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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 8 — Apr. 17, 2006
  • pp: 3204–3213
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Half-quadratic cost function for computing arbitrary phase shifts and phase: Adaptive out of step phase shifting

Mariano Rivera, Rocky Bizuet, Amalia Martinez, and Juan A. Rayas  »View Author Affiliations


Optics Express, Vol. 14, Issue 8, pp. 3204-3213 (2006)
http://dx.doi.org/10.1364/OE.14.003204


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Abstract

We present a phase shifting robust method for irregular and unknown phase steps. The method is formulated as the minimization of a half-quadratic (robust) regularized cost function for simultaneously computing phase maps and arbitrary phase shifts. The convergence to, at least, a local minimum is guaranteed. The algorithm can be understood as a phase refinement strategy that uses as initial guess a coarsely computed phase and coarsely estimated phase shifts. Such a coarse phase is assumed to be corrupted with artifacts produced by the use of a phase shifting algorithm but with imprecise phase steps. The refinement is achieved by iterating alternated minimization of the cost function for computing the phase map correction, an outliers rejection map and the phase shifts correction, respectively. The method performance is demonstrated by comparison with standard filtering and arbitrary phase steps detecting algorithms.

© 2006 Optical Society of America

1. Introduction

Ikr=akr+bkrcos(fr+Δk)+ηkr,
(1)

where r = [x,y]T denotes a pixel position in the image lattice L, ak is the background illumination component, bk is the fringe contrast, f is the unknown phase, Δ = {Δ1, Δ2,…, ΔK} is the phase shift vector and ηk represents additive independent noise. We assume that a normalization process is performed on the fringe pattern so that the illumination components are for filtering out: thus we have that ak ≈ 0 and bk is estimated. Therefore the normalized fringe pattern, Î, is approximated by

Ikr=bkcos(fr+Δk)+η̂kr;
(2)

where it is preferable to left b in the model because we do not want to perform division by zero b values (or noise). However, without lost of generality, in the rest of the paper we assume b ≈ 1. and drop it from the rest of the derivation.

η^ is a remainder residual of the background illumination and the additive noise with low frequency bandwidth and unknown distribution. Such a kind of normalization can be achieved by means of a procedure as the reported in [1

1. J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30, 3018–3020 (2005). [CrossRef] [PubMed]

].

Recently there has been a thorough research on methods for recovering phase from a single closed fringe pattern (see [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

] and references therein). However PS methods are preferably used in stable acquisition conditions (i.e. the temporal dependency of the illumination components, a and b, is eliminated) and the phase shifts, Δ, can be introduced with high accuracy. In such a case the wrapped phase , can be recovered with very simple algorithms; where = W(f) ≝ f + 2πn for an integer n such that ∈(-π,π] (where W denotes the wrapping operator). For instance, for a low level noise, can be computed from 4 fringe patterns (by assuming Δk = /2) with:

f̂r=tan1(I4rI2rI1rI3r).
(3)

In order to compute the real phase, it is needed to unwrap the calculated wrapped phase. Here we denote by f = W -1() the unwrapped phase computed with the unwrapping operator W -1. However the phase unwrapping process is an ill posed problem because a wrapped phase may correspond to multiple unwrapped phases, i.e. there exist many f such that = W(). For this reason, the unwrapping operator W -1 is, in general, implemented as the minimization of a regularized cost function, see for instance [4

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004). [CrossRef] [PubMed]

].

In this paper we address the problem of computing the phase maps from noisy, arbitrary shifted fringe patterns. Before to introduce our approach, we present in section 2 a brief review of standard procedures that consists of estimating the phase shifts follows by the use of a generalized phase shifting method (based on least square procedures) for computing the phase. In section 3, we proposed a robust method for the simultaneous computation of the phase map and the phase shifts. Our method if formulated as the minimization of a half-quadratic cost function that reduces the outliers contribution to the final results. A set of experiments, designed to demonstrate our method performances, are presented in section 4. Our conclusion are finally given in section 5.

