uncertainty analysis of displacements measured by phase-shifting moiré interferometry

Optics Communications 237 (2004) 25–36

www.elsevier.com/locate/optcom

Uncertainty analysis of displacements measured byphase-shifting Moir�e interferometry

Ra�ul R. Cordero *, Ignacio Lira

Department of Mechanical and Metallurgical Engineering, Pontifica Universidad Catolica de Chile, Vicu~na Mackenna,

4860 Santiago, Chile

Received 9 February 2004; received in revised form 27 March 2004; accepted 29 March 2004

Abstract

Phase-shifting Moir�e interferometry is a whole-field optical technique used to measure displacements. Because of its

high sensitivity, its results are affected by even small environmental perturbations. In this paper, the influences of these

perturbations are characterized and compared with other systematic effects. The experiment was a uniaxial tensile test.

In-plane displacement measurements were performed several times under repeatability conditions at different load

stages. The dispersions of these results were evaluated by their standard deviations, which showed fluctuations at twice

the fringe frequency. An uncertainty analysis was carried out to include other systematic effects. We found that the local

displacement uncertainties depended on the sample elongation and on the reference location. An equation was found

for these uncertainties as a function of the total number of fringes occurring.

� 2004 Elsevier B.V. All rights reserved.

PACS: 42.30.M; 42.97.B; 81.70.B

Keywords: Moir�e interferometry; Phase shifting; Uncertainty analysis; Tensile tests

1. Introduction

Phase-shifting Moir�e interferometry (PSMI) is a

well-known high sensitivity optical measurement

technique. It is useful for analyzing very small

deformations, such as those that occur in tensile

tests of engineering materials. The technique in-

* Corresponding author. Tel.: +56-2-686429/5632654501; fax:

+56-2-6865828/5632797656.

E-mail addresses: [email protected], [email protected]

(R.R. Cordero).

0030-4018/$ - see front matter � 2004 Elsevier B.V. All rights reserv

doi:10.1016/j.optcom.2004.03.079

volves three stages: first, a phase-shifting algo-

rithm is used to obtain the wrapped phase map;second, phase unwrapping is performed to obtain

the phase differences across the field of view; and

third, from the unwrapped phase map the relative

displacement field is derived.

The results of these processes are affected by

several random and systematic influences. These

include: optical noise and environmental pertur-

bations; the quality of the diffraction gratings andother optical components of the interferometer;

the care put into beam alignments; the character-

istics of the phase shifting and unwrapping

ed.

mail to: [email protected],

26 R.R. Cordero, I. Lira / Optics Communications 237 (2004) 25–36

processes; etc. Therefore, it is to be expected that

measurements performed on a nominally identical

measurand should yield varying results [1]. It ap-

pears, however, that the uncertainty of PSMI

measurements has not been evaluated. Related

efforts include: the search of optimal phase-shiftingalgorithms [2–4]; the quantification and compen-

sation of systematic errors introduced by these

algorithms [5–8]; the evaluation of the statistical

properties of their results [9,10]; the estimation of

their response under vibration [11]; the determi-

nation of the influence of noise under some par-

ticular conditions [12] and the development of

newer unwrapping algorithms to eliminate phasedistortions and residues in the unwrapped results

[13–16].

In this paper, we perform an uncertainty anal-

ysis on the results obtained in a typical Moir�e ex-

periment. These refer to the measurement of the

in-plane relative displacements undergone by a

sample of aluminum sheet metal subject to a uni-

axial tensile test. The measurements were carriedout under repeatability conditions at four load

stages selected within the elastic region of the test.

At each of these, the phase-difference fields in the

pulling direction were measured 60 times and the

corresponding standard deviation fields were

computed.

We found that the standard deviation fields

presented oscillations that were closely correlatedwith the corresponding fringe patterns, and that

their spatial average increased linearly with the

total elongation of the sample. Internationally

accepted recommendations [17] were then used to

evaluate the uncertainty of systematic effects as-

sociated with the possible deviation of the various

parameters of the measurement model from their

nominal values. We found that the contributionof environmental effects to the displacement

uncertainty was relatively small. Therefore, we

conclude that, under our experimental conditions,

accurate displacement measurements could have

been carried out from just a single image at each

load stage.

