Phase extraction from electronic speckle pattern interferometry addition fringes

Phase extraction fromelectronic speckle pattern interferometryaddition fringes

Andrew J. Moore, John R. Tyrer, and Fernando Mendoza Santoyo

Addition fringes are obtained in real time from electronic speckle pattern interferometry (ESPI) by use ofa twin-pulsed laser when two pulses are fired during a single field of a CCD camera. This enables objectdeformations to be studied in harsh environmental conditions. However, the fringe patterns have poorvisibility because optical noise is additive. To our knowledge automatic phase extraction from additionfringes has not previously been achieved: Low-pass filtering to suppress random speckle noise alsoeliminates the fringes because of their low visibility. Two phase-stepping algorithms that calculatephase from ESPI fringes without the need for a preprocessing filter are presented. In the first ESPIsubtraction fringes are considered, for which an improvement in accuracy is seen, and in the second ESPIaddition fringes are considered, which, we believe, has enabled the phase to be extracted for the firsttime. The algorithms are demonstrated with theoretical data and with experimental ESPI fringepatterns recorded with a cw laser. As presented, they form the first step toward a procedure that can beused with twin-pulsed ESPI.

Key words: Electronic speckle pattern interferometry, subtraction ESPI, addition ESPI, phasestepping.

1. IntroductionElectronic speckle pattern interferometry (ESPI) is alaser-based technique that measures static or dy-namic deformations.' Depending on the optical con-figuration, components of out-of-plane or in-planedeformation may be measured independently and aredisplayed as fringes that represent the loci of theconstant optical path difference in the interferometer.We generated the fringe pattern by correlating twospeckle patterns that are recorded by a TV camera inthe coherent interferometer. We form the firstspeckle pattern by imaging the test object in itsreference state, the second after the object has de-formed. For cw lasers the speckle patterns aregenerally correlated by subtraction. This subtrac-tion removes stationary optical noise, and good visibil-

A. J. Moore and F. Mendoza Santoyo are with the Centro deInvestigaciones en Optica A.C., Apartado Postal I-948, Leon, Gto.37000, Mexico. J. R. Tyrer is with the Department of MechanicalEngineering, Loughborough University of Technology, Loughbor-ough, Leicestershire LEIl 3TU, England.

Received 21 January 1993; revised manuscript received 18March 1994.

0003.6935/94/317312090600/0.c 1994 Optical Society of America.

ity fringes are obtained. For objects undergoingharmonic vibration at a frequency greater than theTV frame rate (25 Hz), the deformation is averagedover each TV frame. The fringe pattern has azero-order Bessel function Jo intensity profile andrepresents contours of constant vibrational amplitude.The rapid dropoff in amplitude and the irregularspacing of zero values of JO make automated fringe-pattern analysis more complicated than for fringeswith a cosinusoidal intensity profile,2 such as thoseobtained from a statically deformed object.

Pulsed lasers have been introduced for harmonicand nonharmonic vibration studies with ESPI.3

This is particularly useful when environmental condi-tions are such that coherent interferometric effectsare destroyed for cw illumination. The object isilluminated at two discrete points in its vibrationcycle, which is equivalent to a static displacementbetween the two laser pulses, and the fringe patternhas a cosinusoidal intensity profile that is moreamenable to automated fringe-pattern analysis.Cookson et al.3 demonstrated correlation of the twospeckle patterns both by subtraction and by addition.The system incorporated a pulsed ruby laser with a10-s pulse-repetition rate. Sophisticated electronicswere necessary to synchronize the laser pulses withthe blanking and scan periods of a vidicon TV camera.

7312 APPLIED OPTICS / Vol. 33, No. 31 / 1 November 1994

VARIABLE SYNC. VARIABLEFREQUENCY UYNIT PULSE

ENRTOR GENERATOR

Fig. 1. Experimental layout for pulsed in-plane phase-steppedESPI. PZT, piezoelectric transducer.

