

]S2 @dS~Ei, Fi!#2 1

12

]2Dw

]C2 @dC~Ei, Fi!#2

1 · · · 11k!

]kDw

]Sk @dS~Ei, Fi!#k

11k!

]kDw

]Ck @dC~Ei, Fi!#k 1 . . . . (7)

Approximating Eq. ~7! to the first-order terms leadsto

dDw 51

kbdS~Ei, Fi!cos Dw 2

1kb

dC~Ei, Fi!sin Dw.

(8)

he linear filtering of the raw data after the arctan-ent calculation and the unwrapping process leads tofiltered phase error:

dDwf 5 f @dDw#

5 fF 1kb

dS~Ei, Fi!cos DwG2 fF 1

kbdC~Ei, Fi!sin DwG . (9)

ith Eq. ~9! it is not straightforward to derive thenalytical expression of a filtered systematic errorhat is due to a variation on a parameter of the mea-urement ~i.e., a, b, f, c!. Considering that the spa-

tial autocorrelation function of systematic error,given by the autocorrelation of the responsible pa-rameter, is narrower than that of the impulse re-sponse of the spatial filter, and following the linearfiltering theory, we estimate the power of the filterederror from

PdDwf 5 PdDw (m52M

m51M

(l52N

l51N

uh~m, l !u2, (10)

PdDw being the power of the original systematic error,given by

PdDw 5 ^dDw2~c, Dw, f, a, b!&, (11)

ith

^dDwn~c, Dw, f, a, b!& 5 *2`

1`

*2`

1`

dDwn~u, v, f, a, b!

3 Pc,Dw~u, v!dudv, (12)

here Pc,Dw~u, v! is the joint density function of c andw, which are independent variables. Owing to the

ndependence of c and Dw, the joint probability den-ity function is equal to the product of each density.he speckle phase c obeys uniform statistics. In theame way we can consider that the phase change Dws uniformly distributed on the interval @0, 2p#.hus we have

PdDw 51

4p2 *2p

1p

*0

2p

dDw2~u, v, f, a, b!dudv. (13)

he dependence of systematic phase errors on ran-om parameters such as a, b and c implies that thepatial autocorrelation of dDw is comparable with theransverse speckle size. Indeed, a and b are relatedo the speckle-field amplitude, and c is related to thepeckle phase. If we consider that the phase isearly constant in a speckle grain, that ensures thathe spatial autocorrelation of dDw is nearly equal tohe speckle correlation length. Choosing a convolu-ion kernel size greater than the speckle size leads tosmoothing of systematic phase errors with a gain of

gh 5 1YF (m52M

m5M

(l52N

l51N

uh~m, l !u2G1y2

.

Thus the linear filtering of the raw phase reduces thesystematic error of the measurement, thus increasingthe accuracy of the process, and it reduces the ran-dom fluctuations, thus increasing the repeatability ofthe process. If we consider a multiple linear filter-ing process, the effective impulse response to be con-sidered is that given by an n time convolution of thempulse response, n being the number of filteringn $ 1!. If we denote hn as the effective impulse

response, we have hn,

where p denotes correlation, hn is of size ~2nM 1 1! 3~2nN 1 1!, and the gain of the linear filtering processwill be

ghn5 1YF (

m52nM

m51nM

(l52nN

l51nN

uhn~m, l !u2G1y2

.