Finally, an equation was found that represents

fairly well the behavior of the displacement un-certainties at all load stages. These varied mainly

in the pulling direction and they depended

strongly on the placement of the reference sys-

tem used to measure the displacements. Our

equation may be used to evaluate the uncertainty

of displacements measured under similar envi-

ronmental conditions and following the same

experimental procedure.

2. Theory

2.1. Moir�e interferometry [18]

In a typical Moir�e interferometer two mutually

coherent plane beams travelling in the y–z-planeimpinge from opposite sides upon a high-fre-

quency grating with lines oriented in the x-direc-tion. The grating divides each incident beam into a

number of diffracted beams that follow preferred

directions called diffraction orders. At diffraction

order m, the diffraction angle with respect to the

grating normal is given by

bm;0 ¼ arcsinðsin a� kf Þ; ð1Þ

where k is the laser wavelength, a is the angle of

incidence, f is the grating frequency and subscript

0 indicates the unstrained grating. The angles of

incidence are adjusted such that the diffracted

beams corresponding to order m ¼ �1 emergejointly and normal to the grating, that is, such that

b�1;0 ¼ 0�. Therefore, according to Eq. (1), if a

He–Ne laser is used (k ¼ 632:8 nm) together with a

grid of f ¼ 1200 lines/mm, both angles of inci-

dence must be equal to 49.4�.The interference of the diffracted beams pro-

duces a null field, that is, a pattern of uniform in-

tensity. But if the grating undergoes a small localstrain e in the y-direction, its frequency changes to

f ð1þ eÞ�1. Therefore, the diffracted beams will

emerge with a small angle 2b�1;e between them,

where

b�1;e ¼ arcsin sin a

�� kg1þ e

�: ð2Þ

The interference of the diffracted beams produces

a pattern of Moir�e fringes. These are labeled with

consecutive numbers N called fringe orders, start-

ing from an arbitrary reference fringe. Their fre-quency is given by

R.R. Cordero, I. Lira / Optics Communications 237 (2004) 25–36 27

dNdy

¼2 sin b�1;e

k: ð3Þ

Combining Eqs. (1)–(3) we obtain

dNdy

¼ 2sin b�1;0

k

�þ f e1þ e

�: ð4Þ

Now, if V is the local value of the relative dis-

placement in the y-direction, we substitute thestrain e ¼ dV =dy in Eq. (4) and integrate. As-

suming e � 1, we obtain

V ¼ 1

2fNð � N0Þ; ð5Þ

where

N0 ¼2

k

Z y

0

sin b�1;0 dy ð6Þ

is the fringe order at position y in the pattern

generated by the unstrained grating. This patternis in general not null, because the grating may not

be perfectly flat.

Between two consecutive fringes the displace-

ment difference is 1=ð2f Þ. Thus, if a grid of

f ¼ 1200 lines/mm is used, the sensitivity is 417

nm/fringe.

2.2. Phase shifting and unwrapping

In general, the intensity I of an interference

fringe pattern is described by

I ¼ Aþ B cosD/; ð7Þwhere A and B are functions of position and

D/ ¼ 2pN is the local phase difference between theinterfering wavefronts. This phase cannot be es-

tablished by just measuring the intensity I , becauseof the sign ambiguity of the cosine, and also be-

cause the terms A and B are normally unknown. A

combination of temporal phase shifting and spa-

tial phase unwrapping algorithms removes this

difficulty.

In the former technique, a CCD camera is usedto capture the diffracted light and to measure

Iðx; yÞ. At least four fringe patterns are generated

at different times by stepping the phases of the

interfering beams in known amounts while the

object is kept nominally unchanged. Several com-

binations are possible. In the conventional four-

frame algorithm [19], phase differences D/B ¼ p=2,D/C ¼ p and D/D ¼ 3p=2 are induced sequentially

between the beams relative to the first pattern

generated with a phase difference D/A. The wrap-

ped phase difference is then obtained as

D/w ¼ arctanID � IBIC � IA

� �; ð8Þ

where IA through ID denote the corresponding in-

tensities.

