Phase extraction from three and more interferograms registered with different unknown wavefront...

Phase extraction from three and moreinterferograms registered with different

unknown wavefront tilts

Oleg Soloviev and Gleb VdovinEI Lab., EWI, TU Delft, Mekelweg 4, 2628CD Delft, the Netherlands

[email protected]

Abstract: We propose phase retrieval from three or more interferogramscorresponding to different tilts of an object wavefront. The algorithmuses the information contained in the interferogram differences to reducethe problem to phase shifting. Three interferograms is the minimum forrestoring the phase over most of the image. Four or more interferograms areneeded to restore the phase over the whole image. The method works withimages including open and closed fringes in any combination.

© 2005 Optical Society of America

OCIS codes: (120.3180) Interferometry, (120.2650) Fringe analysis, (120.5050) Phasemeasurement.

References and links1. D. Malacara, M. Servın, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc.,

New York, Basel, Hong Kong, 1998).2. Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51 –

60 (1996).4. D. Malacara, ed., Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., 1992).5. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).6. K. Okada, A. Sato, and J. Tujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in

phase shifting interferometry,” Opt. Commun. 84(3,4), 118–124 (1991).7. I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,”

Opt. Eng. 34, 183–187 (1995).8. M. Chen, H. Guo, and C. Wei, “Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,”

Appl. Opt. 39(22), 3894–3898 (2000).9. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,”

Appl. Opt. 41(13), 2435–2439 (2002).10. T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt.

Soc. Am. A 3(6), 847–855 (1986).11. J. A. Quiroga, J. A. Gomez-Pedrero, and A. Garcıa-Botella, “Algorithm for fringe pattern normalisation,” Opt.

Commun. 197, 43–51 (2001).12. C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. 26, 1668–

1673 (1987).13. Z. Ge, F. Kobayashi, S. Matsuda, and M. Takeda, “Coordinate-transform technique for closed-fringe analysis by

the Fourier-transform method,” Appl. Opt. 40(10), 1649–1657 (2001).14. K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17),

2886–2893 (2001).15. F. Cuevas, J. Sossa-Azuela, and M. Servin, “A parametric method applied to phase recovery from a fringe pattern

based on a genetic algorithm,” Opt. Commun. 203, 213–223 (2002).16. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application

to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).17. E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–

1376 (1998).

(C) 2005 OSA 16 May 2005 / Vol. 13, No. 10 / OPTICS EXPRESS 3743#7118 - $15.00 US Received 12 April 2005; revised 4 May 2005; accepted 4 May 2005

mailto:[email protected]

18. G. X. Ritter and J. N. Wilson, Handbook of computer vision algoritm in image algebra (CRC Press, Boca Raton,New York, London, Tokyo, 1996).

1. Introduction

The information about a physical quantity being measured interferometrically is contained inthe phase term φ of the fringe pattern described by (1). The goal of analysis [1] consists inextracting this phase term. Many widely used phase-shifting methods are based on phase de-tection of a sinusoidal signal [2, 3, 4] and use at least three phase-shifted interferograms. Thephase shifts, however, should be controlled with high accuracy. Any error in linearity of an ac-tuator response can introduce an error into the calculated phase. A number of methods referredas self-calibrating techniques has been developed to deal with linear and nonlinear piston-shifterrors and tilt-shifts errors. Lai and Yatagai [5] directly calculate phase shifts from additionallinear fringes due to an additional tilted reference mirror moved together with the test object.In the methods suggested by Okada et al [6] and by Kong and Kim [7], the phase distributionand phase shifts are calculated alternatively by iteration of the least-square algorithm. Chen etal [8] have presented a modified method which uses the first-order Taylor series expansion ofthe phase-shifting errors to compensate also the phase error caused by tilt-shift errors. Dobroiuet al [9] cancel the effects of both phase shift errors and the presence of tilts by calculatingaverage phase shifts for sufficiently small blocks of the interferogram space, so that the phaseshifts inside every block can be assumed constant. Based on a contrast-level map, the algorithmiteratively adjusts the phase shifts and tilts.