B. Phase Calculation with Filtered Data

In practice the filtering of the raw phase requires theuse of a robust noise-immune phase-unwrapping al-gorithm. Thus it is common practice to filter theraw data before arctangent calculation @Eq. ~4!#. Inhis way there is no simple equivalence between fil-ered systematic error and systematic error of the

hn~i, j! 5h~i, j! p h~i, j! p · · · p h~i, j!

n times , (14)


pd

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2

filtered data. It can be noted that the filtering of thedata operates on b sin Dw and b cos Dw, which do nothave the same power spectrum density as Dw. Be-cause b sin Dw and b cos Dw have more spatial vari-ations than Dw, their power spectrum contains morehigh frequencies than that of Dw Thus terms b sinDw and b cos Dw change by filtering, and there is nosimple analytical expression for the result of f @b sinDw# and f @b cos Dw#. Thus filtering the data inducesa phase error. If we denote DwR as the phase com-

uted after filtering, then the phase error is given byDwR 5 DwR 2 Dw and is found to be

dDwR 5

atanHf @b sin Dw#cos Dw 2 f @b cos Dw#sin Dw

f @b cos Dw#cos Dw 1 f @b sin Dw#sin DwJ .

(15)

hen the data are corrupted by intensity variationshat are due to systematic errors or to random errors,he actual computed phase is given by

tan DwE 5Sf 1 dSf

Cf 1 dCf, (16)

with Sf 5 f @kb sin Dw#, Cf 5 f @kb cos Dw#, dSf 5@dS~Ei, Fi!#, and dCf 5 f @dC~Ei, Fi!#. This leads to

a systematic phase error

dDw 5 DwE 2 DwR

5 atanS CfdSf 2 SfdCf

kCf2 1 kSf

2 1 CfdCf 1 SfdSfD . (17)

hen approximating the reciprocal of the denomina-or by a first-order binomial expansion and consider-ng a linear approximation of the arctangentunction, we now have

dDw <1k

CfdSf

Cf2 1 Sf

2 21k

SfdCf

Cf2 1 Sf

2

11k2

~Sf2 2 Cf

2!dSfdCf

~Cf2 1 Sf

2!2 11k2

SfCf~dCf2 2 dSf

2!

~Cf2 1 Sf

2!2 .

(18)

Limiting relation ~18! to the two first terms, corre-ponding to a linear approximation, we get

dDw 51k

f @b cos Dw#

f @b cos Dw#2 1 f @b sin Dw#2

3 f @dS~Ei, Fi!#

21k

f @b sin Dw#

f @b cos Dw#2 1 f @b sin Dw#2

3 f @dC~Ei, Fi!#. (19)

The equivalence of Eqs. ~9! and ~19! is not guaranteedwhatever the values of the filter function f @. . .#, thephase change Dw, term b, terms dS~Ei, Fi! and dC~Ei,Fi!. The filtering of the data tends to reduce thesystematic error as indicated by Eq. ~19! and the


resence of terms f @dS~Ei, Fi!# and f @dC~Ei, Fi!#, butt adds a phase error that is due to the random naturef parameter b as indicated by Eq. ~15!. In this casehe total phase error is due to the contributions of theystematic phase error of the measurement and of anrtifact generated by the filtering itself. When con-idering iterative filtering as described by Aebischernd Waldner,16 which consists of recomputing the

sine and the cosine of the phase before the filtering,Eq. ~19! is valid only for the first iteration. Indeed,uch filtering deals with linear filtering of the numer-tor and the denominator of Eq. ~3! and with nonlin-ar filtering of the variations of the denominator andhe numerator introduced by the sine and the cosinealculations. An exact analytical iterative computa-ion can be obtained in the case of the influence oferm b by the following algorithm:

sin Fk 5 f @sin Fk21#, cos Fk 5 f @cos Fk21#, k $ 1,

dDwRk 5 atanSsin Fk cos Dw 2 cos Fk sin Dw

cos Fk cos Dw 1 sin Fk sin DwD ,

Ck 5 atanSsin Fk

cos FkD ,

sin Fk 5 sin Ck,

cos Fk 5 cos Ck,

with sin F0 5 b sin Dw, cos F0 5 b cos Dw. (20)

