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Optics Express

  • Editor: Michael Duncan
  • Vol. 14, Iss. 1 — Jan. 9, 2006
  • pp: 88–102
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Constraints in dual phase shifting interferometry

Abhijit Patil, Rajesh Langoju, and Pramod Rastogi  »View Author Affiliations


Optics Express, Vol. 14, Issue 1, pp. 88-102 (2006)
http://dx.doi.org/10.1364/OPEX.14.000088


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Abstract

The paper presents approaches based on traditional phase shifting, flexible least-squares, and signal processing methods in dual phase shifting interferometry primarily applied to holographic moiré for retrieving multiple phases. The study reveals that these methods cannot be applied straightforward to retrieve phase information and discusses the constraints associated with these methods. Since the signal processing method is the most efficient among these approaches, the paper discusses significant issues involved in the successful implementation of the concept. In this approach the knowledge of the pair of phase steps is of paramount interest. Thus the paper discusses the choice of the pair of phase steps that can be applied to the phase shifting devices (PZTs) in the presence of noise. In this context, we present a theoretical study using Cramér-Rao bound with respect to the selection of the pair of phase step values in the presence of noise.

© 2006 Optical Society of America

1. Introduction

Holographic moiré is an efficient technique for the nondestructive evaluation of rough objects. The method functions by integrating two holographic interferometers within one system. The result is produced in the form of a moiré. However, the complex nature of the interferometer and the beat inherent in the process precludes the use of standard phase shifting procedures to holographic moiré [1–3

1. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

]. It has been shown previously that the design of special phase shifting procedures to holographic moiré allows for simultaneously determining the information regarding the out-of-plane and in-plane displacement components [1–3

1. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

]. In this configuration, the phase terms corresponding to the sum of phases (carrier) and difference of phases (moiré) carry information regarding the out-of-plane and in-plane displacement components, respectively. This capability adds new dimension to the measurement of displacement components and in the nondestructive testing of rough objects.

Information carried by the carrier and moiré can be decoded by either the traditional phase shifting approach or by least-squares fit techniques or, by the introduction of signal processing concepts in phase shifting interferometry. In the traditional approach, we refer to three-frame, five-frame, and seven-frame algorithms [1–2

1. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

] which allow for accommodating dual PZTs. By least-squares fit technique, we extend the least squares fit data reduction method proposed by Morgan [4

4. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef] [PubMed]

] and Grievenkamp [5

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

] to accommodate dual phase shifting devices PZTs. On the other hand, the signal processing approach allows for applying high resolution frequency estimation techniques such as polynomial rooting (annihilation method) [6

6. A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferome-try using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

], MUltiple-SIgnal Classification (MUSIC) [7

7. A. Patil and P. Rastogi, “Subspace-based method for phase retrieval in interferometry,” Opt. Express 13, 1240–1248 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1240. [CrossRef] [PubMed]

], Minimum-Norm (min-norm) [8

8. A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13, 4070–4084 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070. [CrossRef] [PubMed]

], Estimation of Signal Parameters via Rotational Invariance Technique (ESPRIT) [9

9. A. Patil and P. Rastogi, “Rotational invariance approach for the evaluation of multiple phases in interferometry in presence of nonsinusoidal waveforms and noise,” J. Opt. Soc. Am. A 9, 1918–1928 (2005). [CrossRef]

], and the Maximum-Likelihood Estimation (MLE) [10

10. A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. 17, 2227–2229 (2005). [CrossRef]

] to optical configurations including two PZTs.

However, these approaches have inherent limitations and cannot be applied straightforward to configurations such as holographic moiré for the simultaneous extraction of two orthogonal displacement components. For instance, in the traditional approach, the retrieval of information is not straightforward and the phase map corresponding to the carrier yields a phase pattern corrupted by moiré and vice-versa. Hence, additional step needs to be associated to the measurement to yield uncorrupted phase maps. On the other hand, in both the least squares fit technique and the signal processing approach, though the phase steps imparted by the PZTs can be arbitrary, the selection of phase steps in the presence of noise is crucial to the successful implementation of the concept.

In this context, the objective of this paper is to discuss in detail the constraints encountered while following these approaches. We believe that understanding the conditions that are necessary to employing these algorithms is of paramount importance for the successful implementation of the proposed concept. Since, the signal processing approach is the most efficient among these approaches, we present a statistical analysis using Cramér-Rao lower bound [11

11. D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,”IEEE Transactions on Information Theory IT–20, 591–598 (1974). [CrossRef]

] for the selection of the optimal pair of phase steps in the presence of noise. Once the phase steps have been estimated within an allowable accuracy, the Vandermonde system of equations [12

12. M. Marcus and H. Minc, “Vandermonde Matrix,”in A Survey of Matrix Theory and Matrix Inequalities (Dover, New York), pp. 15–16 (1992).