By convention, the arc-tangent returns a prin-

cipal value in the range (�p; p). The process of

adding to this value the correct integral multiple of

2p is called spatial phase unwrapping, for which

different algorithms exist [20]. Conventionally, one

proceeds by calculating the difference between thewrapped phases D/w at adjacent pixels along some

chosen path. If this difference falls outside the range

(�p; p), it is assumed to be due to a 2p phase jump,

which is then added or subtracted to one of the

points in order to bring the difference between the

phases back into the correct range. In this way, an

unwrapped phase difference map, D/u, is obtained

with neither discontinuities nor sign ambiguities.

2.3. Measurement uncertainty [17,21]

In order to evaluate a measurement uncertainty,the measurand must be defined through an ap-

propriate measurement model. When the model is

linear or weakly nonlinear, it allows to express the

standard uncertainty of the output quantity in

terms of the standard uncertainties of the input

quantities by using the so-called law of propaga-

tion of uncertainties, LPU. It states that the

standard uncertainty uðZÞ of a quantity Z modeledas a function F of two input quantities X and Y is

obtained as the square root of

u2ðZÞ ¼ c2X u2ðX Þ þ c2Y u

2ðY Þ þ 2cX cY rXY uðX ÞuðY Þ;ð9Þ

where cX ¼ oF oX and cY ¼ oF oY are the sensi-

tivity coefficients, and rXY is the correlation coef-

ficient between X and Y . If the input quantities areindependent, the correlation coefficient is zero.

The generalization of this law to more than two

input quantities is straightforward.

Fig. 1. Optical setup. I, incident plane beams in the y–z-plane;D, diffracted beams; G, grating; S, specimen and P, pulling

direction.


The standard uncertainties of the input quan-

tities may be evaluated in several ways. To quan-

tities that are modeled in terms of further

quantities, the LPU is applied sequentially. For

quantities that are measured directly only once, or

that are imported from other sources, the standarduncertainty is taken as the standard deviation of

the probability density function (pdf) that is as-

sumed to apply to the quantity involved. For ex-

ample, if a quantity X is assumed to vary

uniformly within a given range of width dX , its

standard uncertainty is taken as

uðX Þ ¼ dXffiffiffiffiffi12

p : ð10Þ

Finally, for a quantity X that is measured several

times under repeatability conditions, the standard

uncertainty associated with the arithmetic average

X of the Xk measured values, with k ¼ 1; . . . ; n, isgiven by

uðX Þ ¼ sXffiffiffin

p ; ð11Þ

where

s2X ¼ 1

n� 1

Xn

k¼1

Xk

�� X

�2 ð12Þ

is the sample variance. If one of the Xks is given as

the measurement result, its uncertainty is just sX .

3. Experimental procedure

An Instron testing machine working in tension

was used to stretch an aluminum sheet metal

specimen. The experiment was carried out in an

optical metrology laboratory under constant

temperature (20 �C) and controlled conditions ofdust and air currents. The setup is depicted in

Fig. 1. A 50-mW He–Ne laser was used for illu-

mination. The beam was cleaned, expanded, col-

limated and divided by a beam splitter into two

beams that were directed by mirrors towards a

holographic commercial grating glued onto the

specimen surface. The sensitivity was along the

vertical pulling direction, indicated by coordinatey. The frequency of the grating was f ¼ 1200

lines/mm.

An 8-bit CCD camera with a resolution of

512� 512 pixels located directly in front of the

specimen captured the diffracted light. The ob-

served area, 12 mm per side, was located at the

geometrical center of the sample. The camera was

connected to a digitizer and frame grabber system

managed with a commercial software that wasused as well to perform the phase-shifting proce-

dure. This was carried out by moving a piezo-

electric device acting on one of the mirrors. We

used the conventional four-frame algorithm with

p=2 phase steps described in Section 2.2.