2. Out of step phase shifting

In this section we presents a brief review of methods for out of step phase shifting, i.e. methods foe computing the phase from fringe patterns with arbitrary phase shifts. In this case, the phase can be computed with the Bruning-Grievenkamp algorithm by using arbitrary (but known) phase shifts [5

5. J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and D.J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974). [CrossRef] [PubMed]

][6

6. J.E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

] (section 14.8.2 in [7

7. J.E. Grievenkamp and J.H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501–598.

]):

minfk=1KrLHkr2(f;Δ),
(4)

where

Hkr(f;Δ)=Ikrcos(fr)cos(Δk)+sin(fr)sin(Δk);
(5)

cos(δIJb)=corrIJI.*JIJ(I.*II2)12(J.*JJ2)12,
(6)

where .* denotes the componentwise product of vectors and ⟨∙⟩ denotes the mean intensity value of the whole interferogram. More recently Cai et al. proposed:

δIJc=2sin1(π2IJ2b)
(7)

for computing the relative phase shift between the interferograms I and J [18

18. L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003). [CrossRef] [PubMed]

]; where b is the illumination contrast component (assumed equal for both interferograms). An alternative method was reported by Larkin in [17

17. K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001). [CrossRef] [PubMed]

]. Therein, Larkin noted that the shifts estimated by his method (but the same is valid for (6), (7) or a similar method) should be refined by using the phase computed with the generalized phase step method (4). That leads one to an iterative procedure that is iterated until an accurate result is computed. This iterative procedure has the disadvantages of computing from scratch at each iteration, both: The phase and the phase shifts. Moreover the problem of adjusting, simultaneously, phase and shifts is ill-posed. To understood this claim, one can consider the case of 3 fringe patterns with unknown phase shifts, in such a case one have more unknowns (illumination components, phase and shifts) than data (that are, besides, corrupted with noise).

Last ill-posed problem could be solved by a minimum last squares procedure that implies to increase the number of computed fringe patterns with the subsequent increment in the complexity of the experimental procedure; or by using regularization techniques that implies to incorporate prior knowledge about the solution.

The estimation by least squares is equivalent to the maximum likelihood estimation when additive Gaussian residuals are assumed, η^ [or η if one assumes model (1)]. As it is well known, the accuracy of least–squares based methods is reduced by small signal to noise rations (snr) and by non-gaussian residuals. A source for deviations from Gaussian residuals is a non-sinusoidal pure fringe profile. A significant deviation of the residual from the Gaussianity produces phase artifacts as fringe harmonics [23

23. C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A 12, 1997–2008 (1995). [CrossRef]

]–[26

26. B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997). [CrossRef] [PubMed]

]. Therefore, as it is noted in [7

7. J.E. Grievenkamp and J.H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501–598.

], the best performance is achieved with uniformly spaced, into the interval (-π, π], phase shifts.

Other authors that have reported methods for the joint estimation of phase and shifts are Okada et al. [27

27. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991). [CrossRef]

] and Marroquin et al. [28

28. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters for multi-phase stepping images,” Opt. Lett. 23, 238–240 (1998). [CrossRef]

]. In particular, in [27

27. K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991). [CrossRef]

] three least-squares fitting procedures are proposed for computing an initial phase, the phase map and the phase steps, respectively. In Ref. [28

28. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters for multi-phase stepping images,” Opt. Lett. 23, 238–240 (1998). [CrossRef]

] there was proposed a method for computing unknown phase steps, therein is shown that the steps are easily computed, with a closed formula, if the quadrature fringe pattern set is known. Thus, the method in [28

28. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters for multi-phase stepping images,” Opt. Lett. 23, 238–240 (1998). [CrossRef]

] proposed a complexus nonlinear optimization procedure for, simultaneously, computing: the quadrature fringe pattern set, the local frequency and the corresponding phase shifts.

3. Adaptive out of step phase shifting

In this section we present a method that unifies in a single procedure the two main ingredients in out of step phase shifting: The computation of the phase shifts and the phase field. Our method is based on the minimization of a regularized cost function and guarantees convergence.

In particular, we present a half-quadratic cost function that uses as data the fringe pattern and, guesses for the phase shifts and the phase map. Then by assuming that such initial guesses are close enough to the correct values, we formulate as unknowns the corrections for the phase shifts and phase field. Half-quadratic regularization is a framework for formulating robust (in the real statistical sense of robust estimators) cost function that can be minimized by iterating solutions of linear systems and/or closed formulas. In such a formulation an outlier rejection mechanism is incorporated in order to weight the contribution of the data to the final estimation [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

][4

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004). [CrossRef] [PubMed]

][29

29. D. Geman and G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

]–[33

33. M. Rivera and J.L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image and Vision Computing 21, 345–357 (2003). [CrossRef]

].