Another popular approach is based on a spatial analysis of the interferogram, e.g. with thediscrete Fourier transform [1, 10, 11, 12]. This method needs only one interferogram but workswell only when the interferogram has a narrow bandwidth, a high-frequency carrier and a lownoise level. The method cannot be used if the interferogram contains closed fringes, thoughdifferent improvements have been proposed to overcome this obstacle [13]. Goldberg andBokor [14] have proposed to use the Fourier-transform method together with the phase-shiftingto determine unknown phase-step increments. The spatial analysis of the interferograms allowsto find the phase steps deterministically, not iteratively, without any a priori information; whiletime-domain based phase shifting provides ease of implementation. Both phase-shifting andFourier transform-based methods calculate the phase wrapped in the interval (−π ,π ]. Somemethods – such as the local phase-tracking technique and genetic algorithms [15, 16, 17] –return unwrapped phase values, but these are computationally expensive.

This paper presents a new algorithm which allows one to reconstruct the phase from threeinterferograms registered with three different arbitrary unknown wavefront tilts. Unlike themethods described in [8, 9], the tilts are purposely introduced, and do not represent a phase-shifter error. On the contrary, the phase shifts are introduced in every interferogram point dueto the tilts. The magnitude of the tilts is larger then in tilt-eliminating methods [8, 9] (severalfringes per interferogram space), but not as large as in FFT-based methods [14]. Our methoduses both the spatial analysis of the interferogram and the phase shifting. This makes it possibleto alleviate usual for spatial- or time-domain-only methods restrictions related to the bandwidth,carrier frequency, noise level, background illumination and intensity variation, and, moreover,accuracy of the phase shifts. In fact, introduced phase shifts are calculated by the algorithm andthus need not be known a priori. The only requirement for these is that they change linearlyover the detector region. Spatial analysis of two interferograms with such a phase differencewill detect its parameters, and then a general phase-shifting algorithm can be used to retrievethe phase.


2. The algorithm

We will describe the interferogram as

I[x] = a[x]+b[x]cos(φ [x]), (1)

where x = (x,y) denotes the position in the recorded image, and square brackets are used toemphasize its discrete nature. We assume the functions a and b to be dependent only on thedetector position x. Sometimes we will omit the argument x for simplicity.

The method starts with recording three interferograms, I0, I1, I2, which differ from each otheronly by a small linear term in the phase

Ii[x] = a+bcos(φi[x]) , i = 0,1,2, (2a)

φ1[x] = φ0[x]+ τ1[x]+ σ1, (2b)

φ2[x] = φ0[x]+ τ2[x]+ σ2, (2c)

where τi[x] = ti ·x are the tilt terms and σi the phase shifts. The interferograms can be obtainedby, for instance, tilting the reference mirror in the interferometer by a small random angle.Again, we assume that a and b can be considered the same for all three interferograms.

For every pixel with the coordinates x, we have three equations (2), and we have five un-knowns to find: a,b,φ0,δ1 = τ1 + σ1,δ2 = τ2 + σ2. In phase shifting interferometry, δ1 andδ2 with τ1,τ2 = 0, are known, and equations (2) can be solved for every pixel to obtain the(wrapped) phase φ0. For instance, the following identity:

(I1 − I2)cosφ +(I2 − I0)cos(δ1 + φ)+ (I0 − I1)cos(δ2 + φ) = 0

can be used to find the phase as

φ = arctanI2 − I1 +(I0 − I2)cosδ1 +(I1 − I0)cosδ2

(I0 − I2)sinδ1 +(I1 − I0)sinδ2. (3)

In our case, the phase shifts δ1 and δ2 are supposed to be unknown and different for everypixel. Moreover, for those pixels where one or both of the phase shifts are equal or close to inte-ger numbers of 2π , the number of equations in system (2) is reduced from three to two, or one.However, as will be shown later, for the rest of the object area, the set of three interferogramscontains enough information to determine the tilt and piston terms, and thus to make system (2)solvable.