The advent of lasers with high repetition rates used inconjunction with CCD cameras reduces the aboveproblem and has renewed interest in pulsed ESPI.4Consider the in-plane sensitive system of Fig. 1. Atwin-cavity, diode-seeded Nd:YAG laser is used. Inthe single-pulse mode, one cavity fires single pulseswith a repetition rate as high as 50 Hz. In thetwin-pulse mode a train of coherent twin pulses, onepulse from each cavity with any time separation, isfired with a repetition rate as high as 50 Hz.4 Thepulses are synchronized to the object vibrations,enabling the first pulse to be fired at the same pointon the vibration cycle. For subtraction fringes,single-pulse operation is used: The first pulse isfired at any point in the charge integration time of aframe-transfer CCD camera, and the second pulse isfired at some other point in the vibration cycle duringthe following charge integration period. Subtract-ing the two images yields fringes of high visibility.In the addition mode, two twin pulses are fired withina single TV frame scan. The addition of the specklepatterns takes place on the CCD sensor. The addi-tion fringe pattern is noisier than for subtractionbecause stationary optical noise is reinforced. How-ever, the small time separation between pulses en-ables correlation fringes to be obtained in extremelyharsh environmental conditions. A method for sub-tracting the two images associated with double pulseshas recently been proposed.5 The two pulses arefired on either side of the transfer gate of an interline-transfer CCD, and the speckle patterns associatedwith each are read out as two consecutive TV fields.Subtraction of the two fields yields a high-visibility,double-pulsed subtraction ESPI fringe pattern. Im-munity to harsh environmental conditions is main-tained because of the short image transfer period ofthe interline-transfer CCD, typically 1 Ls, comparedwith typically 300 [Ls for a frame-transfer CCD.Thus the penalty for improved fringe visibility is aCCD-imposed restriction on the minimum pulse sepa-ration. It should be stressed that in this paper we

are concerned with twin-pulsed addition ESPI fringes,recorded with a frame transfer CCD, for which thepulse separation may be reduced without limit.

The optical phase difference introduced because ofsurface deformation is indirectly detected by theformation of correlation fringes, and the sign ofdeformation (i.e., in the positive or negative x direc-tion in Fig. 1) is lost. Techniques to determine thephase difference have therefore proved useful forautomated analysis of interferograms.6 Phase step-ping is a straightforward and accurate technique andis well suited to full-field ESPI measurements.7 -9The phase difference between the beams of a classicalinterferometer varies smoothly over the detectorarray. However, for ESPI the phase varies ran-domly because of the random phase of individualspeckles. Two methods for suppressing randomspeckle phase have been proposed in the literature.In the first method, random speckle noise is removedfrom the correlation fringes by application of a suit-able filter. Phase is then calculated from the filteredfringes. For the purposes of this paper, this methodis termed phase of difference and is described brieflyin Subsection L.A. The second method to removerandom speckle noise is described in Subsection 1.Band has been termed difference of phase. The phaseat each pixel in the TV image is calculated fromphase-stepped speckle patterns that are recordedbefore and after deformation.8 9 The correlationfringes are not formed.

A. Phase of Difference

One common phase-stepping algorithm requires threeimages to be recorded: Between each recording thephase is stepped by 0, 21T/3, and 4rr/3 rad, respec-tively.9 The intensity at each (x, y) pixel position isdenoted I(x, y), I2(x, y) and I3(x, y) for each of thethree images. The optical phase difference Ai4(x, y)introduced between the beams from surface deforma-tion is given by

4( X A= tan-' (2I - 2)Ai~(, y = an'(2I, - I2 -I) (la)

Another common algorithm involves recording fourimages with 0, r/2-, 'r-, and 37r/2-rad phase shifts.7Four intensity values, I(x, y), I2(x, y), I3(x, y), and14(x, y), are combined at each pixel to give the deforma-tion phase by

Ai(x,y) = tan-'I2 1 - (lb)

By the phase-of-difference method,7 a referencespeckle pattern is digitized, for which the intensity atany (x, y) pixel may be approximated by

Iref(X, y) = Ir + I + 2(IrIo)1/2 cos(.). (2)

Ir(x, y) is the reference beam intensity, I0(x, y) is theobject beam intensity, and +j(x, y) is a composite phaseangle between the two beams that incorporates the

1 November 1994 / Vol. 33, No. 31 / APPLIED OPTICS 7313

random speckle phase term. After the object de-forms, phase-stepped speckle patterns are recorded.Considering Eq. (la), we record three images forwhich the interference expression is

Idef,k+1(Xy) = Ir + I. + 2(IrIo)1/22k,r

X Cos ( + At, + AOk + 3

k = 0,1, 2. (3)