First, we consider that a good guess δ = {δ 1,δ 2,…,δK } of the unknown phase steps, Δ is available. Therefore, we have

Δk=δk+αk,
(8)

where α = {α 1,α 2,…,αK } are the unknown phase step residuals. If we expect small residual phases, then a coarse estimation of the real phase can be computed by means of the minimization of (4) and an unwrapping method [4

4. M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004). [CrossRef] [PubMed]

]. Thus, as in the phase shift case, we have the follows relationship between the real phase, f, and its coarse estimation, ψ:

fr=ϕr+ψr.
(9)

where ϕ is an unknown residual phase field that corrects artifacts produced by noise and the residual steps, α. Thus by substituting (8) and (9) in the normalized model (2), we obtain

Ikrcos(ψr+δk+ϕr+αk).
(10)

Remember that we are assuming bk ≈ 1 for simplifying the presentation, but the fringe contrast term can and should be used in the procedure.

The task here is to compute such a residual phase, ϕr , and therefore to compute the real phase, f. For such a purpose we need to estimate the phase steps residuals, α.

Recently was reported an efficient method for computing the residual phase from a coarse one for a single fringe pattern: Algorithm 1 in [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

]. Such method transforms an, originally, no-linear optimization problem in a sequence of quadratic optimization problems. Here we extend such a formulation for computing the real phase, ψ + ϕ, and the true phase steps, δ + α, from a phase shifted pattern sets. Following [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

], we assume that |ϕr + αk | is relatively small such that the first order Taylor series can be used to define the residual error:

EkrϕαIkrcos(ψr+δk)+(ϕr+αk)sin(ψr+δk)0.
(11)

Note that this residual error can also be obtained from (10) by using the trigonometric identity cos(x + y) = cosxcosy - sinxsiny [as in the Grievenkamp’s residual error, Eq. (5)] and then by assuming y small enough such that cosy ≈ 1 and sinyy; thus cos(x + y) ≈ cosx-ysinx. Precisely, the advantage of formulating our problem as the computation of the small unknown corrections (instead of the whole value) is that the residual error can be approximated by the linear Taylor expansion Ekr . It is precautious to expect that there could be some places (pixels) where, actually, the residual error were large. Large residual error can be the result of a corrupted initial phase (for instance, phase corrupted with fringe harmonics), an unprecise initial phase shifts or data corruption (for instance, occlusions in the illumination). Then we incorporate a mechanism for weighting the contribution of the data in the estimation process. Such outliers rejection mechanism is implemented by a half-quadratic potential controlled by a parameter that avoids an over–detection or under-detection of atypical data.

As was discussed in section 2, if no additional data are used for solving the ill posed problem of, then prior knowledge should be. We assume that unknown real phase f is smooth in the sense of have small second derivatives. Thus we penalize the correction phase ϕ to produces a smooth f. Therefore, we propose to compute the phase correction field, ϕ, the phase step corrections, α, and an outliers detection field, ω, by an alternating quadratic minimization of the cost function:

U(ϕ,α,ω)=k=1KrL[ωr2Ekr2ϕα+μ(1ωr)2]+γ[k=1Kαk2+rLϕr2]
+λq,r,sL[ψq+ϕq2(ψr+ϕr)+ψs+ϕs]2,
(12)

The cost function (12), that extends the previously proposed in [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

], for dealing with a set of fringe pattern as data. Moreover, we have included two terms (weighted by the parameter γ) that enforce small values for the step correction vector, α, and the residual phase field, ϕ. Such terms improved significantly the convergence and stability of the minimization process. The details of the phase refinement procedure are formalized in Algorithm 1. It is important to note in Algorithm 1 that once a residual (ϕ or α) is computed, then the corresponding base variable (ψ or δ) is updated. Such a strategy reduces iteratively the value of the unknown residual and the fitness of the first order Taylor approach, and consequently the algorithm performance.

Algorithm 1 Adaptive out of step phase shifts.

Let g = {g 1,g 2,… ,gK } a fringe pattern set with expected phase steps equal to δ and ψ an initial coarse phase.

