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Optics Express

Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 10 — May. 16, 2005
  • pp: 3743–3753
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Phase extraction from three and more interferograms registered with different unknown wavefront tilts

Oleg Soloviev and Gleb Vdovin  »View Author Affiliations


Optics Express, Vol. 13, Issue 10, pp. 3743-3753 (2005)
http://dx.doi.org/10.1364/OPEX.13.003743


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Abstract

We propose phase retrieval from three or more interferograms corresponding to different tilts of an object wavefront. The algorithm uses the information contained in the interferogram differences to reduce the problem to phase shifting. Three interferograms is the minimum for restoring the phase over most of the image. Four or more interferograms are needed to restore the phase over the whole image. The method works with images including open and closed fringes in any combination.

© 2005 Optical Society of America

1. Introduction

The information about a physical quantity being measured interferometrically is contained in the phase term ϕ of the fringe pattern described by (1). The goal of analysis [1

1. D. Malacara, M. Servín, and Z. Malacara, Interferogram analysis for optical shop testing (Marcel Dekker, Inc., New York, Basel, Hong Kong, 1998).

] consists in extracting this phase term. Many widely used phase-shifting methods are based on phase detection of a sinusoidal signal [2

2. Y. Surrel, “Fringe analysis,” in Photomechanics, P. K. Rastogi, ed. (Springer, 1999).

, 3

3. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996). [CrossRef]

, 4

4. D. Malacara, ed., Optical Shop Testing, 2nd ed. (John Wiley & Sons, Inc., 1992).

] and use at least three phase-shifted interferograms. The phase shifts, however, should be controlled with high accuracy. Any error in linearity of an actuator response can introduce an error into the calculated phase. A number of methods referred as self-calibrating techniques has been developed to deal with linear and nonlinear piston-shift errors and tilt-shifts errors. Lai and Yatagai [5

5. G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991). [CrossRef]

] directly calculate phase shifts from additional linear fringes due to an additional tilted reference mirror moved together with the test object. In the methods suggested by Okada et al [6

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). [CrossRef]

] and by Kong and Kim [7

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). [CrossRef]

], the phase distribution and phase shifts are calculated alternatively by iteration of the least-square algorithm. Chen et al [8

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). [CrossRef]

] have presented a modified method which uses the first-order Taylor series expansion of the phase-shifting errors to compensate also the phase error caused by tilt-shift errors. Dobroiu et al [9

9. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef]

] cancel the effects of both phase shift errors and the presence of tilts by calculating average phase shifts for sufficiently small blocks of the interferogram space, so that the phase shifts inside every block can be assumed constant. Based on a contrast-level map, the algorithm iteratively adjusts the phase shifts and tilts.

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 to emphasize its discrete nature. We assume the functions a and b to be dependent only on the detector position x. Sometimes we will omit the argument x for simplicity.

The method starts with recording three interferograms, I 0, I 1, I 2, which differ from each other only 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]=t i·x are the tilt terms and σi the phase shifts. The interferograms can be obtained by, 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 unknowns to find: a,b,ϕ0,δ1=τ 1+σ 1,δ 2=τ 2+σ 2. In phase shifting interferometry, δ 1 and δ 2 with τ 1,τ 2=0, are known, and equations (2) can be solved for every pixel to obtain the (wrapped) phase ϕ 0. For instance, the following identity:

(I 1-I 2)cosϕ+(I 2-I0)cos(δ 1+ϕ)+(I 0-I 1)cos(δ 2+ϕ)=0

can be used to find the phase as

ϕ=arctanI2I1+(I0I2)cosδ1+(I1I0)cosδ2(I0I2)sinδ1+(I1I0)sinδ2.
(3)

In our case, the phase shifts δ 1 and δ 2 are supposed to be unknown and different for every pixel. Moreover, for those pixels where one or both of the phase shifts are equal or close to integer 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 interferograms contains 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 of the interferogram Id,1 and Id,2:

Id,1=I1-I0=b(cos(ϕ+τ11)-cos(ϕ))

and

Id,2=I2-I 0=b(cos(ϕ+τ 2+σ 2)-cos(ϕ)),

which we can write as

Id,1=2bsinτ1+σ12sin(ϕ[x]+τ1+σ12),
(4a)
Id,2=2bsinτ2+σ22sin(ϕ[x]+τ2+σ22).
(4b)

Then, the set of zero-crossing points for each of the differences consists of two groups: the first one is determined by the term sin τi+σi2 when τi+σi=2, and the second contains the points 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 from the origin. The algorithm proceeds by finding these parameters. Then t can be found as 2πλ(cosθ,sinθ) and σ=2πsλ. This gives the phase shift values δi for every pixel, which can be substituted in equations (2) or directly in the phase formula (3).

