Optics and Lasers in Engineering 45 (2007) 274–280

Advanced iterative algorithm for randomly phase-shifted interferogramswith intra- and inter-frame intensity variations

Zhaoyang Wanga,�, Bongtae Hanb

aDepartment of Mechanical Engineering, The Catholic University of America, Washington, DC 20064, USAbDepartment of Mechanical Engineering, University of Maryland, College Park, MD 20742, USA

Available online 29 March 2006

Abstract

The fundamental concept of the advanced iterative algorithm is reviewed and a detailed procedure for implementation is presented.

Computer simulations are conducted to evaluate its performance for truly random phase-shifted interferograms with intra- and inter-

frame intensity variations. The algorithm is implemented for random phase shifts encountered during thermal warpage measurement of a

microelectronics subassembly subjected to convective thermal loading.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Advanced iterative algorithm; Phase shifting technique; Random phase shifts; Intensity variations

1. Introduction

Phase shifting is one important technique in interfero-metry [1–3]. Conventional phase shifting algorithmsrequire the phase shift amounts to be known; however,errors of phase shifts are common for the phase shiftmodulators in real applications, and such errors canfurther cause substantial errors in the determinations ofphase distributions. In 1991, Okada et al. [4] proposed aleast-squares phase shifting algorithm to cope with theuncertainties in phase shift amounts. Since then, severalself-calibrating phase shifting algorithms based on theleast-squares approach have been proposed to correctuncertainties in phase shift amounts [5–12]. In theseapproaches, both phase shift amounts and phase distribu-tions are treated as unknowns and they are determinedsimultaneously through an iteration process.

When a practically feasible number of frames (say, p5)are used in the iteration process, however, the abovealgorithms do not provide stable convergence or accuratephase shift amounts unless shift intervals are reasonablyuniform and the initial shift estimates are within a fewdegrees of actual phase shifts. They are capable of handling

only small phase-shift errors such as those caused by thenon-linearity of piezo electric devices. In addition, theexisting iterative algorithms generally require the cumula-tive phase shifts to be monotonically ascending ordescending, and thus the phase shifts cannot be completelyarbitrary or random. This impedes the applications of thealgorithms to real-time and dynamic measurements wherethe phase shifts can be completely random due to the rigid-body motions.The advanced iterative algorithm (AIA) is also based on

the least-squares iterative procedure but it copes withthe limitation of the existing algorithms by separating aframe-to-frame iteration from a pixel-to-pixel iteration[13]. The algorithm provides stable convergence andaccurate phase extraction with the number of interfero-grams as few as three even when the phase shifts arecompletely random. The basic concept of AIA wasproposed in Ref. [13] and this paper will focus on thefollowing three objectives:

(1) to provide a detailed procedure for implementing theAIA,

(2) to evaluate the performance of AIA for randomlyphase-shifted interferograms with intra- and inter-frameintensity variations, and

(3) to present results obtained from a real application.

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0143-8166/$ - see front matter r 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.optlaseng.2005.11.003

�Corresponding author. Tel.: +1202 319 6703.

E-mail address: wangz@cua.edu (Z. Wang).

2. Advanced iterative algorithm (AIA)

The intensity of an interferogram can be expressed as[1–3]

I ij ¼ Aij þ Bij cosðfj þ diÞ, (1)

where I is the intensity of the interferogram, the subscript‘‘i’’ denotes the ith phase shifted image ði ¼ 1; 2; . . . ;MÞand ‘‘j’’ denotes the individual pixel location in each imageðj ¼ 1; 2; . . . ;NÞ. In the equation, Aij is the background ormean intensity, Bij is the modulation amplitude, fj is theangular phase information, di is the induced phase shiftamount of each of M ðMX3Þ acquired frames, and N is thetotal number of pixels in each image frame.

From Eq. (1), it is evident that there are (2MN+N+M)unknowns, i.e., Aij, Bij, fj and di, and the number ofavailable equations is (MN). Using a direct method to solvethe unknowns is not possible; instead, an iterative methodcan be employed. The iteration may have two or more stepsin each cycle and different groups of unknowns can bereleased in relevant steps to make the problem solvable.The basic concept of the iterative algorithm is performingan iteration process until the sum of the squared differencesbetween the estimated theoretical intensities and themeasured intensities converge to a pre-defined smallthreshold value. A brief description of the algorithm, itsrelations with conventional algorithms and a detailedprocedure of using the algorithm are presented below.