] can be designed for the determination of phases.

As a case study, we will consider the five-frame algorithm (traditional approach) [2

2. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

] and the annihilation method (spectral estimation approach) [6

6. A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferome-try using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

]. The performance of other estimators such as MUSIC, MIN-NORM and ESPRIT can also be studied in a similar way. The present study is expected to provide the guidelines for applying dual phase shifting procedures to holographic moiré.

The paper is organized as follows. Section 2 presents the three approaches and discusses the constraints associated with each of these approaches. Section 3 presents the Cramér-Rao bound (CRB) analysis for the estimation of phase steps and studies the performance of the annihilation method in estimating the phase steps with respect to noise.

2. Dual phase shifting interferometry: methods and their limitations

2.1. Tradition approach: a five-frame algorithm

Figure 1 shows the configuration for the holographic moiré setup.

Fig. 1. Schematic of holographic moiré.

The intensity equation for moiré is given by [2

2. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

]

I(P)=Idc{1+V[cosφ1(P)+cosφ2(P)]}
(1)

where, Idc is the mean intensity, V is the visibility, and φ1(P) and φ2(P) are the interference phases at a point P on the object surface along the two arms of the interferometer. The sum of phases can be extracted by introducing appropriate phase shifts in the two arms of the interferometer and recording the corresponding intensities. Equation (1) thus becomes

I(P)=Idc{1+V[cos(φ1(P)+α)+cos(φ2(P)+β)]}
(2)

where, α and β are the phase steps introduced in the two arms of the interferometer. In order to recover the sum of phase term Φ+ = φ1(P)+ φ2(P), we apply pairs of phase steps (-2α, -2β), (-α,-β), (0,0), (α,β), and (2α,2β) to the PZTs. Frames I 1,I 2,I 3, I 4, and I 5 corresponding to these phase steps are recorded in the computer [1

1. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

, 2

2. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

]. Assuming, α = β = π/2, the term corresponding to the sum of phases is given by

Φ+=2tan1[2(I2I4)2I3I1I5],forφ1(P)φ2(P)(2χ+1)π
(3)

where, χ is some integer constant. Similarly, in order to recover the difference of phase term, Φ- = φ1(P) - φ2(P), we apply pairs of phase steps (-2α,2β), (-α,β), (0,0), (α, -β), and (2α, -2β) to the PZTs. Frames I 1, I 2, I 3, I 4, and I 5 corresponding to these phase steps are recorded in the computer. Assuming, α = β = π/2, the term corresponding to the difference of phases is given by

Φ=2tan1[2(I2I4)2I3I1I5],forφ1(P)φ2(P)(2χ+1)π
(4)

2.1.1. Constraints in tradition approach

Although α = β = π/2 in Eqs. (3) and (4) minimizes the first order calibration errors in PZTs, the conditions in Eqs. (3) and (4) are the main constraints when it comes to extracting the wrapped phases. For instance, in Eq. (3) whenever, φ1(P) - φ2(P) = (2χ + 1)π, the equation becomes indeterminate. Because of the arctangent operator this indeterminacy manifests itself as a discontinuity and Φ+ jumps by either +2π or - 2π depending upon the sign. Hence, while extracting the wrapped sum of phases, the fringes corresponding to moiré are also seen to modulate the wrapped phase pattern, Φ+.

The above mentioned phenomenon is better understood by simulating the moiré fringes. Let the phase terms recorded at a pixel (,) on the CCD be written as

φ1nʹjʹ=2π(nʹn0)2+(jʹj0)2λ1+Φran1
(5a)
φ2nʹjʹ=2π(nʹn0)2+(jʹp0)2λ2+Φran2
(5b)

where, (n 0, j 0) is the origin for the intensity image of n × j pixels corresponding to phase in Eq. (5a