Fig. 1. Three original interferograms I 0, I 1, I 2 obtained with a Twyman-Green interferometer. The first interferogram corresponds to the original object; additional tilts were introduced in the second and third interferograms. Note that all three interferograms contain closed fringes.

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 zero differences.

To extract the zero-crossing lines, we used a threshold-based method in which the point is considered to be zero-crossing if the corresponding value is closer to zero than some chosen value. Fig. 3 shows the zero lines extracted in this step. Points extracted from each differential image 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, we used an algorithm based on the Hough transform [18

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).

]. The Hough transform maps any point (x,y) onto a sinusoidal function ρ=xcosθ+ysinθ. The key property of the Hough transform is that sinusoidals in Hough space associated with points lying on the same line have a common point of intersection, say (ρ 0,θ 0). This line has a normal (cosθ 0, sinθ 0) and is shifted by distance ρ 0 from the origin.

In practice, the Hough space is limited to (ρ,θ)∊[-R,R]×[0,π) for some R (the maximum distance of any image point from the origin). Then it is quantized with finite steps δρ=Rm and δθ=πn and represented by a matrix of size (2m+1)×n

A=(Ai,j), i=-m,…,m, j=0,n-1,

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

i(x,y)(j)=[xcos(jδθ)+ysin(jδθ)δρ12],,

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

Fig. 2. Differences of the interferograms Id,1=I 1-I 0 and I d,2=I 2-I 0. Black and white correspond to minimum (-255) and maximum (255) levels, respectively. The circle marks the mask. Masking the object edges helps to avoid spurious zero level points.
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.

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 as lines 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 its local maxima, which have values greater than αM for some threshold α (typical values are 0,3–0, 7, see Fig. 7). To reduce influence of the noise, the algorithm replaces the values of nearby 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 accumulator by dividing each cell by its maximum possible value (see Fig. 6). This makes maxima more 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 2sinτi+σi2, to obtain

Îd,1=bsin(ϕ[x]+τ1+σ12)and
(6a)
Îd,2=bsin(ϕ[x]+τ2+σ22),
(6b)

shown in Fig 8. With proper values for the tilt the picture should be as smooth as the original interferogram except in the vicinity of the parallel lines, where sin τi+σi2 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(αβ)sin(ϕ+β)sin(αβ)..

Thus, with α=δ12=τ1+σ12 and β=δ22=τ2+σ22,

bcos(ϕ[x]+τ2+σ22)=Îd,1cos(τ1+σ1τ2σ22)Îd,2sin(τ1+σ1τ2σ22).
(7)

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

ϕ+τ2+σ22=arctan(Îd,1cos(τ1+σ1τ2σ22)Îd,2sin(τ1+σ1τ2σ22),Îd,2).
(8)

The result is presented in Fig. 9.

Fig. 4. The differential images I d,1 and I d,2 in the Hough space. R=354, δρ=1, δθ=π/360. One can see that zero-level points of I d,1 should contain about 8 lines at angle θ about 170π/360, and I d,2 — about 14 lines at angle approximately equal to 70π/360.
Fig. 5. Accumulators’ rows containing the maximum elements. The local maxima corresponds to the sinusoidal intersection points. The highest maxima corresponds to parallel lines in zero-crossing curves. The second raw of pictures shows the maxima values for normalized accumulator.
Fig. 6. Normalized accumulators’ rows containing the maximum elements and a norm function we have used. The norm function is the same for every horizontal line in the Hough space.

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 δ 2, or δ 1δ 2 is equal to 2kπ, k∊ℤ, that is, where the system (2) is badly defined. Although these artifacts introduce a small rms error, they can seriously affect the unwrapped phase. They can be removed either by introducing a fourth interferogram with tilt δ 3 in the algorithm and calculating the phase as the median of the results of three possible tilt combinations (see Fig. 10), or by obtaining a and b from the system (2) and then calculating the phase ϕ by substituting their smoothed by low-pass filter values in equation (1) or just by using robust and noise-immune phase unwrapping algorithms.

To illustrate the error caused by the artifacts, we used the phase extracted with the Fourier transform method from an interferogram with a large tilt (Fig. 11). Note the good correspondence between the two extracted phases.

Fig. 7. Lines detected by the Hough transform. The threshold α for maxima selection was set 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 2sinτi+σi2.

4.2. Initial phase

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

4.3. Optimal tilt range

As the tilt information is derived from spatial analysis based on periodicity, the linear term should change at least several wavelengths over the detector area. The more lines are contained in the zero-level points, the more accurate are the calculated parameters (5). For very small tilt values, the lines in the set of zero-level points will become too thick, even for minimal threshold 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 to approach parallel lines with the same slope, making tilt detection difficult. Thus the algorithm works best for those tilt values for which Fourier-transform based methods fail.