For cw ESPI, Iref represents the intensity recordedwhen the object is in its reference state and Idef is theintensity for any subsequent deformation. For har-monic vibration analysis with pulsed ESPI, 'ref de-notes the intensity recorded for the first pulse and Idefthe intensity for the second pulse; Ati represents theharmonically induced deformation between record-ings. AOk is a phase term describing, for example,rigid-body motion and may vary between phase steps.Correlation fringes will be seen only if AO is small.In many cases AO may be controlled, and for thesubsequent analysis it is assumed that AO = 0.Phase-stepped subtraction fringes are generated by

Ik+l(XY) = Idefk+l - Iref

= 2IM sin(24 + A+ 3k)

x sin 205 + 3 )|'(4)

where IM = 2(IrIo)1/2. The modulus is taken tosuppress negative values (i.e., full-wave rectification,which is equivalent to subtraction with exclusive ORlogic in a dedicated external framestore). It is gener-ally quicker to digitize correlation fringes producedby an external framestore rather than to recordspeckle patterns. The interferometer can be ad-justed to optimize fringe visibility, and three ratherthan four images are recorded. This is the tech-nique we use for both out-of-plane and in-planeESPI.1011 Figures 2(a)-2(c) show fringes repre-sented by Eq. (4) recorded with an in-plane sensitiveinterferometer. Continuous-wave argon-ion illumi-nation was used in an experimental layout similar tothat in Fig. 1. The object studied was a flat 80 mm x80 mm plate that was rotated in its own plane aboutthe viewing axis. Assuming that AW varies slowlywith respect to A, in Eq. (4) we describe two functionsmodulated by each other, the first at a higher spatialfrequency than the second. Hence a low-pass filtermay be applied to reduce the high-spatial-frequencyspeckle noise and Eq. (la) is applied to the filteredfringes. Figure 2(d) shows this result after we ap-plied three iterations of a 7 x 7 pixel local neighbor-hood average filter to the speckle fringes of Figs.2(a)-2(c). Phase values in the 0 < At4 < 2iT rangehave been linearly scaled to 256 gray levels fordisplay. Discontinuities in the gray value occur as aresult of the asymptotic nature of the arctangent

(a)

I'

(b)

a Tr~~~~~~~~i~~~~~i~Z -x,.

(c) (d)Fig. 2. Phase-of-difference method. Subtraction fringes:(b) 27r/3, (c) 47r/3, (d) phase map.

(a) O,

function. These discontinuities are removed, whichis termed phase unwrapping, 7-9 to give the displace-ment unambiguously across the test surface. Notethe mottled nature of the phase map in Fig. 2(d)caused by the low-spatial-frequency speckle noise stillpresent in the filtered images.

B. Difference of PhaseWith the difference-of-phase approach8 9 intensityfringes are not formed. The process involves speckle-phase correlation rather than speckle-intensity corre-lation. Three phase-stepped reference speckle pat-terns are recorded, which may be represented by

Irefk+1(X, y) = Ir + I. + 2(IrIo)1/2 cos( 3 + 2 )

k = 0,1, 2. (5)

This equation corresponds to Eq. (2) incorporating aphase-step term. We may calculate the randomspeckle phase kf at each pixel using Eq. (la) [Fig. 3(a)].A further three phase-stepped images are recordedafter deformation [Eq. (3)], and a second phase map iscalculated [Fig. 3(b)]. This phase map gives therandom speckle phase plus the object deformation,+ + A+. Thus subtracting Fig. 3(a) from Fig. 3(b),maintaining the result modulo 2 Tr, yields the phaseresulting from object deformation alone [Fig. 3(c)].Each pixel has been used as an individual interferom-eter by this method, and the phase data are notmottled by the low-frequency speckle noise thatremains in the filtered fringes. However, phasecannot be calculated accurately at saturated pixels orthose that show zero intensity in any of the six


(a) (b)

(c)Fig. 3. Difference-of-phase method: (a) reference phase map, (b)deformed phase map, (c) difference between (b) and (a), modulo 2rr.