  1. Set, initially ϕ = 0, ω = 1 and given ε > 0;
  2. For all the pixels rR:
  3. while ||g - cosψ|| > ε do
  4. Compute ψr = ψr + ϕr and then set ϕr = 0;
  5. Compute α = argminα U(ϕ = 0, ω, α); {use (13)}
  6. Update δ = δ + α and then set α = 0;
  7. Compute ω = argminω U(ϕ = 0, ω, α = 0); {use (14)}
  8. Compute ϕ = argminϕ U(ϕ, ω, α = 0);
  9. {see Appendix A in [2

    2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

    ]}
  10. end while
  11. Finish with results ψ and δ;

δdc=argmindDI0cos(ψ̂+d)22,

where D = {di = 2πi/N}, for i = 0,1,2,…,N - 1, is a N steps set (we use N = 20 in our experiments). Alternatively, the method in Eqs. (6) or (7) can be used for the same purpose. Taking into account that the accuracy of these methods are reduced if the fringe patterns are corrupted by additive independent noise, see experiments in Fig. 4. Nevertheless how the δdc is estimated, we initialize the coarse phase by assigning: ψψ + δdc . On the other hand, the steps δk ’s can be initialized with the ideal values or, to reduce the risk of large α residuals, can be estimated (as δdc ).

Fig. 1. First row: Synthetic fringe pattern set. Second row: Reconstruction using the computed phase map and the computed phase steps.
Fig. 2. Coarse solution computed with Eq. (3) by assuming ideal steps (π/2). From left to right: Wrapped phase, unwrapped phase computed with the convex algorithm in [4] and its cosine (reconstructed fringe pattern).

In the following we present details of the partial minimizations (steps 5,7 and 8) in Algorithm 1. First, we note that, for ϕ = 0 and ω fixed, cost function (12) can be written as:

U(ϕ=0,ω,α)=r[ωr2k(gkrcosψ̂kr+αksinψkr)2+γαk2]+Q(ϕ=0,ω),

where ψ̂krψr + δk and the potential Q(∙) contains the α–independent terms. Equating to zero the partial gradient with respect to (w.r.t.) α and solving for αk , we obtain a closed formula for computing the α’s optimum coefficients:

αk=rωr2sinψ̂kr(cosψ̂krgkr)γ+rωr2sin2ψ̂kr.
(13)

In a similar way, for ϕ = 0 and α = 0, we obtain a closed formula for computing the ω field:

ωr=μμ+k(gkrcosψ̂kr)2.
(14)

Note that the weight is close to one for the sites (pixels) where the model fits well the data and is close to zero for a large error fit. Finally, ϕ is obtained by solving the linear system that results of equaling to zero the partial gradient w.r.t. ϕ of U(ϕ, α = 0, ω), keeping α = 0 and keeping fixed ω. In particular, we use a Gauss-Seidel scheme similar to the one proposed in the Appendix A of [2

2. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

].

Fig. 3. From left to right: Rewrapped phase (for illustration purposes), unwrapped phase and its cosine: Results computed with the proposed method (first row) and by smoothing, with a thin plate regularization filter [32], the coarsely computed phase in Fig. 2

4. Experiments

First experiment demonstrates the method performance in synthetic noisy test data. First row in Fig. 1 shows the noisy fringe pattern set generated from a synthetic phase that has low and high frequencies: a slight tilt with sharp Gaussian peaks. Then a coarse phase map in Fig. 2 is computed with (3) by assuming regular phase steps equal to π/2 [Fig 4(a)]. The real random phase shifts are plotted in Fig. 4(b). As one can note in the Fig. 2 phase artifacts, correlated with the fringe pattern, corrupt the computed phase. Such a coarse phase is used as initial guess for the proposed method. The results are shown in the first row of Fig. 3. The proposed method recovers the wide bandwidth phase by smoothing spurious artifacts and preserving real high frequencies. Moreover the method recovers effectively the phase steps [see the phaser plot in Fig. 4(c)]. We performed experiments for different size (cardinality) of the fringe pattern set, K. As it is expected, the method performance is improved as K grows given that cost function (12) effectively incorporates redundant information in the fringe pattern set. On the other hand, a simple low–pass filtering computed by the minimization a thin plate potential,

Up(ϕ)=rL(ψrϕ)2+λqrsL[ϕq2ϕr+ϕs]2,

despite redundant information. Thus, the spurious artifacts are not eliminated and real high frequencies are over–smoothed, see second row in Fig. 3. Fig. 4. shows obtained results from an electronic speckle pattern interferometry (ESPI) set. The fringe pattern corresponds to a steel plate under mechanical stress.