For the influence of terms dS~Ei, Fi! and dC~Ei, Fi!the exact analytical iterative computation is de-scribed by this algorithm:

sin Fk 5 f @sin Fk21#, cos Fk 5 f @cos Fk21#,

sin Qk 5 f @sin Qk21#, cos Qk 5 f @cos Qk21#, k $ 1,

dDwk 5 atanSsin Qk cos Fk 2 cos Qk sin Fk

cos Qk cos Fk 1 sin Qk sin FkD ,

Ck 5 atanSsin Fk

cos FkD , Vk 5 atanSsin Qk

cos QkD ,

sin Fk 5 sin Ck, sin Qk 5 sin Vk,

cos Fk 5 cos Ck, cos Qk 5 cos Vk,

with sin F0 5 b sin Dw, sin Q0 5 b sin Dw

1 dS~Ei, Fi!yk,

cos F0 5 b cos Dw, cos Q0 5 b cos Dw

1 dC~Ei, Fi!yk. (21)

n the following we consider a numerical investiga-ion of the approaches developed in Subsections 3.And 3.B.

4. Numerical Results and Discussion

A. Introduction

There are many potential error sources in a phase-shifting speckle interferometer. There is coherentnoise,17 phase-shift calibration,18 phase-shift nonlin-

19

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bpf

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Ci

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p

t

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earity, detector nonlinearity, and so on. The sen-itivity of the phase computation @Eq. ~3!# to thearameters a, b, f, and c depends strongly on thelgorithm used for the calculation. An overview ofhe phase-shifting algorithms can be found in Refs. 8nd 9. To illustrate the results of Section 3, we willonsider here the four-frame algorithm for which Dws evaluated with two sets of four phase-shiftedrames ~f 5 py2!. In this case we have11

Dw 5

atanF~F4 2 F2!~E1 2 E3! 2 ~F1 2 F3!~E4 2 E2!

~F1 2 F3!~E1 2 E3! 1 ~F4 2 F2!~E4 2 E2!G ,

(22)

and k 5 4, b 5 b2. The error source we considerelow is that given by the calibration stage of thehase modulator. If the phase shift between eachrame is given by f 5 2pcVyl, V being the voltage

applied to the phase-shifting device, the measure-ment of the calibration parameter c leads to an un-certainty characterized by its standard deviationvalue sc. This uncertainty is generally due to thefluctuations of the set up: laser instabilities and me-chanical andyor thermal drifts. Because the cali-bration parameter is known with a certain confidencelevel ~i.e., 2sc if c obeys Gaussian statistics!, a sys-tematic phase error is induced by the inequality ofthe phase shift generated by its linear variation fromits actual value in the experimental setup. For ex-ample, we developed a polarizing speckle interferom-eter with a Pockels cell as phase shifter, and wemeasured scyc 5 0.7174% at l 5 632.8 nm.20 If wessume that there is no error on each first step of eachet because the first interferogram is the referencerame of the set and that the phase shift variation onach frame is dfi 5 ~i 2 1!pscyc, then with Eqs. ~1!,2!, and ~22! the intensity variations are found to be

dS~Ei, Fi! 5 0,

dC~Ei, Fi! 5 24p~scyc!b2 sin~2c 1 Dw!. (23)

ith Eq. ~8! the phase error is given by

dDw~c, Dw, c! 5p

2sc

c@cos~2c 1 2Dw! 2 cos 2c#,

(24)

onsidering Eqs. ~13! and ~24!, the rms value, whichs equal to the square root of PdDw in this case, is found

to be sDw 5 pscy~2c!. With the previous value we getsDw 5 0.01127 rad. The investigation of the analysis

eveloped in Section 3 is performed with a computerrogram. A random phase c uniformly distributedn @2p, 1p# is generated on a 512 3 512 matrix. Weompute a deterministic phase change Dw~m, n! withpatial characteristics that will be discussed below.he parameters m, n are the coordinates on a 512 312 pixel mesh. Because the filtering process has annfluence on the accuracy, we compute two sets of 2 3phase-shifted interferograms following Eqs. ~1! and