2.1. Universal algorithm for phase extractions from

interferograms with known phase shifts

The conventional phase shifting algorithms requirephase shift amounts not only to be known but also to beuniformly spaced [1–3]. In these algorithms, it is assumedthat the background intensities and the modulationamplitudes have only pixel-to-pixel (intra-frame varia-tions), i.e., A1j ¼ A2j ¼ � � � ¼ AMj ¼ Aj and B1j ¼ B2j ¼

� � � ¼ BMj ¼ Bj in Eq. (1). With the known phase shiftamount di; the number of unknowns is reduced to (3N).Since the number of available equations is still (MN), thesystem of equations is posed as determined or over-determined if MX3. In this case, the unknowns can bereadily solved.

When di is known and uniformly spaced, an explicitformulation for calculation of phase distributions can bederived. This is the typical principle of conventional phaseshifting algorithms. When the phase shifts are known butrandomly spaced, however, an explicit formulation isgenerally unavailable. In this case, the following classicalleast-squares method, a solver for determined and over-determined system of equations, can be employed to solvethe problem.

Defining a new set of variables as aj ¼ Aij, bj ¼ Bij cosfj

and cj ¼ �Bij sinfj can simplify the intensity equation (1) as

I ij ¼ aj þ bj cos di þ cj sin di (2)

The least-squares error between theoretical and experi-mental interferograms, Sj, accumulated from all the imagesdescribed by Eq. (2), can be written as

Sj ¼XMi¼1

ðIeij � I ijÞ

2¼XMi¼1

ðaj þ bj cos di þ cj sin di � IeijÞ

2;

(3)

where Ieij is the experimentally measured intensity of the

interferogram. The least-squares criteria required for threeunknowns (aj, bj and cj) can be expressed as

qSj

qaj

¼ 0;qSj

qbj

¼ 0;qSj

qcj

¼ 0, (4)

which yields

MPMi¼1

cos di

PMi¼1

sin di

PMi¼1

cos di

PMi¼1

cos2di

PMi¼1

cos di sin di

PMi¼1

sin di

PMi¼1

sin di cos di

PMi¼1

sin2di

26666666664

37777777775

aj

bj

cj

8>><>>:

9>>=>>;

¼

PMi¼1

Ieij

PMi¼1

Ieij cos di

PMi¼1

Ieij sin di

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

. ð5Þ

From Eq. (5), the unknowns aj, bj and cj can be solved.Then, the phase fj can be determined as

fj ¼ tan�1ð�cj=bjÞ. (6)

The above phase shifting algorithm can provide the least-squares determination of the phase distributions when thephase shift amounts are accurately known. Unlike theconventional algorithms, the universal algorithm does notrequire that the phase shift amounts have to be evenly spaced;this is the origin of the terminology, universal. Moreover, theprocessing time of the universal algorithm is of the sameorder as that of the conventional algorithms. It is noted thatwhen M ¼ 4 and di ¼ ði � 1Þp=2 ði ¼ 1; 2; 3; 4Þ, the uni-versal phase shifting algorithm yields the formulationidentical to the most widely used conventional four-framealgorithm.

2.2. Inverse algorithm for determinations of phase shifts

from interferograms with known phase distributions

Using the universal phase shifting algorithm, phasedistributions can be extracted if the phase shift amounts areknown. If phase distributions are known, the phase shiftamounts can be determined in a similar but inverse way. Inthe inverse algorithm, it is assumed that the background

ARTICLE IN PRESSZ. Wang, B. Han / Optics and Lasers in Engineering 45 (2007) 274–280 275

intensities and modulation amplitudes have frame-to-framevariations but they are constants within each individualframe (inter-frame variations), i.e., Ai1 ¼ Ai2 ¼ � � � ¼

AiN ¼ Ai and Bi1 ¼ Bi2 ¼ � � � ¼ BiN ¼ Bi in Eq. (1). Con-sequently, the number of unknowns in the equation isreduced to ð3M þNÞ. If the phase fj is known, the numberof unknowns can be further reduced to (3M). With thenumber of available equations still being (MN), a similaroverdeterministic least-squares method can be employed toobtain the unknowns.