The prior setup adjustment included the mea-

surement of a phase-difference map from a regular

pattern of straight fringes. This pattern was gen-erated by simulating a rigid-body rotation of the

target through tilting slightly one of the mirrors.

This procedure allowed us to verify the absence of

appreciable systematic errors in the performance

of the phase shifting and unwrapping algorithms.

At zero load, a spurious pattern consisting of a

few fringes was observed. As mentioned in Section

2.1, this pattern was due to nonuniformities in theflatness of the specimen. To this initial field, de-

noted as stage 0, the phase-shifting process was

applied 60 times. Afterwards, application of load


gave rise to fringe patterns corresponding to the

in-plane vertical relative components of displace-

ment. At three selected load stages, denoted as 1

through 3, the testing machine was stopped in

order to perform again the phase-shifting process,

also 60 times each. Later, conventional phase un-wrapping as described in Section 2.2 was carried

out in order to produce a total of 240 unwrapped

phase maps.

Since fringe quality and contrast depend on

their frequency [22], and are affected as well by

surface roughness changes [23,24], we limited our

measurements to images containing a maximum of

approximately 50 fringes. With a sensitivity of 417nm/fringe, this was equivalent to relative dis-

placements smaller than about 21 lm. Moreover,

the fields recorded fell within the elastic region of

the material, yielding fringe patterns that were

reasonably steady during the approximately 8 min

it took to perform the temporal phase-shifting

process 60 times.

Force (kN)

0

1

2

1.5

1.0

0.5

0.1 0.2 0.3

Elongatio

Fig. 2. Force-elongation plot of an aluminum tensile test. Fringe

Fig. 3. Moir�e fringes at the load stages indicated in Fig. 2.

4. Results

Fig. 2 shows a plot of the test sequence; it was

constructed from the values of force and specimen

elongation indicated by the display devices of thetesting machine. Binarized fringe patterns associ-

ated to stages i ¼ 0; . . . ; 3 are shown in Fig. 3.

Each unwrapped phase map was obtained as a

gray level image with discrete pixel values between

0 and 255. In all these maps, some pixels where

erroneously assigned a null value. To find and

eliminate these unwrapping faults, we used an

algorithm (based on that described in [14]) thatreplaced the incorrect phase by the median value

of the local region formed by the adjacent pixels.

From symmetry, in a tensile test the center of

the sample should undergo no deformation.

Therefore, the displacements were measured rela-

tive to this location, where the origin of coordi-

nates was placed arbitrarily. The local

displacements are proportional to the fringe orders

3

0.4 0.5

n (mm)

patterns corresponding to stages 0–3 are shown in Fig. 3.

Stage 0 corresponds to the initial (spurious) pattern.


at each point (x; y). At stage i, these were obtainedas

Niðx; yÞ ¼ ai D/u;iðx; yÞ�

� D/u;ið0; 0Þ�; ð13Þ

where D/u;i is unwrapped phase difference and ai isthe conversion factor

ai ¼Nm;i

255

fringes

gray levelð14Þ

and Nm;i is the maximum fringe order occurring at

that stage. The software used to perform the

phase shifting did not give this number automat-

ically. We determined Nm;i by just counting the

number of bright fringes in each pattern. Results

were Nm;0 ¼ 1:6;Nm;1 ¼ 13;Nm;2 ¼ 24 and Nm;3 ¼51, see Fig. 3. Since all stages were within the

elastic region, the Nm;is were essentially propor-tional to the sample elongation and to the applied

force.

From Eq. (5), the displacements were then cal-

culated as

Viðx; yÞ ¼1

2fNiðx; yÞ½ � N0ðx; yÞ�; ð15Þ

where Ni and N0 were evaluated using Eq. (13).

Since the relative displacements were measured

with respect to stage 0 (for which we knew that the

displacements were null), Eq. (15) gave V0 ¼ 0.