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

). In Eq. (5

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

), n 0 is identical in both equations and the centers of the concentric fringes are offset by p 0 in y-direction only. Here, Φran1 and Φran2 represent the random phase terms (0 to 2π) because of the rough nature of the object surface. Assuming the visibility V to be unity, Figs. 2(a) and 2(b) show the fringe pattern (512 × 512 pixels) corresponding to Eq. (2), under the assumption Φran1 = Φran2 = 0 and Φran1 = Φran2 ≠ 0, respectively. The wrapped phase maps corresponding to Φ+ using Eq. (3) for Figs. 2(a) and 2(b) are shown in Figs. 3(a) and 3(b), respectively. The figures show that the information carried by the sum of phases is corrupted by the moiré fringes. The figures are plotted without taking into consideration the constraints in Eq. (3). Figure 4 shows a typical plot along a row in Fig. 3(a). From the plot it can be observed that whenever, φran1 - φran2 = (2χ + 1)π, the discontinuity (shown by R1) is ±2π in Φ+. However, this discontinuity can be removed by processing the wrapped phase term Φ+ using a computer. An efficient way to perform this task is to resample the wrapped phase using the cosine operator. Since, at φ1(P) - φ2(P) = (2χ + 1)π, there is a jump of ±2π, from basic trigonometry we get cos [φ1(P) - φ2(P) ± [2χ +1)π] = cos [φ1(P) - φ2(P)]. Hence, the discontinuity is removed. In Fig. 4 f(x) represents the continuous function obtained from f(x) = cos(Φ+). However, to make the procedure compatible with most of the commercially available unwrapping softwares, the computer generated phase steps can be imparted to f(x). For instance, phase shifts of 0, π/2, π, and 3π/2 can be applied to extract the wrapped phase. Figures 5(a)–5(b) shows that the discontinuities due to constraints in Eq. (3) are removed.

Fig. 2. Moiré fringes (512 × 512 pixels) corresponding to Eq. (1). In (a) the random phases Φran1 = Φran2 = 0, while in (b)Φran1 = Φran2 ≠ 0.
Fig. 3. The map corresponding to the wrapped sum of phases φ1 + φ2 obtained using Eq (3) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b).

The same phenomenon is also observed for the three-frame [1

1. P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

] and seven-frame [2

2. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

] algorithms. The point which needs to be emphasized here is that the straightforward adaptation of these algorithms to dual phase stepping is not possible. Moreover, these algorithms are sensitive to nonsinusoidal wavefronts (a consequence of detector nonlinearity or multiple reflections inside the laser cavity, or the phase shifter itself). These algorithms also do not offer any flexibility either in the selection of phase step values α and β or in the use of non-collimated wavefronts for phase shifting.

2.2. Flexible least-squares method

The use of least squares fitting techniques has assumed greater significance in simple phase shifting interferometry [4

4. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef] [PubMed]

, 5

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

] because of its ability to allow arbitrary phase steps. The concept of least-squares fit has primarily been shown to be effective in configurations involving a single PZT. It can, however, be extended to holographic moiré shown in Fig. 1, albeit, with some constraints. The equation for holographic moiré defined by Eq. (2

2. P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

) can be written as

In=Idc+γcosαn+δsinαn+μcosβn+νsinβnforn=0,1,2,N1
(6)

where, γ = Idc Vcosφ1(P), δ = -Idc Vsinφ1(P), μ = Idc Vcosφ2(P), and v = -Idc Vsinφ2(P). Here, N is the number of data frames. Since Eq. (6) is linear with respect to unknown coefficients Idc, γ, δ, μ, and v, we can use least-squares technique to minimize E(P), defined as

Fig. 4. Typical plot for wrapped sum of phases φ12 (in radians) shown in Fig. 3(a). In plot the pixels in the central row from pixel (256, 0) till pixel (256, 127) is shown. In the plot, R1 shows the discontinuity in phase since φ1 - φ2 = ±2π.
Fig. 5. The map corresponding to the wrapped sum of phases φ12 obtained using Eq. (3) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b).
E(P)=n=0N1[Idc+γcosαn+δsinαn+μcosβn+νsinβnIn]2
(7)

For the best fit, the error function in Eq. (7

7. A. Patil and P. Rastogi, “Subspace-based method for phase retrieval in interferometry,” Opt. Express 13, 1240–1248 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1240. [CrossRef] [PubMed]

) should be minimized (minE(P)). This is done by setting the first derivative of E(P) with respect to the unknown coefficients Idc, γ, δ, μ, and v, written as ∂E/∂Idc, ∂E/γ, ∂E/∂δ, ∂E/∂μ, and ∂E/∂v, respectively, equal to zero. The resulting equations can be written in the matrix form as B = A -1 X, where,