recorded speckle patterns. Thus the phase data arespeckled with pixels of invalid phase. These invalidpixels can be marked and excluded when phaseunwrapping takes place.12 Note that some noise isstill present in the phase data, partially because ofspeckle decorrelation. In general the difference ofphase is considered to give more accurate results thanthe phase of difference, because data from saturatedpixels or those with zero intensity are not filteredback into the image. Typically an accuracy of 27r/30rad is quoted for the difference of the difference ofphase9 compared with 2r/ 15 rad for the phase ofdifference.7 However, as shown in Fig. 3(c), thedifference of phase often requires the phase data to besmoothed with a special filter. Also the techniquerequires more time for data acquisition and phasecalculation and a larger image memory than thephase-of-difference technique. The phase-of-differ-ence approach calculates phase from correlationfringes. For subtraction fringes this approach en-ables a dedicated framestore to be used and theinterferometer to be optimized before the images arerecorded. For addition fringes, this solution is theonly one proposed so far. The constituent specklepatterns cannot be separated to apply the differenceof phase method.

2. Phase Extraction from Subtraction Fringes withoutLow-Pass FilteringThe advantages of digitizing fringes rather thanrecording speckle patterns were discussed in Subsec-tion 1.B. They include fewer images and less compu-tation to extract the deformation phase. Indeed foraddition ESPI only correlation fringes can be digitized.The disadvantages of low-pass filtering to suppress

speckle noise were also discussed: reduced accuracyfor subtraction fringes and failure with additionfringes. These considerations raise an interestingquestion: Does a solution exist for correlation fringesthat enables each pixel in the image to be treated asan individual interferometer? Such a solution wouldbe applied directly to each pixel of the correlationfringe pattern without the need for a preprocessingfilter. This solution would offer improved accuracyfor subtraction fringes and we believe provide theability to phase step addition fringes for the first time.Subtraction fringes are considered in this section andaddition fringes in Section 3.

Modifying Eq. (4) to include a phase-step of arbi-trary size gives

Ik+l(XY A= IM Cosc1 + Aqi + )- cos( )

k = 0, 1, .. .,m-1. (6a)

Considering the terms in k+1 2 to remove the modu-lus, we found a three-step solution (m = 4). Todemonstrate the technique, four images correspond-ing to Eq. (6a) were generated. They are analogousto the fringes of Figs. 2(a)-2(c), although this time thefringes move through one-fourth of their width.Figure 4(a) shows one fringe pattern for k = 0. Ajvaries linearly with horizontal position x in theimage, i.e., ip(x) = 2rnx/256 for a 256 x 240 pixelimage, where n = 2 represents the number of opticalwaves. (x, y) and IM are randomly generated in therange of -ir < < r rad and 0 < IM < 127.5 graylevels. The terms in 1k+12 are

I12 = IM2 COS 2 + - 2IM2cos cos( + A)

+ IM2 COS 2(,} + All),

I22 = IM2 COS2 + 2IM2 cos l sin(l + Al)

+ IM2 sin2(l, + Al,),

I32 = IM2 cos 2 + 2 IM2 cos l cos( + A+,)

+ IM2 COS2((l + Alj),

I42 = IM2 cos 2 l, - 2Im2 cos l, sin(l,

+ IM2 sin2 (l, + A,).

+ A+,)

It is easiest to consider the solution in two stages.First, the combined random speckle and displace-ment phase l, + A are calculated from

+,(x, y) + A(x, y) = tan 12 - I42 (7)

Clearly this is Eq. (lb) with terms in Ik+ 12 rather than

I+1. The random speckle phase must be subtractedat each pixel. To do this, a and b are defined asfollows:

a = I2 + I22 + I32 + I42 = 2IM2(2 COS2 l + 1),

b = [(32 - J12)2 + (22 - I42)2]1/2 = 4 IM2 cos .


(a) (c)

(b) (d)

Fig. 4. Solution for theoretical subtraction fringes: (a) subtraction fringes, k = 0; (b) comparison phase map; (c) selected phase values; (d)removal of marked pixels from (c).

