Optics and Lasers in Engineering 32 (1999) 147}155

Comparison of results of interferometric phaseextraction algorithms for three-dimensional

#ow-"eld tomography

Yao Wei, Dapeng Yan!,1, Soyoung Stephen Cha",*!Department of Applied Physics, Nanjing University of Science and Technology, Nanjing 210094, China

"Department of Mechanical Engineering, University of Illinois at Chicago, 842 West Taylor Street, Chicago,IL 60607-7022 USA

Received 22 September 1999; accepted 4 November 1999

Abstract

A comparison of the results of interferometric phase information retrieval and subsequenttomographic reconstruction from deformed wavefront projections by phase unwrapping ispresented. In interferogram processing, conventional fringe tracking and Fourier transformmethods have been utilized for comparison. With these methods by injecting carrier fringes, theprojection data of any axial cross section can be extracted in all projection directions toimplement three-dimensional tomographic reconstruction. The results of experiments ofa simulated temperature "eld prove that the phase extraction based on the Fourier transformmethod produces tomographic reconstruction much superior to the conventional fringe track-ing technique. ( 2000 Published by Elsevier Science Ltd. All rights reserved.

Keywords: Computational tomography; Interferometry; Image processing; Phase unwrapping

1. Introduction

In interferometric tomography for reconstructing #ow "elds, the conventionalphase extraction based on fringe tracking is still employed for obtaining projectiondata. Even though the recent advances in various interferogram reduction techniques

*Corresponding author. 2039 ERF, 842 W. Taylor St., Chicago, IL 60607-7022, USA. Tel.: 001-312-996-9612; fax: 001-312-413-0447.

1Currently visiting scholar at The University of Illinois at Chicago.

0143-8166/99/$ - see front matter ( 2000 Published by Elsevier Science Ltd. All rights reserved.PII: S 0 1 4 3 - 8 1 6 6 ( 9 9 ) 0 0 0 5 8 - 5

including phase stepping can o!er better accuracy, this simple method, however,o!ers better immunity to severe noise with extremely broken or connected fringes.In fringe tracking, a few associated problems need to be resolved. First, a great dealof useful information can be inevitably lost in the course of interferogramimage preprocessing and data reduction to make the tomographic reconstructionless reliable. This information loss occurs during fringe binarization, thinning,and skeleton extraction. It is generally accepted that the inaccuracy of the fringetracking method close to a fraction of 2p in phase, or equivalently a fraction ofthe fringe width in spatial resolution. When interferograms with a sinusoidalgray-scale distribution is reduced to some single-pixel line fringes, the informationin the intermediate locations is lost. This can be overcome partially by injection ofcarrier fringes. However, fringe breaking and merging due to noise often causeincorrect tracking to seriously a!ect the subsequent phase extraction and tomo-graphic reconstruction. Second, it is more likely that the fringe tracking methodmay not obtain projection data continuous enough to obtain closely spacedaxial cross-sectional reconstruction in computational tomography. The spacingbetween adjacent reconstructed cross-sectional planes should be equal to ormore than the width of a fringe. This problem can also be alleviated by injectingcarrier fringes.

Recently, Takeda et al. [1] reported a method of fringe analysis based on theFourier transform. Macy [2] applied the two-dimensional (2-D) Fourier-transformmethod to analyze holographic interferograms, after which phase unwrapping hasbecome an important topic in optical interferometry for nondestructive evaluation.The Fourier transform method provided potential as an e!ective tool in interferomet-ric #ow-"eld tomography. For quantitative tomography for reconstructing three-dimensional (3-D) #ow "elds, its potential has yet to be tested. The Fourier transformmethod becomes less accurate when the number of fringes in an interferogramreduces. This problem can also be alleviated by injection of carrier fringes. As in thefringe tracking method, too many fringes pose fringe breaking and merging to increasethe noise and inaccuracy in phase extraction especially in phase unwrapping. Phasestepping approaches [3,4], which are also known as phase shifting techniques, needmore than three interferograms. However, these multi-frame techniques are di$cult toimplement for #ow-"eld tomography, especially for a transient #ow. In most cases of#ow diagnostics, only a single interferogram can be obtained due to various restrictingconditions.