Defining a new set of variables for each frame asa0i ¼ Aij , b0i ¼ Bij cos di and c0i ¼ �Bij sin di, the intensityequation (1) can be expressed as

I ij ¼ a0i þ b0i cosfj þ c0i sinfj. (7)

The least-squares error, S0i, accumulated from all thepixels in the ith image, can be expressed as

S0i ¼XN

j¼1

ðI ij � IeijÞ

2¼XN

j¼1

ða0i þ b0i cosfj þ c0i sinfj � IeijÞ

2:

(8)

When fj is known, the least-squares criteria of theunknowns a0i, b0i, and c0i yield

NPNj¼1

cosfj

PNj¼1

sinfj

PNj¼1

cosfj

PNj¼1

cos2 fj

PNj¼1

cosfj sinfj

PNj¼1

sinfj

PNj¼1

sinfj cosfj

PNj¼1

sin2fj

266666666664

377777777775

a0i

b0i

c0i

8>><>>:

9>>=>>;

¼

PNj¼1

Ieij

PNj¼1

Ieij cosfj

PNj¼1

Ieij sinfj

8>>>>>>>>>><>>>>>>>>>>:

9>>>>>>>>>>=>>>>>>>>>>;

. ð9Þ

The unknowns a0i, b0i and c0i can be obtained from Eq. (9)and the phase shifts di can be determined from theunknowns as

di ¼ tan�1ð�c0i=b0iÞ. (10)

The above algorithm can be regarded as an inverseprocedure of the universal phase shifting algorithm. In atypical phase shifting analysis, the inverse algorithm can beemployed to verify the phase shift amounts, and the newdetected phase shift amounts can be used to adjust theinput of phase shifts for phase shifting analysis. Thisprocedure can help increase the accuracies of phaseextractions. This is the concept of the AIA for randomphase shifting.

2.3. Iterative strategy for random phase shifting

The combination of the two algorithms (universal phaseshifting algorithm and inverse phase shifting algorithm)yields an iteration procedure that can determine phasedistributions and phase shifts simultaneously from ran-domly phase-shifted interferograms [13]. The AIA containsthree steps in each iteration cycle. In the kth iteration cycle:

Step 1: Employ the universal phase shifting algorithm,i.e., Eqs. (5) and (6), to calculate phase distributions fk

j ðj ¼

1; 2; . . . ;NÞ based on phase shifts dk�1i ði ¼ 1; 2; . . . ;MÞ

obtained in the second step of previous iteration cycle,i.e., the (k�1)th one. For the first iteration cycle, an initialestimation of phase shifts is not required because they canbe any random values.

Step 2: With known phase distributions fkj ðj ¼

1; 2; . . . ;NÞ, utilize the inverse phase shifting algorithm,i.e., Eqs. (9) and (10), to determine updated phase shifts,dk

i ði ¼ 1; 2; . . . ;MÞ.Step 3: Check to see whether the iteration results satisfy

the converging criteria. It is the relative phase shiftamounts that will converge, so the converging criteria are

jðdki � dk

1Þ � ðdk�1i � dk�1

1 Þjoe; i ¼ 2; . . . ;M, (11)

where e is the pre-defined threshold of accuracy, e.g., 10�6.If the converging criteria are satisfied, then fk

j ðj ¼

1; 2; . . . ;NÞ and dki ði ¼ 1; 2; . . . ;MÞ are the final phase

distributions and phase shifts, respectively. Otherwise, theiteration will repeat until the criteria are satisfied.