A=[Ncosαnsinαncosβnsinβncosαncos2αncosαnsinαncosαncosβncosαnsinβnsinαncosαnsinαnsin2αnsinαncosβnsinαnsinβncosβncosαncosβnsinαncosβncos2βnsinβncosβnsinβncosαnsinβnsinαnsinβnsinβncosβnsin2βn]
(7)

B = [Idc γ δ μ v]T, and X = [∑InIncos αnInsin αnIncos βnInsinβn]T. Here the simulation is carried out from n = 0 to n = N- 1. The solution to the above matrix equation results in the determination of the unknown coefficients Idc, γ, δ, μ, and v, and subsequently, in the determination of φ1 and φ2. The sum and difference of phases can then be obtained using

Fig. 6. The map corresponding to the wrapped difference of phases φ12 obtained using Eq. (4) for the fringe map in (a) Fig. 2(a) and (b) Fig. 2(b). The phase map shows the recurrence of the carrier fringes while extracting the moiré.
Fig. 7. Typical plot for wrapped difference of phases φ12 (in radians) shown in Fig. 6(a). In plot the pixels in the central row from pixel (256, 0) till pixel (256, 511) is shown. In the plot, R2 shows the discontinuity in phase since φ12 = ±2π.
Fig. 8. The figure shows the final wrapped difference of phases obtained after removing the φ12 = ±2π discontinuity. Figure (a) shows the wrapped difference of phases for the fringe map in (a) Fig. 2(a) while (b) shows the wrapped phase for fringe in Fig. 2(b).
Φ+=tan1δμ+νγδνμγ
(9a)
Φ=tan1γν+μδγμνδ
(9b)

2.2.1. Constraints in flexible least-squares method

The main constraint in flexible least-squares method is that the phase steps should be carefully selected such that matrix A is non-singular. It has also been observed that the use of sequential phase steps such as nα and nβ in matrix A results in the determinant to be zero. In practical situations although the successive phase steps cannot be exact multiples of nα and nβ, the resulting matrix in this case will be nearly singular or poorly conditioned. Therefore, if the least squares technique is applied, then only few arbitrary phase steps should be selected such that the matrix A is non-singular. For instance, if thirteen data frames are acquired and, α = 40° and β = 20°, then five data frames corresponding to frames n = 0,2,5,8 and 10 can be selected in a non-regular order. This would result in matrix A to be non-singular. It is also observed that even if the matrix A is not well conditioned, mathematical tools such as MATLAB [13] may still compute the inverse of the matrix A. Therefore, it is always advisable to check whether the matrix is well conditioned or not.

Hence, the well know generalized data reduction technique proposed by Morgan [4

4. C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef] [PubMed]

] and Grievenkamp [5

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

] cannot be extended straightforward to holographic moiré, since utmost care is required in the selection of the pair of phase steps. The other method by which the generalized holographic moiré can be realized is by designing the Vandermonde system of equations. A Vandermonde matrix usually arises in the polynomial least squares fitting, Lagrange interpolating polynomials, or in the statistical distribution of the distribution moments [14–16]. The solution of N × N Vandermonde matrix requires N 2 operations. The advantage of Vandermonde matrix is that its determinant is always nonzero (hence invertible) for different values of nα and nβ. Hence, the matrix for determining phases φ1 and φ2 can be written in the form

[exp(jα0)exp(jα0)exp(jβ0)exp(jβ)1exp(jα1)exp(jα1)exp(jβ1)exp(jβ1)1exp(jκαN)exp(jκαN)exp(jβN)exp(jβN)1][**Idc]=[I0I1IN1]
(10)

where, (α00), (α11),..,and (αN-1N-1) are phase steps for frames I 0, I 1, I 2,…,and I N-1, respectively. The phase distribution φ1 and φ2 are subsequently computed from the argument of ℓ and ℘, where ℓ = 0.5Idc Vexp(j1) and ℘ = 0.5Idc Vexp(φ2). Here, ∗ denotes the complex conjugate.

2.3. Signal processing approach

In the signal processing approach, a parallelism is drawn between the frequencies that are present in the spectrum and the phase steps imparted to the PZTs. The problem therefore reduces to estimating the frequencies or in other words the phase steps. Once the phase steps have been estimated, the Vandermonde system of equation shown in Eq. (10) can be applied for the extraction of phase distributions. Patil et al. [6–10

6. A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferome-try using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

] have introduced signal processing algorithms to configurations involving single and multiple PZTs. The advantage of the these approaches lies in their ability to compensate for the errors arising due to non-sinusoidal wavefronts and PZT miscalibrations. These algorithms also allow the use of spherical wavefronts and arbitrary phase steps between 0 and π radians. The number of data frames that these methods require is equal to at least twice the number of frequencies that are present in the spectrum. Hence, for dual PZTs and for harmonic κ = 1 (assuming sinusoidal wavefronts), we need at least N = 10 data frames. Since, discussion of these signal processing algorithms is beyond the scope of this paper, let us consider, as a case study, the case of one such algorithm based on the design of an annihilation filter.