The following method is proposed to find the unknown terms. Consider the differences ofthe interferogram Id,1 and Id,2:

Id,1 = I1 − I0 = b(cos(φ + τ1 + σ1)− cos(φ))

andId,2 = I2 − I0 = b(cos(φ + τ2 + σ2)− cos(φ)),

which we can write as

Id,1 = −2bsinτ1 + σ1

2sin

(φ [x]+

τ1 + σ1

2

), (4a)

Id,2 = −2bsinτ2 + σ2

2sin

(φ [x]+

τ2 + σ2

2

). (4b)

Then, the set of zero-crossing points for each of the differences consists of two groups: the firstone is determined by the term sin τi+σi

2 when τi + σi = 2kπ , and the second contains the points


Fig. 1. Three original interferograms I0, I1, I2 obtained with a Twyman-Green interferome-ter. The first interferogram corresponds to the original object; additional tilts were intro-duced in the second and third interferograms. Note that all three interferograms containclosed fringes.

where φ + τi+σi2 = 2kπ , where k is an integer. The first group is not dependent on a,b,φ and

forms parallel lines. It can be fully described by three parameters:

θ , λ , s, (5)

where θ is the common normal, λ is the separation distance, and s is the distance fromthe origin. The algorithm proceeds by finding these parameters. Then t can be found as2πλ (cosθ ,sinθ ) and σ = 2πs

λ . This gives the phase shift values δi for every pixel, which canbe substituted in equations (2) or directly in the phase formula (3).

The next section will describe the details of the practical implementation of the algorithmand the method for extracting tilt and piston values from the set of zero-crossing points.

3. Practical implementation and examples

In this section we describe and illustrate one of possible implementations of the algorithm.We begin with the three interferograms shown in Fig. 1. Subtracting the interferograms, we

obtain the images shown in Fig. 2. Note the regions of middle gray, corresponding to zerodifferences.

To extract the zero-crossing lines, we used a threshold-based method in which the point isconsidered to be zero-crossing if the corresponding value is closer to zero than some chosenvalue. Fig. 3 shows the zero lines extracted in this step. Points extracted from each differentialimage form two sets: one with parallel lines, and one repeating the fringes of a ”half-tilted”interferogram.

To determine the angle of parallel lines, their separation and shift relative to the origin, weused an algorithm based on the Hough transform [18]. The Hough transform maps any point(x,y) onto a sinusoidal function ρ = xcosθ + ysinθ . The key property of the Hough transformis that sinusoidals in Hough space associated with points lying on the same line have a com-mon point of intersection, say (ρ0,θ0). This line has a normal (cosθ0,sinθ0) and is shifted bydistance ρ0 from the origin.

In practice, the Hough space is limited to (ρ ,θ )∈ [−R,R]× [0,π) for some R (the maximumdistance of any image point from the origin). Then it is quantized with finite steps δ ρ = R

m andδθ = π

n and represented by a matrix of size (2m+ 1)×n

A =(Ai, j

), i = −m, . . . ,m, j = 0,n−1,


Fig. 2. Differences of the interferograms Id,1 = I1 − I0 and Id,2 = I2 − I0. Black and whitecorrespond to minimum (−255) and maximum (255) levels, respectively. The circle marksthe mask. Masking the object edges helps to avoid spurious zero level points.

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Fig. 3. Points in interferogram differences which are close to zero are used as zero levels.The threshold was chosen to be 1, the minimal possible level that is greater then 0.

which is called an accumulator. The rows of A represent the angle θ ; the columns, the radiusvector ρ . The sinusoidals for every feature pixel in Fig. 3 are “drawn” in the accumulator asfollows. Initially, every cell of the accumulator is set to zero. Then, for every row number j, thecolumn number i(x,y)( j) is calculated by

i(x,y)( j) =[

xcos( jδθ )+ ysin( jδθ )δρ

− 12

],

where the square brackets denote the integer part. The element A i(x,y)( j), j is then incremented byone. The result of the Hough transform is shown in Fig. 4. The intersections of a large numberof sinusoidals are clearly visible.