3. Performance analysis of AIA

It is evident from Eq. (1) that the unknowns of a series ofphase-shifted interferograms include background intensi-ties Aij, modulation amplitudes Bij , phases fj, and phaseshifts di. The total number of unknowns (2MN+N+M) islarger than that of available equations (MN). Similar to theexisting iterative algorithms, the AIA divides the procedureinto two steps to solve the unknowns and the number ofunknowns is reduced in each step based on the relevantassumptions.The first step assumes that background intensities and

modulation amplitudes do not have inter-frame variations;the variations within a frame remain the same for allframes (intra-frame variations only). In the second step, thebackground intensity and modulation amplitude of eachframe are assumed to be uniform but they are allowed tohave frame-to-frame variations (inter-frame variationsonly) to compensate the released unknowns in the firststep. Although it is usually not the case for realinterferograms, such assumption in the second step isreasonable since the number of unknowns is much smallerthan that of available equations. In fact, it is theassumption of uniform background intensity and modula-tion amplitude that makes the iteration converge muchfaster with a much higher accuracy than the existing

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algorithms [13]. An extensive computer simulation wasconducted to verify validity of the assumption.

3.1. Random pattern generation

The base interferogram used in the simulation isshown in Fig. 1, which represents a monotonicallyincreasing phase map with a linear phase gradient (orquadratic phase distribution). Numerous sets of phase-shifted interferograms containing both random and sys-tematic noise were produced by using the followingformula:

I ij ¼ CiC00j Aþ C0iC

00j B � cosðfj þ diÞ þNij

ði ¼ 1; 2; . . .M; j ¼ 1; 2; . . .NÞ, ð12Þ

where Ci ¼ ð1þ 0:3riÞ;C0i ¼ ð1þ 0:3r0iÞ, C00j ¼

exp �1:4 � ðjx � L=2Þ2 þ ðjy � L=2Þ2j k

=L2n o

, di ¼ 2pr00i ,

Nij ¼ nR0ijðAþ BÞ with n ¼ 18, ri, r0i, Rij and R0ij are random

rational numbers ranging from –1 to 1, r00i is a random

rational number ranging from 0 to 1, A and B are set to127.5 for 255 gray level images, jx and jy are the x and y

coordinates of the jth pixel, and L is the width of the square

images, L2 ¼ N.In Eq. (12), Ci and C0i represent random inter-frame

variations where the inter-frame background intensitiesand modulation amplitudes are randomly chosen between–30% and 30% of the nominal value, C00j representssystematic intra-frame variations that follow a Gaussiandistribution where the maximum intensities and amplitudesare twice (results from the coefficient of –1.4) of theminimum values, Nij represents the random electronicnoise, and fj is the phase. It is to be noted that n ¼ 1

8 for therandom image noise is a conservative estimate.

A set of phase-shifted images for the case of four framesare shown in Figs. 2(a)–(d) Fig. 2(e) illustrates the intensitydistributions along the diagonal AA0. The random andsystematic noises are evident.

3.2. Performance analysis

For the performance analysis, a total of 300 sets ofinterferograms (i.e., 100 sets for each case of 3, 4, and 5frames) were used. Because the calculation accuracy ofphase distributions is directly governed by the determina-tion accuracy of phase shift amounts [14], the errors of thedetected phase shift amounts were evaluated in thesimulations. The error is defined as

e ¼1

M � 1

XM�1i¼1

jðdiþ1 � diÞ � ðdeiþ1 � de

i Þj; (13)

where di is the real (or theoretical) phase shift amount usedfor the simulations and de

i is the algorithm-detected phaseshift amount. Because phase shift amounts are relative

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Fig. 1. Computer-generated interferogram with a quadratic phase

distribution.

Fig. 2. Computer-generated interferograms with random and systematic

noise. (a)–(d) Fringe patterns, (e) intensity distributions along the diagonal

AA0.

Z. Wang, B. Han / Optics and Lasers in Engineering 45 (2007) 274–280 277

values, the first phase shift amounts, d1 and de1, were always

subtracted from all phase shifts.Table 1 summarizes the results from twelve representa-

tive cases. The results clearly show that the AIA producesexcellent results in spite of the true random nature of phaseamounts and noise. The trends in accuracies and con-vergences prevailed in all 300 cases in the simulations. It isalso important to note that the AIA provides a very fastconvergence. It typically takes 5–15 iteration cycles toachieve the convergence threshold, and each cycle takesmuch shorter than 1 s for a set of 512 pixels by 512 pixelsinterferograms on a typical PC with Intel P4 processor.Further reduction of processing time is possible using asmall portion of areas of the interferograms for theiteration process.