The displacement fields for the four stages areshown in Fig. 4(a). These maps were obtained

from one of the 60 measured phase maps in the

corresponding set, chosen at random. As expected,

the displacements were almost independent of the

x-coordinate, normal to the pulling direction. In

Fig. 4. For each load stage: (a) relative displacement maps Vi in

Fig. 4(b) the displacements for x ¼ 0 have been

plotted. An almost linear increase with distance

from the origin is apparent. The maximum dis-

placement was about 10 lm in y ¼ 6 mm at stage

3, equivalent to a strain of about 1.6� 10�3.

Therefore, our assumption to derive Eq. (5),e � 1, was justified.

5. Uncertainty analysis

5.1. Phase dispersions

Ideally, the unwrapped and cleaned gray levelphase values D/u;i at any given pixel of the images

corresponding to the same stage i should have

been equal. However, the combined effect of en-

vironmental perturbations and optical noise in-

troduced dispersions that at each pixel appear to

have been random. As shown in Fig. 5, the his-

tograms of errors over line x ¼ 0 for the four

stages are nearly Gaussian. For each pixel in thisline, these errors were calculated as the gray level

D/u;i;kð0; yÞ at map k ¼ 1; . . . ; 60, minus the aver-

age D/u;ið0; yÞ obtained for the same pixel over the

60 maps acquired under stage i.At stage i, to characterize the phase dispersions

at each point, their standard deviations were cal-

culated as the square root of the variances

s2i ðx; yÞ ¼1

n� 1

Xn

k¼1

D/u;i;kðx; yÞh

� D/u;iðx; yÞi2;

ð16Þ

lm, (b) distribution of relative displacements over x ¼ 0.

Stage 0 Stage 1 Stage 2 Stage 3

0.5 0 01 1 −0.5−6 −3 3 1−1 −1 −1 −0.5−0.5 0.50.50 6 0

2000

6000

10000

5000

10000

15000

4000

8000

1200012000

8000

4000

(gray levels) (gray levels) (gray levels) (gray levels)

Fig. 5. Histograms of the gray levels errors for each load stage at all pixels over x ¼ 0. Note that the first plot has different histogram

categories.


where n ¼ 60. The units of these standard devia-

tions are gray levels.

We found that the si fields presented spatialoscillations of fairly uniform amplitude that were

closely correlated with the fringe orders, that is, we

observed coincidence between the bright and dark

fringe positions and the minimums of si. In other

words, si showed spatial fluctuations at twice the

fringe frequency. This correlation can be explained

by the sinusoidal character of the intensity I in Eq.

(7). Small changes in the phase differences pro-duced by external vibrations generated non-linear

differential fringe displacements. Thus, at the po-

sitions of the intensity maximums and minimums,

where its gradient was zero, the phase changes

induced by the vibrations were smaller than those

generated at other points.

Fig. 6. For the initial pattern (stage 0): (a) wrapped phase map; (b)

x ¼ 6 mm.

For example, the image in Fig. 6(a) shows one

of the wrapped phase maps corresponding to the

initial pattern (stage 0). The gray level disconti-nuities in this image occur at the positions of the

dark fringes. A plot of s0 over line x ¼ 6 mm, at

the right border of the image, is shown next to it,

Fig. 6(b). The correlation mentioned above is ap-

parent. The complete s0 map for this stage and its

corresponding contour plot are shown in Fig. 7.

From the similarity between Fig. 6(a) and

Fig. 7(b), we conclude that the effect of vibrationwas a systematic error in the response of the phase-

shifting algorithm at twice the fringe frequency.

This type of spatial oscillations has also been no-

ticed in computer simulation experiments [11].

The correlation effect was observed also in the

patterns associated to the higher load stages, 1–3,

distribution of the standard deviation s0 (in gray levels) over

Fig. 7. (a) Standard deviation map for load stage 0 and (b) its corresponding contour plot.


but with greater difficulties. In these cases, to ap-preciate the correlation it was necessary to per-

form low-pass filtering on the si fields in order to

remove high frequency variations, probably linked

to optical noise. Thus, we conclude that the effect

of noise increased with the number of fringes or, in

other words, with the sample elongation.