2.3.1. Annihilation filter method

In the annihilation filter method, we transform the discrete time domain signal In in Eq. (2) into a complex frequency domain by taking its Z-transform. Let the Z-transform of In be denoted by I(z). The objective is to design another polynomial P(z) termed as annihilation filter which has zeros at frequencies associated with I(z). This in turn would result in I(z)P(z) = 0. The phase steps α and β are estimated by extracting the roots of the polynomial P(z).

In brief, let us consider the presence of multiple-order harmonics in the wavefront and hence, Eq. (2) for κth order harmonics can be written as

In=Idc+k=1κakexp[jk(φ1+)]+k=1κakexp[jk(φ1+)]
k=1κbkexp[jk(φ2+)]+k=1κbkexp[jk(φ2+)];
(11)

where, ak and bk are the complex Fourier coefficients of the κth order harmonic; j = √-1. Let us rewrite Eq. (11) in the following form

In=Idc+k=1κʹkukn+k=1κʹk*(un*)n+k=1κkϑkn+k=1κk*(ϑk*)n
(12)

where, ℓʹk = akexp(jkφ1), uk = exp(jkα), ℘k = bkexp(jkφ2), and ϑk = exp(jkα). The task now reduces to designing an annihilation filter of the form

P(z)=(z1)k=1κ(zuk)(zuk*)(zϑk)(zϑk*)
=k=04κ+2Pkzk
(13)

The multiplication in the frequency domain corresponds to discrete convolution in the time domain. Thus, the discrete convolution of the polynomial P(z), represented as P n, with the intensity signal, represented as I n, vanishes identically, and can be written as

k=04κ+2PkInk=0m{4κ+2,4κ+3,,N1}
(14)

Hence, finding the roots of the polynomial P(z) yields the phase steps α and β. Since the measurement is sensitive to noise this method necessitates acquiring additional data frames as phase steps cannot be estimated reliably at lower SNR’s with minimum required (N = 10) data frames. To reduce the effect of noise, a denoising procedure is suggested in Reference [6] so that phase steps can be estimated even at lower SNR’s. The reader is referred to Reference [6

6. A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferome-try using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

] for details on the annihilation technique and the denoising method.

2.3.2. Constraints in signal processing approach

3. Optimizing phase shifts by signal processing approach

This section provides useful guidelines to optimizing the selection of phase step values obtained by signal processing approach. This study can be performed by deriving the CRB for holographic moiré. The CRB provides valuable information on the potential performance of the estimators. The CRBs are independent of the estimation procedure and the precision of the estimators cannot surpass the CRBs. In this context, we will first derive the CRB for the phase steps α and β as a function of SNR and N. Although the phase steps α and β can be arbitrary for pure intensity signal, the central question is: what is the smallest difference between the phase steps α and β that can be retrieved reliably by any estimator as a function of SNR and N? Hence, we need to determine all the allowable values of phase steps α and β at a particular SNR and N. The reason why we are interested in computing the smallest phase step difference is to provide an experimentalist with a fair idea of phase steps which must be selected at a particular SNR and N, so that the phase values φ1 and φ2 can be estimated reliably. Suppose that inadvertently, the two phase steps are chosen very close to one another and the measurements are performed at a low value of SNR. In such as case, the phase values φ1 and φ2 cannot be estimated reliably. However, for the same value of SNR, if the two phase steps α and β are far apart, the phase values φ1 and φ2 can be reasonably estimated. Hence, there is a need to set up guidelines for the selection of the phase steps.

In order to respond to this query, we will derive the mean square error (MSE) for difference in phase steps α – β as a function of SNR which will indicate the theoretical Cramér-Rao lower bounds to the MSE. The Cramér-Rao bound can then be compared with the MSE obtained by any estimator (in the present case we will study the annihilation filter method). For this, we perform 500 Monte-Carlo simulations at each SNR and the separation between the phase steps α and β is varied from 0 to 100%.