In this paper, use of the two techniques for processing #ow interferograms, that is,the fringe tracking method and the Fourier transform method, are presented [3}5].Comparison of the tomographic reconstruction with the extracted phase information,which is our ultimate purpose, is also presented. The results of the current study basedon simulation of experiments, which provides a means for assessing accuracy, as wellas real experiments for which accuracy assessment is not possible have proved that theprojection data extraction based on the Fourier transform method has merits in datacontinuity, accuracy, and automation of analysis. Especially for interferometric to-mography of 3-D #ow "elds, its use is recommended to substitute the conventionalfringe tracking method.

148 Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155

2. Review of principles on interferometric tomography

Interferometric tomography requires three distinct data processing procedures: thatis, acquisition of projection interferograms, reduction of interferograms for phaseinformation extraction, and computational tomographic reconstruction. The detailsof mathematics of interferometric projection and computational reconstruction is wellexplained in the previous publications by various authors [6}9]. As indicated, here weemphasize the investigation on the e!ect of interferogram processing techniques, forwhich the fringe tracking and Fourier-transform methods were considered.

The fringe tracking method is based on geometrical interpretation of interfero-grams. An interferogram can be essentially described by a 2-D continuous sinusoidaldistribution of gray scale [8]

I(x, y)"A(x, y)M1#<(x, y)cos [2p( f0x

x#f0y

y)!u(x, y)]N (1)

where A(x, y) is the background illumination,<(x, y) is the contrast of fringes, f0x

andf0y

are carrier-fringe (reference) frequencies in x- and y-directions, and u(x, y) is thephase di!erence between the two coherence light beams forming the interferogram. Inthe fringe tracking method, minimum or maximum contour lines are traced based onthe intensity value of I(x, y) to assign values of p-interval. Also, frequently fringe sidesare tracked to more accurately locate phase contour lines, where the gradients aresteepest. This approach thus loses information at those locations between two con-tour lines. The details of fringe tracking can be found in [5].

In quantitative measurement of a #ow "eld, most interferograms contain referencefringes with a limited width of relatively high frequency. If so, Eq. (1) can be rewrittenas

I(x, y)"A(x, y)#c(x, y)exp 2pi[ f0x

x#f0y

y]N

#cH*(x, y)exp M!2pi[ f0x

x#f0y

y]N (2)

where

c(x, y)"(1/2)MA(x, y)<(x, y)exp[ipu(x, y)]N. (3)

When 2-D Fourier-transform is performed with Eq. (2)

I( fx, fy)"A( f

x, fy)#C( f

x!f

0x, fy!f

0y)#CH( f

x!f

0x, fy!f

0y) (4)

where C and CH are Fourier-transforms of c and cH, respectively. Fig. 1 demonstratesthe frequency spectrum of an interferogram as an example.

The procedure of the Fourier-transform method includes three steps. The "rst stepis to determine the location and the width of the spectrum of C( f

x!f

0x, fy!f

0y),

which is in the domain of positive frequency. CH( fx!f

0x, fy!f

0y) and A( f

x, fy) are

then "ltered out using a "lter function of H( fx!f

0x, fy!f

0y), which is manually

de"ned by the operator. The signal is now translated to the origin in the frequencyplane, which can be denoted by C@( f

x, fy). The translation of the positive-order

frequency spectrum is to remove the reference frequency. The second step is toperform 2-D inverse Fourier-transformation with C@( f

x, fy) to obtain the signal

Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155 149

Fig. 1. Example of the frequency spectrum of an interferogram.

information described by Eq. (3). The phase can then be found by

u(x, y)"tan~1Im[c(x, y)]

Re[c(x, y)], (5)

where Im and Re represent imaginary and real parts, respectively. The phase u(x, y)thus found is wrapped in [!p, p]. The true phase uk (x, y) needs to be found. The thirdstep is to unwrap u(x, y) along x- and y-direction, respectively, to restore its originalvalue of uk (x, y). That is, when a jump from 2p to zero is encountered, 2p is added tothe base phase value before encountering the jump in the wrapped phase map. 2p isdeducted from the prior base values when a jump is encountered from zero to 2p.uk(x, y) calculated through 2-D integration represents the true shape of the deformedwavefront.