After the Hough transform has been calculated, the parameters (5) are found as follows.Find the maximum element M in the A and use θM and ρM corresponding to its position aslines normal θ and shift from origin s. To find the separation distance λ , consider the row


of the accumulator corresponding to θM (Fig. 5) and find the average distance between itslocal maxima, which have values greater than αM for some threshold α (typical values are0,3− 0,7, see Fig. 7). To reduce influence of the noise, the algorithm replaces the values ofnearby positioned maxima with their average.

The maxima at the center can potentially reach higher values than maxima near the edges,due to the circular shape of the aperture. To compensate the effect, we can normalize the accu-mulator by dividing each cell by its maximum possible value (see Fig. 6). This makes maximamore uniform and facilitates locating them with the threshold method.

To test the values found for the added tilts one can divide the differential interferogram (4)by the first term −2sin τi+σi

2 , to obtain

Id,1 = bsin(

φ [x]+τ1 + σ1

2

)and (6a)

Id,2 = bsin(

φ [x]+τ2 + σ2

2

), (6b)

shown in Fig 8. With proper values for the tilt the picture should be as smooth as the originalinterferogram except in the vicinity of the parallel lines, where sin τi+σi

2 is close to zero.Now the phase φ can be found. Replacing the formula (3), one can use a simpler identity

sin(φ + α) = cos(φ + β )sin(α −β )+ cos(α −β )sin(φ + β )

to write

cos(φ + β ) =sin(φ + α)− cos(α −β )sin(φ + β )

sin(α −β ).

Thus, with α = δ1/2 = τ1+σ12 and β = δ2/2 = τ2+σ2

2 ,

bcos(

φ [x]+τ2 + σ2

2

)=

Id,1 − cos( τ1+σ1−τ2−σ22 )Id,2

sin( τ1+σ1−τ2−σ22 )

. (7)

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Fig. 4. The differential images Id,1 and Id,2 in the Hough space. R = 354, δρ = 1, δθ =π/360. One can see that zero-level points of Id,1 should contain about 8 lines at angle θabout 170π/360, and Id,2 – about 14 lines at angle approximately equal to 70π/360.


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0102030405060

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Fig. 5. Accumulators’ rows containing the maximum elements. The local maxima corre-sponds to the sinusoidal intersection points. The highest maxima corresponds to parallellines in zero-crossing curves. The second raw of pictures shows the maxima values fornormalized accumulator.

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0.150.2

0.250.3

0.35

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Fig. 6. Normalized accumulators’ rows containing the maximum elements and a norm func-tion we have used. The norm function is the same for every horizontal line in the Houghspace.

This, together with (6b), allows us calculate the phase φ + δ2/2 as

φ +τ2 + σ2

2= arctan

( Id,1 − cos( τ1+σ1−τ2−σ22 )Id,2

sin( τ1+σ1−τ2−σ22 )

, Id,2

). (8)

The result is presented in Fig. 9.

4. Discussion

4.1. Artifacts in the extracted phase

The extracted phase in Fig. 9 contains visible errors along the lines where either δ 1, or δ2, orδ1 − δ2 is equal to 2kπ , k ∈ Z, that is, where the system (2) is badly defined. Although theseartifacts introduce a small rms error, they can seriously affect the unwrapped phase. They canbe removed either by introducing a fourth interferogram with tilt δ 3 in the algorithm and calcu-lating the phase as the median of the results of three possible tilt combinations (see Fig. 10), orby obtaining a and b from the system (2) and then calculating the phase φ by substituting theirsmoothed by low-pass filter values in equation (1) or just by using robust and noise-immunephase unwrapping algorithms.