Two scenarios are possible. First, the MSE obtained from the estimator is below the CRB. We attribute this observation to the non-reliability in the estimated phase steps and discard those values of separation between α and β in which the MSE is below the CRB. Second, the MSE of the estimator is above the CRB. In such a case, we infer that closer the MSE of the estimator is to the CRB, the more efficient is the estimator. Of course, for an unbiased estimator, the MSE obtained by the estimator can never reach the theoretical CRB for limited number of sample points. In the present study, we derive the CRB for the phase steps and compare the performance of the annihilation filter method with the CRB.

3.1. Cramér-Rao bound for holographic moiré

Let Iʹ = I + η) denote the vector consisting of measured intensities in the presence of noise η, where I = [I 0 I 1 I 2I N-1]T. The vector Iʹ is characterized by the probability density function p(Iʹ;Ψ) = p(Iʹ), where Φ is the set of unknown parameters in the moiré fringes. In the present case, Φ = (I dc1 I dc2,V 1,V 21212,α,β)T, where subscripts 1 and 2 refer individually to the two arms of the interferometer. Note that in Eq. (2), it was assumed that I dc1 = I dc2 = Idc and V 1 = V 2 = V. Therefore, I dc1 and I dc2 and are the average intensities while V 1 and V 2 are the fringe visibilities in the two arms. Now, if Φ^ is an unbiased estimator of the deterministic Ψ, then the covariance matrix of the unbiased estimator, E{Ψ^ Ψ^ T}, where E is an expectation operator, is bounded by its lower value given by [11

11. D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,”IEEE Transactions on Information Theory IT–20, 591–598 (1974). [CrossRef]

]

E{Ψ̂Ψ̂T}J1
(15)

where, J is the Fisher Information matrix. This matrix is defined as [9]

J={[Ψlogp(Iʹ)][Ψlogp(Iʹ)]T}
(15)

In other words, if ψr (any rth element of rth) is an unbiased estimator of deterministic ψr, based on Iʹ, whose CRB is given by

E{Ψ̂r2}Jr,r1,forr=1,2,3,,8
(17)

where, J -1 r,r is the (r, r) element in matrix J -1. Assuming η is the additive white Gaussian noise with zero mean and variance σ2, the probability density function p(Iʹ;Ψ) is defined by the mean and variance of the noise. Thus, the joint probability of the vector Iʹ is given by

pIʹΨ=(12πσ)Nexp[12σ2n=0N1(IʹnIn)2]
(18)

It is well understood that the CRB for each unknown parameter can be determined by observing the diagonal elements of the inverse of the Fisher Information matrix, J -1. Simplifying Eq. (18) by taking the logarithmic function (note that log p is asymptotic to p(Iι; Ψ)), we obtain

logp=constant12σ2n=0N1(IʹnIn)2
(19)

Differentiating Eq. (19) with the rth element in Ψ, say ψr, we obtain,

logpψr=1σ2n=0N1Inψr(IʹnIn)
(20)

Similarly, Eq. (19) can be differentiated with respect to ψs. Therefore Eq. (15) for the rth and sth element in Ψ can be written as

Jr,s=E{1σ4n=0N1Inψr(IʹnIn)l=0N1Ilψr(IʹlIl)}
(21)

Equation (21) can be simplified into the following compact form

Jr,s=1σ2n=0N1InψrInψr
(22)

Finally, the lower bounds are given by variance, var(ψ^ r) ≥ J -1,r,r. In the present example the typical Fisher Information matrix will be

J=[JIdc1,Idc1JIdc1,Idc2JIdc1,V1JIdc1,V2JIdc1,φ1JIdc1,φ2JIdc1,αJIdc1,βJIdc1,Idc1JIdc2,Idc2JIdc2,V1JIdc2,V2JIdc2,φ1JIdc2,φ2JIdc2,αJIdc,βJV1,Idc1JV1,Idc2JV1,V1JV1,V2JV1,φ1JV1,φ2JV1,αJV1,βJV2,Idc1JV2,Idc2JV2,V1JV2,V2JV2,φ1JV2,φ2JV2,αJV2βJV1,Idc1Jφ1,Idc2Jφ1,V1Jφ1,V2Jφ1,φ1Jφ1,φ2Jφ1,αJφ1,βJφ2,Idc1Jφ2,Idc2Jφ2,V1Jφ2,V2Jφ2,φ1Jφ2,φ2Jφ2,αJφ2,βJα,Idc1Jα,Idc2Jα,V1Jα,V2Jα,φ1Jα,φ2Jα,αJα,βJβ,Idc1Jβ,Idc2Jβ,V1Jβ,V2Jβ,φ1Jβ,φ2Jβ,αJβ,β]
(23)

where, the subscripts in Eq. (23) represent derivatives with respect to the unknown parameters. Hence, to determine the allowable values of phase steps α and β as function of SNR and N, let us first look at the following simple calculation. Suppose that, we have a variable q = ΨT U, which represents the difference between the phase steps α and β, then the variance of q is given by