3. Results from computer simulation of experiments

For testing the e!ects of interferogram processing algorithms, we conducted com-puter simulation of experiments, which can possibly be only an e!ective means forassessing reconstruction accuracy with the initially known "eld to simulate. Fortesting, a rather simple "eld was assumed. Fig. 2(a) is the interferogram which wasgenerated from an axially symmetric temperature "eld employed for testing. Theinterferogram generated can be represented by

g(x, y)"127.5#127.5]cosGCx#(x.!9

/2) expG![y!(y

.!9/3)]2

(y.!9

/3) HD2p

(x.!9

/4)H,(6)

where g(x,y) is gray-scale value of point (x, y), and x.!9

and y.!9

are the x andy coordinates where the maximum value occurs. The fringes exhibit a sinusoidal

150 Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155

Fig. 2. Computer simulation of experiments without noise: (a) interferogram generated from a knowntemperature "eld; (b) wrapped phase map extracted by the Fourier-transform method; (c) result after phaseunwrapping; (d) projection phase data at a speci"c cross-section.

distribution of high frequency in the x-direction, which are reference fringes. Thefringe pattern shows a normal distribution of slow variation in gray scale in they-direction, which represents the in#uence of the axially symmetric temperature "eld.The simulated interferogram area was 128]128 pixels. The displacement of fringes inthe y-direction with the normal distribution without the reference fringe had a max-imum displacement of two fringes.

The Fourier-transform method and the subsequent phase unwrapping was appliedto Fig. 2(a) to extract projection phase information. Fig. 2(b) shows the wrapped phasemap of the deformed object wavefront. Fig. 2(c) is the phase map after phaseunwrapping where the gray scale represents the shape of the actual wavefront. Fig.2(d) illustrates the projection data along a vertical line, which is located at 1.7 cm from

Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155 151

Fig. 3. Plot of tomographic reconstruction of the simulated experiment with the Fourier-transform methodunder no noise.

the left boundary. With this projection data at the cross 1.7 cm from the left boundary,obtained from the initially assumed temperature "eld, tomographic reconstructionwas performed based on analytical inversion [8]. The reconstruction result usingthese extracted data is shown in Fig. 3. The "elds in Fig. 2(a) is actually horizontal butrotated by 903 in Fig. 3 for plotting purpose. The reconstructed temperature "eldagrees almost exactly with the initially assumed "eld. The Fourier-transform methodworked well for this noiseless case.

In the Fourier-transform method, noise "ltering is performed judiciously by theoperator in the frequency domain. When a proper noise "ltering window is selected,the Fourier-transform method can be made fairly immune to noise, which is veryimportant for the phase unwrapping process. We also tested noise-ridden interfero-grams as in practical experiments. Fig. 4(a) is the result of the interferogram in Fig. 2(a)convoluted with a random noise which can be described as

gn(x, y)"[*n(x, y)#1.0]g(x,y) (7)

where *n(x, y) is a random number of a Gaussian distribution for noise with a stan-dard distribution of 1.0. Fig. 4(b) shows a wrapped phase map modulo 2p of thedeformed object wavefront. Fig. 4(c) is the phase map after phase unwrapping. Owingto the in#uence of the noise and the simple phase unwrapping procedures, the qualityof image, which shows the phase values in gray scale, decreased to some extentespecially near the image boundary. Fig. 5 shows the temperature "eld which wasreconstructed with the projection data. The reconstructed of the cross-section waslocated at 1.7 cm from the left boundary as in the case of the reconstruction without

152 Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155

Fig. 4. Computer simulation of experiments with noise: (a) interferogram generated from a knowntemperature "eld; (b) wrapped phase map extracted by the Fourier transform method; (c) result after phaseunwrapping.

noise. The reconstructed "eld was very close to the originally assumed "eld. However,the projection data near the image boundaries had di$culties to provide meaningfulreconstructions. Here we used a line scanning approach in phase unwrapping. Cur-rently, various types of phase unwrapping algorithms are available. More accuratephase unwrapping algorithms is thus needed to alleviate the error propagationproblem especially near image edges.