4. Application

The AIA was implemented for warpage measurement ofan electronic packaging subassembly using real-timeTwyman–Green interferometry [15]. The subassembly wasa typical chip package with heat sink; it was fabricated byusing the tape automated bonding (TAB) technology [16].In the manufacturing process, a small active chip (5mm by5mm for this case) was joined to a patterned metal on apolymer tape by using thermo-compression bonding. Then,a protecting encapsulant was dispensed on the top surfaceof the chip. The chip/tape subassembly was later adhesivelybonded to an aluminum heat sink to form a final package.

When the subassembly is subjected to a temperatureexcursion, the encapsulant bends the chip due to relativelylarge thermal expansion or contraction of the encapsulant.This bending should be portrayed accurately, since excessstresses caused by the chip bending could produce cracks inthe adhesive layer.

When deformation measurements are required duringthermal cycling, it is necessary to implement the methodwith an environmental chamber that provides convection

heating and cooling. The air inside the chamber must becirculated vigorously to achieve the heating/cooling raterequired for the test. Consequently, the environmentalchamber experiences vibrations, which are normally trans-mitted to the specimen. Twyman–Green interferometrymeasures tiny displacements and those inadvertent vibra-tions can cause fringes to dance at the vibration frequencies.Because the vibration periods are much shorter than thetime required for capturing phase-shifted interferograms,accurate phase shift amounts required by the conventionalphase shifting algorithm cannot be accommodated. In otherwords, the fringe patterns documented at a given tempera-ture contain random phase shifts.Fig. 3 illustrates the optical configuration of the

experimental setup. Four randomly shifted fringe patternsdocumented while operating the oven are shown in Figs.4(a)–(d), which represent the bending deformation of thebackside of the chip. The intensity distributions along thediagonal BB0 are shown in Fig. 4(e). The inter- and intra-frame intensity variations are apparent.

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Table 1

Simulation results of the AIA for randomly phase-shifted interferograms with intra- and inter-frame intensity variations

Simulation cases Real AIA detected Error (e)

d1 d2 d3 d4 d5 d1 d2 d3 d4 d5

3-frame 1 0.0000 2.3227 5.2253 – – 0.0000 2.3145 5.2089 – – 0.0082

2 0.0000 5.1971 1.8130 – – 0.0000 5.2099 1.8335 – – 0.0103

3 0.0000 1.7691 5.4418 – – 0.0000 1.7894 5.4673 – – 0.0128

4 0.0000 4.9566 2.2100 – – 0.0000 4.9754 2.2173 – – 0.0152

4-frame 5 0.0000 5.9739 5.4972 3.1338 – 0.0000 5.9754 5.4920 3.1253 – 0.0038

6 0.0000 3.1338 5.4972 5.9739 – 0.0000 3.1253 5.4920 5.9754 – 0.0062

7 0.0000 5.5910 0.0443 1.2738 – 0.0000 5.5881 0.0285 1.2763 – 0.0114

8 0.0000 0.0313 5.3455 2.1242 – 0.0000 0.0341 5.3686 2.1328 – 0.0125

5-frame 9 0.0000 5.5950 2.1714 0.8638 1.1917 0.0000 5.5942 2.1752 0.8696 1.1948 0.0025

10 0.0000 0.2328 5.3793 0.5304 2.6671 0.0000 0.2359 5.3822 0.5328 2.6604 0.0032

11 0.0000 2.7586 4.3954 4.0307 4.9340 0.0000 2.7528 4.3840 4.0158 4.9497 0.0114

12 0.0000 5.3982 5.7681 2.0846 2.0771 0.0000 5.4160 5.7854 2.0754 2.0645 0.0121

Fig. 3. Optical configuration of Twyman–Green interferometry.