To illustrate the error caused by the artifacts, we used the phase extracted with the Fourier


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2� ,

170, 118, 66, 14, 38,1792, 140

845,

201, 172, 144, 115, 87, 84, 58, 30, 1, 27, 56, 58, 84, 112, 140, 169

�

Fig. 7. Lines detected by the Hough transform. The threshold α for maxima selection wasset to 0.45 for the first set of zero-level points and to 0.25 for the second one.

Fig. 8. Interferogram differences divided by −2sin τi+σi2 .

transform method from an interferogram with a large tilt (Fig. 11). Note the good correspon-dence between the two extracted phases.

4.2. Initial phase

If the phase originally has a large linear carrier, its fringes are close to the set of parallel linesthemselves, and this can interfere with phase shift detection. In this case some initial estimateson introduced tilts are needed.

4.3. Optimal tilt range

As the tilt information is derived from spatial analysis based on periodicity, the linear termshould change at least several wavelengths over the detector area. The more lines are containedin the zero-level points, the more accurate are the calculated parameters (5). For very small


Fig. 9. Extracted phase φ + τ2+σ22 .

tilt values, the lines in the set of zero-level points will become too thick, even for minimalthreshold values (Fig. 12), and the Hough transform can give a wrong value for the line slope.On the other hand, for large tilts, the zero-level points due to the second terms in (4) begin toapproach parallel lines with the same slope, making tilt detection difficult. Thus the algorithmworks best for those tilt values for which Fourier-transform based methods fail.

4.4. Ambiguity in the detection of phase-shift signs

The Hough transform detects the normal θ only in the range from 0 to π . It is clear from theequations (4), that four different pairs of phase shifts {±δ 1,±δ2} produce parallel lines withidentical parameters (5). Thus the algorithm does not see the difference between interferogramswith “positive” and “negative” tilts, and formula (8) provides, generally speaking, four differentphase distributions. Two of them are the “real” phase accurate to the sign, and two the “faulty”phase, which can be recognized by its almost binary shape (see Fig. 13). The algorithm cannotdetermine the sign of the real phase, as the interferograms (2) are the same for φ 0,δ1,δ2 and−φ0,−δ1,−δ2, and some additional information is necessary. In practice, one can limit allpossible tilts only to those increasing in y-direction, which correspond to θ ∈ (0,π) and excludethe tilt signs ambiguity.

4.5. Computational effectiveness and accumulator resolution

The accuracy of the extracted tilts depends on the angle and radius resolution of the accumula-tor. To fill the accumulator with angle step δθ = π/n, one needs O(n×N) operations, where Nis the number of points in the zero-level set. To speed up the calculation one can first estimatethe tilt with low resolution, and then fill only the small region of the accumulator near the max-imum with high resolution. Note that too high resolution in ρ makes the algorithm too sensitiveto noise and to quantization of the interferograms. For the zero-level points from Fig. 12, forinstance, low resolution in ρ and high resolution in θ is necessary.


Fig. 10. The fourth and fifth interferograms, the phase calculated for other tilt values, andthe resulting phase obtained by median.

Fig. 11. An interferogram with a linear carrier, the phase extracted with Fourier-transformmethod used as a reference, and wrapped difference between the reference and the medianphase from Fig. 10.

4.6. Robustness

The algorithm does not depend on a and b as long as they are the same for all the interferograms.In regions where b is close to zero or in cases with high noise level of the detector, spuriouszero-level points appear and decrease the accuracy. The algorithm also fails on overexposedinterferograms or on interferograms recorded with a non-linear camera.

5. Conclusions

We have proposed a new algorithm for phase retrieval from three interferograms that havedifferent unknown tilt terms. The method is independent of the carrier frequency and the exact


Fig. 12. A small tilt will give a wrong slope.

Fig. 13. If one of the tilt signs is incorrect, formula (8) produces unrealistic phase map.

control of the phase shifts. The method can be used in applications which use inexpensive ornon-linear actuators or a control system with such characteristics.

Acknowledgments

The authors would like to thank Mirjam J. Nieman for checking English usage.


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