Fig. 9. Extracted phase ϕ+τ2+σ22.

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 the equations (4), that four different pairs of phase shifts {±δ 1δ 2} produce parallel lines with identical parameters (5). Thus the algorithm does not see the difference between interferograms with “positive” and “negative” tilts, and formula (8) provides, generally speaking, four different phase 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 cannot determine the sign of the real phase, as the interferograms (2) are the same for ϕ 0,δ 1,δ 2 and -ϕ 0,-δ 1,-δ 2, and some additional information is necessary. In practice, one can limit all possible tilts only to those increasing in y-direction, which correspond to θ∊(0,π) and exclude the 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 accumulator. To fill the accumulator with angle step δθ=π/n, one needs O(n×N) operations, where N is the number of points in the zero-level set. To speed up the calculation one can first estimate the tilt with low resolution, and then fill only the small region of the accumulator near the maximum with high resolution. Note that too high resolution in ρ makes the algorithm too sensitive to noise and to quantization of the interferograms. For the zero-level points from Fig. 12, for instance, 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, and the resulting phase obtained by median.
Fig. 11. An interferogram with a linear carrier, the phase extracted with Fourier-transform method used as a reference, and wrapped difference between the reference and the median phase 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, spurious zero-level points appear and decrease the accuracy. The algorithm also fails on overexposed interferograms or on interferograms recorded with a non-linear camera.

5. Conclusions

We have proposed a new algorithm for phase retrieval from three interferograms that have different unknown tilt terms. The method is independent of the carrier frequency and the exact control of the phase shifts. The method can be used in applications which use inexpensive or non-linear actuators or a control system with such characteristics.

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.

Acknowledgments

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

References and links

1.

D. Malacara, M. Servín, 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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

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). [CrossRef]

9.

A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, “Tilt-compensating algorithm for phase-shift interferometry,” Appl. Opt. 41(13), 2435–2439 (2002). [CrossRef]

10.

T. Kreis, “Digital holographic interference-phase measurement using the Fourier-transform method,” J. Opt. Soc. Am. A 3(6), 847–855 (1986). [CrossRef]

11.

J. A. Quiroga, J. A. Gómez-Pedrero, and á. García-Botella, “Algorithm for fringe pattern normalisation,” Opt. Commun. 197, 43–51 (2001). [CrossRef]

12.

C. Roddier and F. Roddier, “Interferogram Analysis Using Fourier Transform Techniques,” Appl. Opt. 26, 1668–1673 (1987). [CrossRef] [PubMed]

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). [CrossRef]

14.

K. A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 40(17), 2886–2893 (2001). [CrossRef]

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). [CrossRef]

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). [CrossRef]

17.

E. Yu and S. S. Cha, “Two-dimensional regression for interferometric phase extraction,” Appl. Opt. 37(8), 1370–1376 (1998). [CrossRef]

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).

OCIS Codes
(120.2650) Instrumentation, measurement, and metrology : Fringe analysis
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Research Papers

History
Original Manuscript: April 12, 2005
Revised Manuscript: May 4, 2005
Published: May 16, 2005

Citation
Oleg Soloviev and Gleb Vdovin, "Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Express 13, 3743-3753 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-10-3743


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References

  1. D. Malacara, M. Servín, 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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  9. A. Dobroiu, D. Apostol, V. Nascov, and V. Damian, �??Tilt-compensating algorithm for phase-shift interferometry,�?? Appl. Opt. 41(13), 2435�??2439 (2002). [CrossRef]
  10. T. Kreis, �??Digital holographic interference-phase measurement using the Fourier-transform method,�?? J. Opt. Soc. Am. A 3(6), 847�??855 (1986). [CrossRef]
  11. J. A. Quiroga, J. A. Gómez-Pedrero, and �?. García-Botella, �??Algorithm for fringe pattern normalisation,�?? Opt. Commun. 197, 43�??51 (2001). [CrossRef]
  12. C. Roddier and F. Roddier, �??Interferogram Analysis Using Fourier Transform Techniques,�?? Appl. Opt. 26, 1668�??1673 (1987). [CrossRef] [PubMed]
  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). [CrossRef]
  14. K. A. Goldberg and J. Bokor, �??Fourier-transform method of phase-shift determination,�?? Appl. Opt. 40(17), 2886�??2893 (2001). [CrossRef]
  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). [CrossRef]
  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). [CrossRef]
  17. E. Yu and S. S. Cha, �??Two-dimensional regression for interferometric phase extraction,�?? Appl. Opt. 37(8), 1370�??1376 (1998). [CrossRef]
  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).

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