Z. Wang, B. Han / Optics and Lasers in Engineering 45 (2007) 274–280278

Using the AIA, the phase shift amounts for theinterferograms in Fig. 4 were determined and they were0.0, 0.613 (35.11), 0.185 (10.61) and 2.606 (149.31).Extremely random phase shifts were resulted. The wrappedand unwrapped phase maps are shown in Figs. 4(f) and (g),respectively; it is to be noted that the defects (black area) inthe phase map were corrected during phase unwrapping.Figs. 4(h) and (i) show the warpage displacement along thediagonal BB’ and the 3D warpage of the package,respectively.

The absolute accuracy (measured displacement) offeredby the AIA was verified by the conventional phase shiftalgorithm. After documenting the patterns in Figs.4(a)–(d), the oven was turned off and another set of fourfringes were documented by using a pre-calibrated linearphase modulator; each fringe has accurate phase shifts of 0,p/2, p, and 3p/2, respectively. The conventional four-frame

algorithm was used to determine the phases and thecorresponding displacements, and the results are plotted inFig. 4(h). The displacements obtained by the conventionalalgorithm match extremely well to those obtained by theAIA, which confirms the validity of the AIA for randomlyphase-shifted interferograms with intra- and inter-frameintensity variations.

5. Discussions

The existing iterative algorithms assume point-by-pointvariations of background intensities and modulationamplitudes in the second step. When phase shifts are trulyrandom, the iteration may converge to a local minimumunless a large number of frames are utilized. Theassumption of the constant intra-frame backgroundintensities and modulation amplitudes employed in the

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Fig. 4. Warpage measurement of a TAB package. (a)–(d) Fringe patterns, (e) intensity distributions along the diagonal BB0, (f) Wrapped phase map,

(g) unwrapped phase map, (h) warpage along the diagonal BB0, and (i) 3D warpage.

Z. Wang, B. Han / Optics and Lasers in Engineering 45 (2007) 274–280 279

second step of the AIA eliminates the problems associatedwith the local minimum and always lead the least squaresto a global convergence minimum, where the actual phaseshifts and phases are located. Consequently, the AIAconverges very fast and is capable of processing randomlyphase-shifted interferograms even when the noise level inthe interferograms is high.

As discussed earlier, the AIA is capable of processingphase-shifted interferograms with truly arbitrary phaseshifts. Because phase shift amounts can be completelyrandom, the cumulative phase shift amounts do not have tobe monotonically ascending or descending and the algo-rithm does not require an accurate initial estimation ofphase shifts. In addition, the total frame number ofinterferograms can be as few as three. Consequently, thecostly and accurate devices for phase shifting are no longerrequired. For the same reason, the advanced iterative phaseshifting algorithm can be used to calibrate the existingphase shifting devices.

Although the AIA provides a fast iteration convergence,it takes relatively longer time than the conventional phaseshifting algorithms [1–3]. Taking advantage of stableconvergence provided by the AIA, only a small portionof the interferogram can be used for the iteration process.In this way, the computational time of AIA can be reducedsignificantly to be in the same order of that of theconventional phase shifting algorithms.

In the AIA, the nonuniformities of background inten-sities and modulation amplitudes are handled by least-squares approaches, thus the algorithm inevitably producesmore or less errors associated with such nonuniformities.For interferograms with excessively large variations ofbackground intensities and modulation amplitudes, theAIA could yield prohibitive errors. It should be also notedthat the errors may also depend on the phase distributionsif coupled with the nonuniformities. As illustrated pre-viously, however, the errors in the results are virtuallyignorable for the level of nonuniformities encountered inroutine practice of interferometric techniques.

6. Conclusions

The concept of the advanced iterative algorithm wasreviewed and the detailed procedure for implementationwas presented. Computer simulations were conducted toevaluate the performance of AIA and the results verifiedthe effectiveness of the algorithm in analyzing truly

random phase-shift interferograms with intra- and inter-frame intensity variations. The absolute accuracy of theAIA was verified by a controlled experiment. The AIAproduces accurate phase information with the number offrames as few as three even when the noise level in theinterferograms is high. The method is suitable for real-timeand dynamic measurements. With the method, the costlyand accurate phase-shifting devices are no longer requiredfor the steady-state measurement.

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