We may calculate the random speckle phase by solv-ing the quadratic in cos l,, i.e.,

a (a2 - 2b2)1/2

k(xy) = cos 2b(8)

From Eq. (8) a total of four solutions for l, exists ateach pixel. To select the correct solution, we com-pared the four solutions with the phase value calcu-lated by applying Eq. (7) to the four low-pass filteredfringe patterns represented in Fig. 4(a). This com-parison phase map is shown in Fig. 4(b). The near-est of the four solutions to the value from thecomparison phase map was selected as the correctsolution [Fig. 4(c)]. No phase was calculated atpoints where IM < 5 gray levels (5000 pixels).These points are represented by zero intensity in Fig.4(c) and by a unique phase value in the phase map.Phase calculated at these points will have reducedaccuracy. These marked pixels were replaced by theaverage of their nonmarked neighbors, and then a3 x 3 median filter was applied [Fig. 4(d)]. The rmsdifference between the theoretical phase distributionand the values represented in Figs. 4(b) and 4(d) are0.093 and 0.034 rad, respectively. (The phase errorassociated with 8-bit digitization for our system is0.004-rad rms.) Some error is introduced in Fig.4(d) by the procedure used to remove marked pixels.

If IM takes a constant value (127.5 gray levels) toeliminate marked pixels, the rms error falls to 0.019rad. The improvement in accuracy achieved withthe new technique is seen visually in the removal ofmottling between Figs. 4(b) and 4(d).

To demonstrate the technique experimentally, re-sults were recorded with an out-of-plane interferom-eter for a cantilever clamped at the left-hand side anddeformed toward the viewer [Fig. 5(a)]. Four phase-stepped subtraction fringes were generated by theframestore, and the phase was calculated as describedabove. To test that one of the four solutions at each

(a) (b)Fig. 5. Solution for experimental subtraction fringes: (a) se-lected phase values, (b) selected phase values with the alternativecomparison phase map.


pixel cannot be selected to agree with any comparisonphase map, the test shown in Fig. 5(b) was performed.The comparison phase map was calculated with theimages reversed, i.e., I,, 12, I3, and 14 represent phasesteps of 3r/2, r, iT/2, and 0 rad, respectively. Thephase profile does not vary smoothly between discon-tinuities as in Fig. 5(a), indicating that a uniquesolution can be found from the four values at eachpixel. Ultimately this solution will be selected with-out reference to a comparison phase map, as dis-cussed in Section 4.

3. Phase Extraction from Addition FringesFrom Eqs. (2) and (3) addition fringe intensity may beapproximated by

k+ 1 = Iref + defk+1

= 2IB + IM cos(+) + cos(+ + A4 + 2krr)

k =0, 1, . .. ,m - 1, (9)

where 2 B = 2(I, + Ir). Equation (9) was used togenerate addition fringes [Fig. 6(a)]. 2 B = 127.5gray levels, and IM = 63.75 gray levels. Figure 6(b)shows the result of applying a single iteration of a 7 x7 pixel neighborhood average filter to each of threesuch phased-stepped fringe patterns and calculatingthe phase with Eq. (la). The fringe pattern is re-moved by the filter, and phase information is lost,which has prevented phase stepping of addition fringesin the past. However, a two-step (m = 3) solutionfor Al, of a form presented in Section 2 can be used.First, Eq. 1(a) is applied to the unfiltered data tocalculate the random speckle phase plus the deforma-tion phase, l, + Al,. It then remains to eliminate therandom speckle phase, which is achieved in a waysimilar to that in Section 2, i.e.,

1l,(xy) = cos' I (I1 + I2 + I3 - 6 IB)

(10)-1 II + I2 + I3 - 6 IB[3(I2 - I3)2 + (2I - 12 - I3)2]1/2

The terms 2IB and IM cos(l,) cannot be isolated fromEq. (9). However, 2 B can be estimated if it isassumed that the statistical average of both cosineterms in Eq. (9) is zero. Thus for each image themean intensity in the (256/n) x (240/n) pixel neigh-borhood of each pixel was evaluated and substitutedinto Eq. (10). From Eq. (10) two solutions for l, existat each point. Once again the correct solution ateach pixel was chosen by reference to a comparisonphase map [Fig. 6(c)]. The comparison phase mapwas calculated by applying Eq. (la) to subtractionfringes, which were calculated with Eq. (4) with m =3. The fringes are the same as in Fig. 4(a) but stepthrough one-third of their width. Random specklenoise was removed with three iterations of a 7 x 7neighborhood average filter. Note that the mini-