E[qq̂2]=E[(qq̂)(qq̂)T]
=E[UT(ΨΨ̂)(ΨΨ̂)TU]
=UTE[(ΨΨ̂)(ΨΨ̂)T]U
UTJ1U
(24)

The problem, therefore, narrows down to selecting a matrix U such that the minimum difference between α and β is computed. Since all the parameters are unknown, the matrix U is given by U =[0 0 0 0 0 0 - 1 1]T .

Fig. 9. The plots show the MSE for β–α with respect to β–α as percentage of α, and SNR. The plots are shown for data frames N = 11, 15, and 20. The plots a, c, and e, are without the denoising procedure, while plots in b, d, and f are with the denoising step. In the plot the red shade represents the Cramér-Rao lower bound while the yellow shade represents the MSE obtained using the annihilation filter method. MSE is represented in log scale.
Fig. 10. The plots show the bounds for retrieving the phase steps α and β for 0 to 70 dB SNR. The values of β in (b), (d), and (e) are selected as 1.2, 1.5, and 1.8 times of α = π/6, respectively. The line with circle dots shows the error bounds obtained using the annihilation method while the simple line represents the bounds given by the CRB.

3.2. Results of CRB analysis

We now perform 500 Monte-Carlo simulations at each SNR and study the performance of the annihilation filter method in retrieving the difference of the phase steps at a pixel point. During the analysis the phases φ1 and φ2 are kept the same as described earlier in Eqs. (5) and (5b), respectively. Additive white Gaussian noise having SNR between 0 and 70 dB is considered during the simulations. Our analysis consists of two parts.

During the first part, we determine the MSE of β – α as a function of the SNR and the difference between the phase steps α and β as percentage of α. During the analysis we choose α = π/6 and vary the difference of α and β between 0 to 100% of α. Given the fixed value of α, this analysis indicates the reliable values of β which can be selected for a particular SNR and N. It is important to note that the value of α can be selected anywhere between 0 and π radians. However, to keep the analysis simple we have selected just one value of α in the present study. The allowable values of β can be determined by comparing the plot of MSE of β – α, obtained using the annihilation filter method for 500 iterations at each SNR, with respect to the mean square error obtained from the CRB given by Eq. (23). The values in the plot where the MSE of β – α, obtained from the annihilation filter method, goes below the MSE obtained using the CRB, are considered as prohibited for the selection of separation between the phase step values α and β. The test is carried out for data frames N = 11, 15, and 20. The study shows that separation between the phase steps can be reduced as the number of data frames increases. Moreover, the incorporation of denoising procedure substantially improves the allowable lower range of phase step separation. For instance, Fig. 9(f) shows that for the separation between α and β as 70% of a, and for SNR 30 dB, the MSE is 0.01 rad2 using the annihilation method.

4. Conclusion

To conclude, this paper has presented three approaches namely, the traditional approach, the least-squares method, and the signal processing (annihilation filter method) approach in dual phase shifting interferometry. The traditional approach suggests that an additional processing step needs to be incorporated to extract the wrapped phases. The study of the least squares fit and the signal processing approaches reveals that a proper choice of the pair of phase steps is of paramount importance. The analysis using Cramér-Rao bound can act as a guideline to select the optimal pair of phase steps in the presence of noise. We believe that a thorough understanding of the issues associated with each of these approaches will pave the way for a real-time and automated simultaneous measurement of two components of displacement in holographic moiré.

Acknowledgments

The financial support of the Swiss National Science Foundation is gratefully acknowledged.

References and links

1.

P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. 31, 1680–1681 (1992). [CrossRef] [PubMed]

2.

P. K. Rastogi, “Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré,” Appl. Opt 32, 3669–3675 (1993). [CrossRef] [PubMed]

3.

P. K. Rastogi, “Phase shifting four wave holographic interferometry,” J. Mod. Opt. 39, 677–680 (1992). [CrossRef]

4.

C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. 7, 368–370 (1982). [CrossRef] [PubMed]

5.

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

6.