Fig. 4. Phaser plots: (a) Ideal phase shifts (with steps equal to π/2), (b) real phase shifts, (c) phase shifts computed with the proposed method and (d) phase shift computed with the method in [18].

5. Conclusion

We have presented a phase shifting robust method for irregular unknown phase steps. The method is based on a half-quadratic regularized phase refinement strategy. The method takes advantage of the redundant information in the phase shifted fringe pattern set for smoothing out artifacts produced by miss–calibrated phase steps and for preserving real high frequencies. The method, implemented as successive quadratic minimizations of a nonlinear cost function, guarantees convergence to a local minimum and is computationally efficient. The method performance was demonstrated by experiments.

The authors were partially supported by CONACYT, Mexico: M. Rivera (grants 40722 and 46270) and R. Bizuet (Scholarship).

Fig. 5. Real data experiment: (a) An original ESPI fringe pattern, (b) computed wrapped phase with a standard four steps (assuming phase steps equal to π/2), (c) refined phase computed with the proposed algorithm (rewrapped for illustration purposes) and (d) computed phase shifts.

References and links

1.

J. A. Guerrero, J. L. Marroquin, M. Rivera, and J. A. Quiroga, “Adaptive monogenic filtering and normalization of ESPI fringe patterns,” Opt. Lett. 30, 3018–3020 (2005). [CrossRef] [PubMed]

2.

M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22, 1170–1175 (2005). [CrossRef]

3.

M. Rivera, J.L. Marroquin, S. Botello, and M. Servin, “A robust spatio-temporal quadrature filter for multi-phase stepping,” Appl. Opt. 39, 284–292 (2000). [CrossRef]

4.

M. Rivera and J.L. Marroquin, “Half-quadratic cost functions for phase unwrapping,” Opt. Lett. 29, 504–506 (2004). [CrossRef] [PubMed]

5.

J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White, and D.J. Brangaccio, “Digital wavefront measuring interferometer for testing optical surfaces and lenses,” Appl. Opt. 13, 2693–2703 (1974). [CrossRef] [PubMed]

6.

J.E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23, 350–352 (1984).

7.

J.E. Grievenkamp and J.H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501–598.

8.

C.J. Morgan, “Least squates estimation in phase-measurement interferometry,” Opt. Lett. 7368–370 (1982). [CrossRef] [PubMed]

9.

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

10.

C. S. Vikram, W. K. Witherow, and J. D. Trolinger, ”Algorithm for phase-difference measurement in phase-shifting interferometry,” Appl. Opt. 32, 6250–6252 (1993). [CrossRef] [PubMed]

11.

C. Wei, M. Chen, and Z. Wang, “General phase-stepping algorithm with automatic calibration of phase steps,” Opt. Eng. 38, 1357–1360 (1999). [CrossRef]

12.

H. van Brug, “Phase-step calibration for phase-stepped interferometry,” Appl. Opt. 383549–3555 (1999). [CrossRef]

13.

M. Chen, H. Guo, and C. Wei,“Algorithm immune to tilt phase-shifting error for phase-shifting interferometers,” Appl. Opt. 39, 3894–3898 (2000). [CrossRef]

14.

W. Li and X. Su, “Real-time calibration algorithm for phase shifting in phase-measuring profilometry,” Opt. Commun. 40, 761–766 (2001).

15.

O. Soloviev and G. Vdovin, “Phase extraction from three and more interferograms registered with different unknown wavefront tilts,” Opt. Express 13, 3743–3753 (2005) [CrossRef] [PubMed]

16.

K.A. Goldberg and J. Bokor, “Fourier-transform method of phase-shift determination,” Appl. Opt. 402886–1894 (2001) [CrossRef]

17.

K. Larkin, “A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns,” Opt. Express 9, 236–253 (2001). [CrossRef] [PubMed]

18.

L. Z. Cai, Q. Liu, and X. L. Yang, “Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps,” Opt. Lett. 28, 1808–1810 (2003). [CrossRef] [PubMed]

19.

A. Patil, B. Raphael, and P. Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 29, 1381–1383 (2004). [CrossRef] [PubMed]

20.

A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferometry using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004) [CrossRef] [PubMed]

21.