2!, the first set without any intensity deviation ~saideference set! and the final set by taking account ofhe inequality of the phase shift ~said erroneous set!.he parameters a and b are computed with interme-iate variables a1 and a2 with Rayleigh statistics, and

we have a 5 a12 1 a2

2 and b 5 2a1a2. In thissimulation the correlation length of the speckle fieldsis equal to one pixel. We used two methods for fil-tering the data: first, that described in Subsection3.A, denoted case A, starting with a 3 3 3 averagefilter and filtering iteratively; finally, that describedin Subsection 3.B, denoted case B, following the pro-cedure described in Ref. 16 and starting with a 3 3 3average filter. For case A with the erroneous set, wecompute and unwrap the numerical phase Dwnum;then for each iteration we apply the filtering toDwnum, and we compute the numerical phase errordDwA

num 5 Dwnum 2 Dw. At the same time we com-pute the filtering of the analytical phase error, andwe compute the theoretical rms of the filtered error,i.e., sDwyghn

. For case B, at each iteration, we com-ute ~1! the reference phase DwR

num with the refer-ence set and the numerical phase error from dDwR

num

5 DwRnum 2 Dw, ~2! the erroneous phase DwE

num withthe erroneous set and the numerical phase error thatis due to the calibration parameter dDwB

num 5DwE

num 2 DwRnum, and ~3! the analytical phase errors

from Eqs. ~20! and ~21!. The iteration number goesfrom 0 ~i.e., phase change computed with raw datawithout any filtering! to 30. Because the phasechange may not remain invariant by the filtering,we consider two kinds of spatial characteristics ofthe phase change: a low-spatial-frequency phasechange and a high-spatial-frequency phase change.

B. Results for a Low-Frequency Phase Change

We start by computing a deterministic phase changeDw with low spatial frequencies,

Dw~m, n! 5 a~m2 1 n2!. (25)

The parameter a controls the fringe number and thusthe cut-off spatial frequency of the phase change. Thevalues of a, m, and n are adjusted so that we get fourfringes over the field of view, corresponding to a rela-tively low-spatial-frequency signal. Figure 1 showsthe simulated wrapped phase map. Figure 2 showsthe computed systematic error with raw data and theapproximated systematic error @Eq. ~24!# for a varia-ion on parameter c of 2scyc 5 1.4348% ~i.e., in the 2sc

range!. The rms error of the simulation is ;sDw 5.01126 rad. Thus the numerical values corroboratehe analytical one, and this shows that the linear ap-roximation is an efficient tool for studying the behav-or of systematic errors in ESPI. The upper part ofig. 3 shows the 0° profile of the numerical phase error

n case A and the analytical one with 30 iterations.he lower part of Fig. 3 shows the 0° profile of theumerical and analytical @Eq. ~21!# phase errors, whichre due to the calibration parameter in case B, for 30terations. Figure 4 shows the 0° profile of the nu-

erical and the analytical @Eq. ~20!# phase errors,


ht

Tdro

2

which are due to the filtering process in case B, for 30iterations. The upper part of Fig. 5 shows the rmsvalues of the numerical and the analytical errors ofcases A and B. For case A the theoretical value of therms computed from sDwyghn

is represented. It can beseen that with the chosen phase change the contribu-tion of the systematic error is reduced in case A com-pared with case B. The lower part of Fig. 5 shows therms value that is due to the filtering process in case B.It can be seen that in case B the contribution of theparameter b to the phase error is much greater thatthat of the calibration parameter. In case B the phaseerror is mainly due to the filtering process and to thehigh-frequency nature of parameter b. The resultsshow, in the case of a low-frequency phase change, thegood agreement between numerical simulation andtheory of Section 3.

C. Results for a High-Frequency Phase Change

In this case the phase change Dw is chosen to be

Dw~m, n! 5 a~m2 1 n2! 1 a expF2~m 2 m0!