Because of the high signal-to-noise ratio and

high spatial resolution of Moir�e interferometry,fringe patterns displaying high contrast and ex-

cellent visibility are obtained. By contrast, speckle-

based methods, such as electronic speckle pattern

interferometry (ESPI), yield noisier patterns.

Therefore, we speculate that in the latter methods

it would be difficult to detect spatial oscillations in

the unwrapped phase standard deviations. For the

same reason, the relative influence of vibrations on

i = 3 i = 2 i = 1

Nm,i (fringes)

i = 0

60 50 10 40 30 20

⟨si ⟨(gray level)

2.0

1.5

1.0

0.5

(a)

Fig. 8. At the four load stages, average standard deviation of: (a) phas

maximum fringe order Nm;i observed at each load stage.

the uncertainty of displacements measured byESPI should be smaller than in the case of PSMI.

We note also that the histograms in Fig. 5 in-

dicate that the dispersion of errors decreased as the

tensile load was increased. This effect can be

shown as well by calculating the spatial averages

hsii of the standard deviation fields corresponding

to each stage, and plotting these averages as a

function of the corresponding maximum fringeorders Nm;i. Results are shown in Fig. 8(a). The

decreasing trend of hsii with Nm;i can be explained

by considering the number of pixels between frin-

ges. In patterns with fewer fringes, the quantity of

pixels available to detect fringe shifts caused by

external perturbations was greater than in those

with a higher number of fringes. Therefore, the

former patterns, associated to low load stages,

i = 2

i = 1

i = 0

i = 3

0.02

0.04

0.06

Nm,i (fringes)

ai⟨si ⟨(fringes)

60 50 10 40 30 20

(b)

e difference and (b) fringe order. The independent variable is the


were highly sensitive to environmental influences.

In fact, at stages 0 and 1 it was even possible to

observe small fringe movements on the computer

screen with the naked eye. On the contrary, at

stages 2 and 3 the fringes appeared to be rather

steady.However, the situation changes drastically if the

standard deviation averages hsii are expressed in

terms of the fringe orders through multiplication

by the conversion factors ai. Since these factors areproportional to the total number of fringes at each

stage, a linearly increasing trend in aihsii with in-

creasing Nm;i, or equivalently, with increasing

elongation, was obtained, see Fig. 8(b). The slopeof this trend was very small, however, giving a

maximum average standard deviation of only

about 0.06 fringes for the 60 patterns of stage 3.

This means that, under our experimental condi-

tions, the variations induced by external pertur-

bations on the fringe order were small. Therefore,

we conclude that any one of the 60 phase or dis-

placement maps at a given stage, such as the onesdepicted in Fig. 4, was a good representation of the

corresponding ‘‘true’’ map.

5.2. Fringe order standard uncertainty

To evaluate the standard uncertainty uðNiÞ of

the fringe orders at each stage, we applied the LPU

to a model based on Eq. (13), but now including aquantity q, with estimated value zero, to account

for gray level quantization:

Niðx; yÞ ¼ ai D/u;iðx; yÞ�

� D/u;ið0; 0Þ þ q: ð17Þ

In this model all input quantities were assumed

to be independent.

The estimated values of the conversion factors

ai were evaluated from Eq. (14). The application of

the LPU to this equation gives

uðaiÞ ¼uðNm;iÞ255

fringes

gray level; ð18Þ

where the standard uncertainty uðNm;iÞ was evalu-ated using Eq. (10) and assuming a maximum

counting error of 1%. Then

uðNm;iÞ ¼2 0:01Nm;ið Þffiffiffiffiffi

12p fringes: ð19Þ

The estimated value of the unwrapped phase

D/u;i at each point (x; y) of the field, including the

origin (0,0), was the gray level value obtained at

that point after performing the unwrapping. As

discussed in Section 5.1, the standard uncertainties

of D/u;iðx; yÞ were mainly caused by environmentalvibrations and optical noise. We took these un-

certainties to be equal to the corresponding values

of the standard deviations siðx; yÞ. Division by the

square root of 60 was not carried out since, in

PSMI displacement measurements, normally just

one phase map is obtained at each stage. There-

fore, we decided to evaluate the uncertainties

corresponding to one of the measured displace-ment maps at each stage, chosen at random. See

the last paragraph of Section 2.3.