A. Patil, R. Langoju, and P. Rastogi, “An integral approach to phase shifting interferome-try using a super-resolution frequency estimation method,” Opt. Express 12, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

7.

A. Patil and P. Rastogi, “Subspace-based method for phase retrieval in interferometry,” Opt. Express 13, 1240–1248 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1240. [CrossRef] [PubMed]

8.

A. Patil and P. Rastogi, “Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm,” Opt. Express 13, 4070–4084 (2005), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070. [CrossRef] [PubMed]

9.

A. Patil and P. Rastogi, “Rotational invariance approach for the evaluation of multiple phases in interferometry in presence of nonsinusoidal waveforms and noise,” J. Opt. Soc. Am. A 9, 1918–1928 (2005). [CrossRef]

10.

A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. 17, 2227–2229 (2005). [CrossRef]

11.

D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,”IEEE Transactions on Information Theory IT–20, 591–598 (1974). [CrossRef]

12.

M. Marcus and H. Minc, “Vandermonde Matrix,”in A Survey of Matrix Theory and Matrix Inequalities (Dover, New York), pp. 15–16 (1992).

13.

http://www.mathworks.com/.

14.

http://mathworld.wolfram.com/VandermondeMatrix.html.

15.

K. M. Hoffman and R. Kunze, “Linear Algebra, 2nd ed.,” (Englewood Cliffs, New Jersey: Prentice Hall), 1971.

16.

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

17.

Abhijit Patil, Benny Raphael, and Pramod Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. 12, 1381–1383 (2004). [CrossRef]

OCIS Codes
(090.2880) Holography : Holographic interferometry
(120.3180) Instrumentation, measurement, and metrology : Interferometry
(120.4120) Instrumentation, measurement, and metrology : Moire' techniques
(120.5050) Instrumentation, measurement, and metrology : Phase measurement

ToC Category:
Holography

Citation
Abhijit Patil, Rajesh Langoju, and Pramod Rastogi, "Constraints in dual phase shifting interferometry," Opt. Express 14, 88-102 (2006)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-14-1-88


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References

  1. P. K. Rastogi, "Phase shifting applied to four-wave holographic interferometers," Appl. Opt. 31, 1680-1681 (1992). [CrossRef] [PubMed]
  2. P. K. Rastogi, "Phase-shifting holographic moiré: phase-shifter error-insensitive algorithms for the extraction of the difference and sum of phases in holographic moiré," Appl. Opt 32, 3669-3675 (1993). [CrossRef] [PubMed]
  3. P. K. Rastogi, "Phase shifting four wave holographic interferometry," J. Mod. Opt. 39, 677-680 (1992). [CrossRef]
  4. J. E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
  5. 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), <a href= http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681>http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681</a>. [CrossRef] [PubMed]
  6. A. Patil and P. Rastogi, "Subspace-based method for phase retrieval in interferometry," Opt. Express 13, 1240-1248 (2005), <a href= http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1240>http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-4-1240</a>. [CrossRef] [PubMed]
  7. A. Patil and P. Rastogi, "Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm," Opt. Express 13, 4070-4084 (2005), <a href= http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070>http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-11-4070</a>. [CrossRef] [PubMed]
  8. A. Patil and P. Rastogi, "Rotational invariance approach for the evaluation of multiple phases in interferometry in presence of nonsinusoidal waveforms and noise," J. Opt. Soc. Am. A 9, 1918-1928 (2005). [CrossRef]
  9. A. Patil and P. Rastogi, "Maximum-likelihood estimator for dual phase extraction in holographic moiré," Opt. Lett. 17, 2227-2229 (2005). [CrossRef]
  10. M. Marcus and H. Minc, "Vandermonde Matrix," in A Survey of Matrix Theory and Matrix Inequalities (Dover, New York, 1992), pp. 15-16.
  11. <a href= http://www.mathworks.com/>http://www.mathworks.com/</a>.
  12. <a href= http://mathworld.wolfram.com/VandermondeMatrix.html>http://mathworld.wolfram.com/VandermondeMatrix.html</a>.
  13. K. M. Hoffman and R. Kunze, "Linear Algebra, 2nd ed.," (Prentice Hall, 1971).
  14. G. Lai and T. Yatagai, "Generalized phase-shifting interferometry," J. Opt. Soc. Am. A 8, 822-827 (1991). [CrossRef]
  15. Abhijit Patil, Benny Raphael, and Pramod Rastogi, "Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search," Opt. Lett. 12, 1381-1383 (2004). [CrossRef]

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