2 1 n2

2sn2 G .

(26)


With the same values of a, m, n, used in Eq. ~25!, theigh-frequency part is introduced by the Gaussianerm. The parameters a and sn control the ampli-

tude and the width of the Gaussian peak. The pa-rameter m0 controls the position of the maximum ofthe peak along the horizontal axis of the grid. Fig-ure 6 shows the simulated wrapped phase map. Fig-ure 7 shows the computed systematic error with rawdata and the approximated systematic error @Eq.~24!# for the variation on parameter c discussed here.

he statistics are the same as those calculated fromata of Fig. 2. The upper part of Fig. 8 shows thems values of the numerical and the analytical errorsf cases A and B for the influence of the calibration

Fig. 3. 0° profile of phase error of iteration 30, influence of thecalibration parameter ~solid curve, numerical; circles, analytical!.

Fig. 4. 0° profile of phase error of iteration 30, case b, influence ofthe filtering process ~solid curve, numerical; circles, analytical!.

Fig. 1. Simulated low-frequency phase change.

Fig. 2. Influence of the calibration parameter, 0° profile of rawphase error ~solid curve, numerical; circles, analytical!.

t

c

parameter. For case A the theoretical value of therms computed from sDwyghn

is represented as well.It can be seen that for case A the rms value of thenumerical phase error increases after two iterations,meaning that the filtering of the unwrapped phase

Fig. 5. Cases A and B: rms phase error due to the calibrationparameter ~case A: solid curve, numerical; circles, analytical; as-erisks, sDwyghn

. Case B: dotted–dashed curve, numerical;crosses, analytical!.

Fig. 6. Simulated high-frequency phase change.

map tends to distort the high-frequency part of Dw.The lower part of Fig. 8 shows the rms value that isdue to the filtering process in case B. It can be seenthat the contribution of the parameter b to the phaseerror increases after five iterations showing, as in

Fig. 7. Influence of the calibration parameter, 0° profile of rawphase error ~solid curve, numerical; circles, analytical!.

Fig. 8. Cases A and B: rms phase error due to the calibrationparameter ~Case A: solid curves, numerical; circles, analytical;asterisks, sDwyghn

. Case B: dotted–dashed curve, numerical;rosses, analytical!.


2

case A, that filtering the data tends to distort theuseful high-frequency part of the signal. Figures 9and 10 show the 0° profile of the numerical and theanalytical phase errors for the first iteration. It canbe seen on the upper part of Fig. 9 that, for case A, thedistortion of the high-frequency part of the signalalready occurs in the zone of the high spatial fre-quency between pixels 100 and 175. For case A withan iteration number greater than 5 the distortion ofthe high-frequency part of the signal is quite signif-icant and tends to cause a great increase in the rmsvalue of the phase error that is now much larger thanthe contribution that is due to the calibration param-

Fig. 9. 0° profile of phase error of iteration 1, influence of thecalibration parameter ~solid curve, numerical; circles, analytical!.

Fig. 10. 0° profile of phase error of iteration 1, case B, influence ofthe filtering process ~solid curves, numerical; circles, analytical!.


eter. Figures 11 and 12 show the 0° profiles in caseB for 30 iterations. It can be seen that the filteringprocess on one hand tends to reduce the contributionof parameter c and on the other hand tends to in-crease the distortion of the signal and thus increasethe total rms phase error. The difference of thevalue of the iteration number that produces a signaldistortion in cases A and B can be explained by con-sidering the power density spectrum of on one handthe unwrapped phase change Dw and on the otherhand the sine and the cosine of the phase change. Incase A the filtering procedure is applied on the un-wrapped phase change, whereas in case B cos Dw andsin Dw have to be considered. The power spectrumdensity of Dw contains more high-frequency compo-nents than those of cos Dw and sin Dw because of itshigher amplitude: This is why procedure A appearsto be more quickly sensitive to the high-frequencypart of the phase change.