mum intensity for the addition fringes is shifted by Trrad from the zero position for subtraction fringes.Thus a T rad offset was added to the comparisonphase map calculated from subtraction fringes. Thephase selected at each pixel is displayed in Fig. 6(d).Once again a smooth phase profile is seen comparedwith the mottled effect in Fig. 6(c). The comparisonphase map of Fig. 6(c) contains a rms phase error of0.1333 rad. This error is larger than that for thecomparison phase map of Fig. 4(c) because Eq. (la)with fewer images is more sensitive to residual specklenoise in the filtered fringes than Eq. (7). The rmsphase error for Fig. 6(d) is 0.0144 rad. Fringes of thetype shown in Fig. 6(a) demonstrate the techniquebut are unrealistic in that the background and modu-lation amplitudes are constant. Figure 6(e) showsaddition fringes calculated with 2 B and IM varyingrandomly in the 0 < 2IB < 127.5 and 0 < IM < 63.75gray-level range. The same comparison phase mapwas used [Fig. 6(c)] to select the phase values for Fig.6(f). The phase distribution now contains manymarked pixels (- 18,000) resulting from both low-intensity modulation IM < 5, and points at which nosolution to Eq. (10) exists. The rms error of un-marked pixels is 0.724 rad. Despite the high level ofnoise a considerable improvement is seen over Fig.4(b).

To demonstrate the technique on experimentaldata, addition fringes were calculated from the dataset in Subsection 1.A, which enabled the subtractionfringes of Figs. 2(a)-2(c) to be used to generate thecomparison phase map [Fig. 2(d)] shifted by r rad.The mean value Of Iref + Idefk4 was calculated for each kand linearly scaled to 127.5 gray levels to give theaddition fringes shown in Figs. 7(a)-7(c). Fringesmay be discerned but are clearer when switchingbetween images on the image processor. Note thataddition fringes from the pulsed Nd:YAG system alsophotograph poorly but are more clearly visible on theTV monitor when they are in motion. The phasevalue selected from the two values at each pixel isshown in Fig. 7(d). With 6 < n < 7 a 43 x 43 pixelwindow was used to calculate 2IB at each pixel. Theneed to estimate 2 B from local intensity valuesrather than the whole image is indicated by theillumination beam profile in Figs. 7(a)-7(c).

4. DiscussionBoth the algorithms presented require a correctsolution to be chosen from the values calculated ateach pixel. For subtraction this choice is one of fourvalues and for addition one of two values. A simpleselection procedure, based on a comparison phasemap, was introduced. For subtraction fringes, calcu-lating the comparison phase map requires no addi-tional information, and consequently the techniqueproposed in Section 2 uses fewer images and reducesthe time for data acquisition compared with thedifference in the phase method. The accuracy isimproved noticeably with regard to the phase-of-difference method. A rms phase error of 0.093 rad in


(a) (d)

(b) (e)

U

(c) (f)

Fig. 6. Solution for theoretical addition fringes: (a) addition fringes for IB andIM constant, k = 0; (b) phase from filtered addition fringes;

(c) comparison phase map; (d) selected phase values; (e) addition fringes for IB and IM random, k = 0; (f ) selected phase values.

the comparison phase map of Fig. 4(b) caused bymottling corresponds to the rms phase error intro-duced by a 16% linear error in the size of the appliedphase step substituted into Eq. (lb). This error wasreduced to 0.034 rad by the new technique, which isequivalent to reducing the corresponding phase-steperror to 6%. The procedure is, however, computa-tionally intensive when compared with both thedifference-of-phase and phase-of-difference methods.In total the algorithm requires 15 s to run on a33-MHz 486 IBM clone personal computer. Experi-mental data always contain phase errors caused byspeckle decorrelation and by points at which the

modulation amplitude is insufficient to calculate thephase with the desired accuracy [Fig. 5(a)]. Theseerrors are precisely those associated with the differ-ence of phase method [Fig. 3(c)], and hence an experi-mental accuracy of 27r/30 can be expected with thenew algorithm.