Finally, for the standard uncertainty of the

quantization correction q we used a uniform pdf of

width equal to 1 gray level. Thus

uðqÞ ¼ 1ffiffiffiffiffi12

p gray levels: ð20Þ

Since the displacements were mostly indepen-

dent of the x-coordinate, the same was true also

for the standard uncertainties of the fringe orders.

Fig. 9(a) depicts the standard uncertainty map of

uðN3Þ for load stage 3. Fig. 9(b) shows the same

values along line x ¼ 0. Disregarding the high

frequency variations, probably due to optical

noise, it may be seen that the uncertainty increasedwith the distance from the origin. Similar behavior

was obtained for the uncertainties uðNiÞ at the

other two load stages.

Fig. 10 depicts the different contributions to the

square of uðN3Þ over x ¼ 0 as a function of the

square of N3. For clarity, these contributions have

been labeled as CðX Þ � cXuðX Þ for generic quan-

tity X , where cX is the corresponding sensitivitycoefficient. It may be seen that u2ðN3Þ was almost

proportional to N 23 , and that the most important

contributor was the conversion factor a3. Similar

plots were obtained for the other stages.

5.3. Displacements standard uncertainty

To evaluate the standard uncertainties of the

displacements Vi at each stage we used the LPU

based on model (15). The input quantities are the

Fig. 11. Contributions to the square of uðV2Þ over x ¼ 0 as a

function of the square of V2. Line 1: C2ðf Þ, line 2:

C2ðf Þ þ C2½N0ð0; yÞ�, line 3: C2ðf Þ þ C2½N0ð0; yÞ� þ C2½N2ð0; yÞ�.

Fig. 9. For load stage 3: (a) standard uncertainty map of fringe order uðN3Þ, (b) standard uncertainty uðN3Þ over x ¼ 0.

Fig. 10. Contributions to the square of uðN3Þ over x ¼ 0 as a

function of the square of N3. Line 1: C2ða3Þ, line 2:

C2ða3Þ þ C2ðqÞ, line 3: C2ða3Þ þ C2ðqÞ þ C2½D/u;3ð0; 0Þ�, line 4:

C2ða3Þ þ C2ðqÞ þ C2½D/u;3ð0; 0Þ� þ C2½D/u;3ð0; yÞ�.


fringe orders Ni and the grating frequency f . Thesequantities were assumed to be independent. The

standard uncertainties of Ni were evaluated as ex-

plained in Section 5.2. For f we used the manu-

facturer’s datum (1.2 lines/lm� 0.5%) and we

considered also the eventual damage that the grid

may have undergone during its gluing, the possi-

bility of some failure in its capacity to follow thespecimen deformation, and eventual errors in the

collimation of the incident beams. The latter

caused deviations in the sensitivity vector from its

estimated value. A uniform pdf of width

1.5� 0.012 lines/lm seemed reasonable, where the

1.5 is a ‘‘safety factor’’. Thus, we took

uðf Þ ¼ 18ffiffiffiffiffi12

p lines

mm: ð21Þ

Fig. 11 depicts the different contributions to the

square of uðV2Þ over x ¼ 0 as a function of the

square of the displacement V2. Again, these con-tributions have been labeled as CðX Þ for generic

quantity X . It may be seen that the contribution of

the N0 fringes was relatively small. Actually, this

influence decreased with the number of fringes in

the patterns, such that at stage 3 it was almost

imperceptible. For all stages, the most important

contributor to the displacement standard uncer-

tainties were the corresponding fringe orders Ni.Fig. 12 compares the displacement standard un-

certainties at stages 1 through 3 over x ¼ 0. It may

be seen that these uncertainties increased with the

stage (or equivalently, with the number of fringes in

the patterns) and that at each stage, the smallest

Fig. 12. Squares of uðViÞ over x ¼ 0 as a function of the squares of Vi at load stages i ¼ 1; 2; 3.


values were obtained at positions close to the origin,

for which Vi ¼ 0. In other words, the displacement

uncertainty was strongly dependent on the location

of the reference system. Furthermore, disregarding

the high frequency oscillations caused by optical

noise (that can be minimized by low-pass filtering),it may be seen that, at each stage, the squares of the

uncertainties increased almost linearly with the

squares of the corresponding displacements.