For comparison of the results, phase information was also extracted with theconventional fringe tracking method from the interferograms of the both caseswithout and with noise as seen in Figs. 2(a) and 4(a), respectively. The "eld was thenreconstructed. Fig. 6 is the line image obtained from Fig. 2(a) for the noiseless case,through binarization and skeletonization of the interferogram. The process which weadopted did not employ any pre-processing of interferograms. As seen, when the

Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155 153

Fig. 5. Plot of tomographic reconstruction of the simulated experiment with the Fourier-transform methodwith noise.

Fig. 6. Fringe line image obtained from the interferogram under no noise with the fringe tracking method.

simple approach is applied for fringe tracking, the quality of the processed dataappears to be slightly inferior to that of the Fourier-transform method for thenoiseless case. The "eld, which was reconstructed with the projection data for thecross section at the same distance from the boundary as before, practically produced

154 Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155

a very similar plot to that of the Fourier-transform method in Fig. 3. The reconstruc-tion was very close to the original accurate "eld in this case too. For the case withnoise-a!ected interferograms, the success of fringe tracking depended heavily onimage pre-processing. If only the simple algorithms of conventional noise "ltering,including median "ltering, gray-scale transform and logarithm transform, were ap-plied, they did not have much e!ect on the success of the results. The most di$cultproblem was broken or inter-connected fringes. With the simple fringe tracking, thesubsequent reconstruction could not produce any useful results for the case withnoise.

4. Conclusions

The phase information extraction from projection interferograms for tomographicreconstruction of #ow "elds, which is based on the Fourier-transform method, hasmerits of data continuity, accuracy, and automation of analysis over the conventionalfringe tracking method. Our results of both computer simulation and real experimentshave in general proved our claim. For 3-D #ow-"eld tomography, the use of theFourier-transform method is recommended instead of the fringe tracking method. Forthe Fourier-transform method, however, how to provide a suitable frequency domain"lter and how to conduct phase unwrapping are challenges. These e!ects are synergis-tic and error propagation can be multiplied, depending on the adopted approaches.The appropriate algorithms for interferogram reduction and the synergistic e!ects oninterferometric tomography need to be further investigated. The ensuing investigationwill report these "ndings in the future.

References

[1] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringe pattern analysis for computer-based tomography and interferometry. J Opt Soc Am 1981;72:156}60.

[2] Macy WW. Hologram analysis using 2-D Fourier-transform. Appl Opt 1983;22:3898}901.[3] Robinson DN, Reid GT. Interferogram analysis. Institute of Physics Publishing, 1993. p. 145}89.[4] Judge TR, Bryanston-Cross PJ. A review of phase unwrapping techniques in fringe analysis. Opt

Lasers Engng 1994;21:199}239.[5] Becker F, Yu YH. Digital fringe reduction techniques applied to the measurement of three-dimen-

sional transonic #ow "elds. Opt Eng 1985;24:429}34.[6] Yu E, Cha SS. Two-dimensional regression for interferometric phase extraction. Appl Opt

1998;37:1370}6.[7] Cha S. Interferometric tomography for three-dimensional #ow "elds via envelope function and

orthogonal series decomposition. Opt Eng 1988;27:557}63.[8] Vest. Holographic interferometry. New York: Wiley, 1979. p. 315}21.[9] Cha SS, Sun H. Tomography for reconstructing continuous "elds from ill-posed multidirectional

interferometric data. Appl Opt 1990;29:251}8.

Y. Wei et al. / Optics and Lasers in Engineering 32 (1999) 147}155 155