D. Discussion

The results of Subsections 4.B and 4.C show that b isa determinant parameter of the filtering process of

Fig. 11. 0° profile of phase error of iteration 30, case B, influenceof the calibration parameter ~solid curves, numerical; circles, an-alytical!.

Fig. 12. 0° profile of phase error of iteration 30, case B, influenceof the filtering process ~solid curve, numerical; circles, analytical!.

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data issued from phase-shifting speckle interferome-try. Indeed, in the case of a low-frequency phasechange this parameter tends to increase the rms ofthe total phase error of the computed phase map to alevel that is much larger than that induced by somesystematic error sources that can occur in the setup.Its contributions decrease asymptotically with theiteration number. So it is preferable to filtering thedata following the procedure A, because in this casephase-change filtering reduces the systematic errorswithout introducing any subsequent contribution.It can be noted that applying procedure A requires arobust noise-immune phase-unwrapping algorithm.In the case of a high-frequency phase change, proce-dure B is clearly better than A, because the latterintroduces a signal distortion even if the iterationnumber remains small. Filtering following proce-dure B tends to decrease the contribution of the sys-tematic error that is due to the calibration parameter,but at the same time the contribution that is due toparameter b induces a distortion on the high-requency part of the signal so that the iteration num-er cannot reach a high value. With the examplehosen the iteration number should be less than 10.t should be noted that the notion of high-frequencyignal is closely related to the correlation fringe den-ity of the specklegram. The convolution kernel sizeas to be chosen taking account of the speckle size,hich can be larger than the pixel size, and takingccount of the spatial characteristics of the phasehange. Indeed if the specklegram is constitutedith vertical fringes, then it would be better to choose

.. N. Inversely, if the fringes are horizontal,hen the choice must be M ,, N. If there are someegions of the specklegram where the fringe densitys high, then it would be preferable to adapt the ker-el size to the part of the image that is being filtered

n order to limit the distortion of the signal. Finally,t should be noted that procedure B, used with a highteration number as described in Ref. 16, is an effi-ient tool for a qualitative analysis ~if only a visual-zation is needed! of the wrapped phase change,ecause it tends to emphasize the 2p phase jumps,hereas for a quantitative analysis, used for a mod-

ling validation for example,20 procedure B has to becomputed with a small iteration number. In thislast case, if the iteration number reaches a highvalue, then the cumulative measurement needed tothe validation can be corrupted by a nonnegligibleerror that will falsify the comparison between exper-imental results and numerical simulation.

5. Conclusions

We have presented a theoretical and numerical in-vestigation of the systematic phase errors in phase-shifting speckle interferometry. The theoreticalinvestigation has analyzed the behavior of some sys-tematic errors induced by intensity variations in twocases of data-computing techniques. In the firstcase the filtering is done by means of computing andunwrapping the phase change with raw data andthen filtering the result. The investigation has

shown that this technique is quite advantageouswhen the phase change has low spatial frequencies,because this technique leads to a high smoothing ofthe systematic errors and thus increases the accuracyof the method. It should be noted that this tech-nique requires a robust noise-immune phase-unwrapping algorithm. In the final case of filtering,the technique starts with filtering the raw data be-fore the arctangent calculation and then unwrappsthem. This technique reduces the contribution ofthe systematic error, but it takes the disadvantage tobe sensitive to the modulation parameter of thephase-shifted frames. This induces a phase errorthat is due to the high-frequency nature of this pa-rameter. In the case of high-frequency phasechange this technique is better than the first, but theiteration number has to be small, certainly less than10 as a general rule. The algorithms that have beenproposed to describe the influence of both the modu-lation parameter and the systematic error have beenvalidated by a numerical simulation that shows goodagreement with the theoretical analysis. The accu-racy available by means of this filtering technique islimited by the contribution of the modulation param-eter that appears to be quite larger than that of thesystematic error.

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