Straight line fitting to the plots of Fig. 12 pro-

duced, for i ¼ 1; 2; 3,

uðViÞ2 ¼ aþ bV 2i ; ð22Þ

where

a ¼ ð0:431� 10�3 � 25:4� 10�6Nm;i

þ 1:18� 10�6N 2m;iÞ lm2 ð23Þ

and b ¼ 51� 10�6 is the common slope.Eq. (22) represents fairly well the behavior of

the standard uncertainty of the displacements and

can therefore be taken as our final result. This

equation allows to estimate the standard uncer-

tainties of displacements measured from other

patterns, obtained under similar environmental

conditions and following the same experimental

procedure, by just counting the maximum fringeorder observed.

6. Summary and conclusions

An investigation was carried out to evaluate the

uncertainty of in-plane displacements measured

with phase-shifting Moir�e interferometry (PSMI).

The experiment was a uniaxial tensile test of an

aluminum sheet metal sample. Measurements were

performed 60 times under repeatability conditions

at four load stages selected within the elastic re-

gion of the test.

At each pixel, environmental perturbations andoptical noise produced nearly Gaussian disper-

sions in the unwrapped phase differences. These

dispersions were characterized by their standard

deviations. It was found that these standard devi-

ations presented minimum values at the positions

of the dark and bright fringes. From this fact, we

were able to conclude that the effect of vibration

was a systematic error in the response of the phase-shifting algorithm at twice the fringe frequency.

This effect was easier to visualize at low load

stages. At stages with greater number of fringes the

standard deviations of the unwrapped phase dif-

ferences showed high frequency variations, prob-

ably linked with optical noise. Therefore, we

conclude that the influence of noise increased with

the number of fringes.To compare the dispersions in the unwrapped

phase differences at the different stages, the spatial

averages of the standard deviation maps were

calculated. It was found that these averages in-

creased almost linearly with the total elongation of

the sample. However, it was found also that the

variability in the phase difference data, induced by

vibrations or optical noise, was quite small.Therefore, we conclude that under our experi-

mental conditions, a single unwrapped phase map

at each stage was sufficient to obtain a reliable

picture of the corresponding ‘‘true’’ displacement

map.


An uncertainty analysis of the measured dis-

placement maps at each stage was also carried out.

The effects of the possible deviations of the input

quantities with respect to their nominal values

were considered. It was found that, at each stage,

the squares of the displacement standard uncer-tainties increased almost linearly with the squares

of the sample elongation. Therefore, we conclude

that the uncertainty was dependent on the place-

ment of the reference system used to measure the

displacements.

Finally, disregarding high frequency variations,

we compared the standard uncertainties of our

whole-field displacement measurements, at thedifferent load stages. We found an equation that

allows estimating the standard uncertainties of the

displacements measured from any fringe pattern,

by just counting its maximum fringe order. This

equation should be applicable to other patterns

obtained under similar environmental conditions

and following the same experimental procedure.

Acknowledgements

The experiment described in this paper was

performed whilst the first author was on a visit to

the Laboratoire des Syst�emes M�ecaniques et

d’Ing�enierie Simultan�ee (LASMIS) of the Univer-

sit�e de Technologie de Troyes (France) under thedirection of Dr. Manuel Franc�ois. The visit was

made possible through the ECOS/CONICYT

(Chile) C01E04 research grant. We thank Dr.

Ram�on Rodr�ıguez-Vera, from Centro de Investi-

gaciones en Optica (CIO, M�exico), and the referee,

for helpful suggestions to the final draft of this

paper. The supports of Fondecyt (Chile) 1030399

research grant and of MECESUP PUC/9903 pro-ject are also gratefully acknowledged.

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uncertainty analysis of displacements measured by phase-shifting moiré interferometry

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