A comparison phase map cannot be calculated foraddition fringes generated in the twin-pulsed mode.At this stage the reference speckle pattern was re-corded independently to enable the comparison phasemap to be generated from filtered subtraction fringes.Ultimately some other procedure is required. Kujaw-inska et al. 1 3 proposed an algorithm to determine the


(a) (b)

(c) (d)Fig. 7. Solution for experimental addition fringes. Additionfringes: (a) 0, (b) 27r/3, (c) 4rr/3, (d) selected phase values.

phase from the four speckle patterns recorded for thephase-of-difference method, Subsection 1.A, withoutforming correlation fringes. Some similarities existbetween this method and the solution for additionfringes proposed in Section 3, in that two solutionsexist at each pixel. The solution that maintained thesmoothest Ai profile within a 3 x 3 pixel neighbor-hood was selected,13 which is analogous to the low-pass filter required with the phase-of-differencemethod, in that the displacement phase is assumed tovary slowly with respect to the random speckle phase.However, the filter is applied to the phase data ratherthan to the fringe function. The phase value that isselected has been calculated from the intensity modu-lation at a single pixel. Alternatively the correctsolution could be seeded from one or several points ofknown phase, measured independently by, for ex-ample, a laser vibrometer. It is envisaged that asuitable procedure will be implemented in due course.

For addition fringes the advantages of the methodproposed in Section 3 are self-evident when we com-pare Figs. 6(b) and 6(f) and have enabled phaseextraction from ESPI addition fringes for the firsttime, to our knowledge. The method is considerablyeasier than for subtraction fringes, because the cor-rect solution needs to be selected from only twopossible values. The advantage of addition opera-tion is the ability to take measurements in harshenvironments. In such conditions the assumptionthat environmental disturbances may be ignored [i.e.,setting A\O 0 in Eq. (4)] while recording threetemporally phase-stepped fringe patterns is no longervalid. An associated problem is how to translate amirror in one beam of the interferometer in the shorttime period between twin laser pulses. One solution

to these problems that we are currently investigatingis the spatial separation of the three phase-steppedinterferograms. All three interferograms are thenrecorded with one twin pulse, enabling the algorithmto operate on data recorded in harsh conditions.One possible solution is to insert a grating betweenthe test object and the CCD13 to present the camerasimultaneously with three phase-stepped additionfringe patterns.

5. ConclusionsTwo phase-stepping algorithms for ESPI correlationfringes have been presented. Both eliminate theneed for a preprocessing filter to be applied to theintensity fringes. For subtraction fringes the algo-rithm of Section 2 was shown to improve the accuracyof the measured phase by eliminating the mottlingassociated with the phase-of-difference method.Theoretical data showed a reduction in rms phaseerror from 0.093 to 0.034 rad, which suggests that anaccuracy of 2Tr/30 can be achieved experimentallywith this method (comparable with the difference-of-phase method) rather than 27r/15 usually attributedto the phase of difference with a low-pass filter. Thenew algorithm requires fewer images to be recordedthan the difference-of-phase method. The techniquewas demonstrated for theoretical and experimentalfringes, generated with full-wave rectification to re-move negative values. The algorithm is also validwhen negative values from the subtraction have beenremoved by squaring the result.

In Section 3 we demonstrated the calculation ofphase from the ESPI addition fringes, which to ourknowledge has not been achieved until now. Thereis no exact solution to separating the backgroundintensity term, and so a locally measured value wassubstituted into the solution. The technique wasdemonstrated for theoretical and experimental fringes.In the theoretical example a rms phase error of 0.724rad at unmarked pixels was found. Twin-pulsedaddition ESPI is primarily used to study transientdeformations and vibrations in harsh environmentalconditions. Temporal phase stepping, by which thealgorithm was demonstrated in this paper, will not besuitable in such conditions, and a method to separatethe phase-stepped images spatially is currently beinginvestigated. For both techniques the correct phasevalue was chosen from a comparison phase map thatwas calculated from subtraction fringes. This proce-dure cannot be used for addition fringes recordedwith the twin-pulsed system, and an alternativealgorithm to select the correct phase values must beinvestigated also.

This research was funded by the Science andEngineering Research Council, UK. The authorsthank David Kerr for helpful comments on the manu-script and Ken Topley and Vince Scouthern forpreparation of the photographs. They are all in theDepartment of Mechanical Engineering, Loughbor-ough University of Technology. Fernando Mendoza


Santoyo acknowledges financial support from Con-sejo Nacional de Ciencia y Tecnologia.

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Phase extraction from electronic speckle pattern interferometry addition fringes

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