A. Patil and P. Rastogi, “Nonlinear regression technique applied to generalized phase-shifting interferometry,” J. Mod. Opt. 52, 573–582 (2005). [CrossRef]

22.

14. C.J. Tay, C. Quan, and L. Chen, “Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift,” Appl. Opt. 44, 1401–1409 (2005). [CrossRef] [PubMed]

23.

C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A 12, 1997–2008 (1995). [CrossRef]

24.

K. Hibino, B.F. Oreb, D.I. Farrant, and K.G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” J. Opt. Soc. Am. A 12, 761–768 (1995). [CrossRef]

25.

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

26.

B. Zhao and Y. Surrel, “Effect of quantization error on the computed phase of phase-shifting measurements,” Appl. Opt. 36, 2070–2075 (1997). [CrossRef] [PubMed]

27.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84, 118–124 (1991). [CrossRef]

28.

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters for multi-phase stepping images,” Opt. Lett. 23, 238–240 (1998). [CrossRef]

29.

D. Geman and G. Reynolds, “Constrained restoration and the recovery of discontinuities,” IEEE Trans. Image Process. 14, 367–383 (1992).

30.

M.J. Black and A. Rangarajan, “Unification of line process, outlier rejection, and robust statistics with application in early vision,” Int. J. Comput. Vis. 19, 57–91 (1996). [CrossRef]

31.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computer imaging,” IEEE Trans. Image Process. 6, 298–311 (1997). [CrossRef] [PubMed]

32.

M. Rivera and J.L. Marroquin, “Adaptive rest condition potentials: Second order edge-preserving regularization,” Comput. Vision Image Understand. 88, 76–93 (2002). [CrossRef]

33.

M. Rivera and J.L. Marroquin, “Efficient half-quadratic regularization with granularity control,” Image and Vision Computing 21, 345–357 (2003). [CrossRef]

OCIS Codes
(100.2650) Image processing : Fringe analysis
(100.3020) Image processing : Image reconstruction-restoration
(100.3190) Image processing : Inverse problems
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Image Processing

History
Original Manuscript: February 22, 2006
Revised Manuscript: April 4, 2006
Manuscript Accepted: April 11, 2006
Published: April 17, 2006

Citation
Mariano Rivera, Rocky Bizuet, Amalia Martinez, and Juan A. Rayas, "Half-quadratic cost function for computing arbitrary phase shifts and phase: Adaptive out of step phase shifting," Opt. Express 14, 3204-3213 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-8-3204


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References

  1. J. A. Guerrero, J. L. Marroquin, M. Rivera, J. A. Quiroga, "Adaptive monogenic filtering and normalization of ESPI fringe patterns," Opt. Lett. 30, 3018-3020 (2005). [CrossRef] [PubMed]
  2. M. Rivera, "Robust phase demodulation of interferograms with open or closed fringes," J. Opt. Soc. Am. A 22, 1170-1175 (2005). [CrossRef]
  3. M. Rivera, J.L. Marroquin, S. Botello and M. Servin, "A robust spatio-temporal quadrature filter for multi-phase stepping," Appl. Opt. 39, 284-292 (2000). [CrossRef]
  4. M. Rivera and J.L. Marroquin, "Half-quadratic cost functions for phase unwrapping," Opt. Lett. 29, 504-506 (2004). [CrossRef] [PubMed]
  5. J.H. Bruning, D.R. Herriot, J.E. Gallagher, D.P. Rosenfeld, A.D. White and D.J. Brangaccio, "Digital wavefront measuring interferometer for testing optical surfaces and lenses," Appl. Opt. 13, 2693-2703 (1974). [CrossRef] [PubMed]
  6. J.E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
  7. J.E. Grievenkamp and J.H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara ed. (John Wiley & Sons, Inc. New York, 1992) pp. 501-598.
  8. C.J. Morgan, "Least squates estimation in phase-measurement interferometry," Opt. Lett. 7368-370 (1982). [CrossRef] [PubMed]
  9. G, Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822- (1991) [CrossRef]
  10. C. S. Vikram, W. K. Witherow, and J. D. Trolinger, "Algorithm for phase-difference measurement in phaseshifting interferometry," Appl. Opt. 32, 6250-6252 (1993). [CrossRef] [PubMed]
  11. C. Wei, M. Chen, Z. Wang, "General phase-stepping algorithm with automatic calibration of phase steps," Opt. Eng. 38, 1357-1360 (1999). [CrossRef]
  12. H. van Brug, "Phase-step calibration for phase-stepped interferometry," Appl. Opt. 383549-3555 (1999). [CrossRef]
  13. M. Chen, H. Guo, and C. Wei,"Algorithm immune to tilt phase-shifting error for phase-shifting interferometers," Appl. Opt. 39, 3894-3898 (2000). [CrossRef]
  14. W. Li and X. Su, "Real-time calibration algorithm for phase shifting in phase-measuring profilometry," Opt. Commun. 40, 761-766 (2001).
  15. O. Soloviev and G. Vdovin, "Phase extraction from three and more interferograms registered with different unknown wavefront tilts," Opt. Express 13, 3743-3753 (2005) [CrossRef] [PubMed]
  16. K.A. Goldberg and J. Bokor, "Fourier-transform method of phase-shift determination," Appl. Opt. 402886-1894 (2001) [CrossRef]
  17. K. Larkin, "A self-calibrating phase-shifting algorithm based on the natural demodulation of two-dimensional fringe patterns," Opt. Express 9, 236-253 (2001). [CrossRef] [PubMed]
  18. L. Z. Cai, Q. Liu, and X. L. Yang, "Phase-shift extraction and wave-front reconstruction in phase-shifting interferometry with arbitrary phase steps," Opt. Lett. 28, 1808-1810 (2003). [CrossRef] [PubMed]
  19. A. Patil, B. Raphael and P. Rastogi, "Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search," Opt. Lett. 29, 1381-1383 (2004). [CrossRef] [PubMed]
  20. A. Patil, R. Langoju, and P. Rastogi, "An integral approach to phase shifting interferometry using a superresolution frequency estimation method," Opt. Express 12, 4681-4697 (2004) [CrossRef] [PubMed]
  21. A. Patil and P. Rastogi, "Nonlinear regression technique applied to generalized phase-shifting interferometry," J. Mod. Opt. 52, 573 - 582 (2005). [CrossRef]
  22. 14 C.J. Tay, C. Quan, L. Chen, "Phase retrieval with a three-frame phase-shifting algorithm with an unknown phase shift," Appl. Opt. 44, 1401-1409 (2005). [CrossRef] [PubMed]
  23. C. Rathjen, Statistical properties of phase-shift algorithms, J. Opt. Soc. Am. A 12, 1997-2008 (1995). [CrossRef]
  24. K. Hibino, B.F. Oreb, D.I. Farrant, and K.G. Larkin, "Phase shifting for nonsinusoidal waveforms with phaseshift errors," J. Opt. Soc. Am. A 12, 761-768 (1995). [CrossRef]
  25. Y. Surrel, "Design of algorithms for phase measurements by the use of phase stepping," Appl. Opt. 35, 51-60 (1996). [CrossRef] [PubMed]
  26. B. Zhao and Y. Surrel, "Effect of quantization error on the computed phase of phase-shifting measurements," Appl. Opt. 36, 2070-2075 (1997). [CrossRef] [PubMed]
  27. K. Okada, A. Sato, and J. Tsujiuchi, "Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry," Opt. Commun. 84, 118-124 (1991). [CrossRef]
  28. J. L. Marroquin, M. Servin and R. Rodriguez-Vera, "Adaptive quadrature filters for multi-phase stepping images," Opt. Lett. 23, 238-240 (1998). [CrossRef]
  29. D. Geman and G. Reynolds, "Constrained restoration and the recovery of discontinuities," IEEE Trans. Image Process. 14, 367-383 (1992).
  30. M.J. Black and A. Rangarajan, "Unification of line process, outlier rejection, and robust statistics with application in early vision," Int. J. Comput. Vis. 19, 57-91 (1996). [CrossRef]
  31. P. Charbonnier, L. Blanc-F´eraud, G. Aubert and M. Barlaud, "Deterministic edge-preserving regularization in computer imaging," IEEE Trans. Image Process. 6, 298-311 (1997). [CrossRef] [PubMed]
  32. M. Rivera and J.L. Marroquin, "Adaptive rest condition potentials: Second order edge-preserving regularization," Comput. Vision Image Understand. 88, 76-93 (2002). [CrossRef]
  33. M. Rivera, and J.L. Marroquin, "Efficient half-quadratic regularization with granularity control," Image and Vision Computing 21, 345—357 (2003). [CrossRef]

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