## Constraints in dual phase shifting interferometry

Optics Express, Vol. 14, Issue 1, pp. 88-102 (2006)

http://dx.doi.org/10.1364/OPEX.14.000088

Acrobat PDF (773 KB)

### 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

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

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

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

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

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]

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]

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]

## 2. Dual phase shifting interferometry: methods and their limitations

### 2.1. Tradition approach: a five-frame algorithm

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*is the mean intensity,

_{dc}*V*is the visibility, and φ

_{1}(

*P*) and φ

_{2}(

*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

_{+}= φ

_{1}(

*P*)+ φ

_{2}(

*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

**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]

_{-}= φ

_{1}(

*P*) - φ

_{2}(

*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

#### 2.1.1. Constraints in tradition approach

_{1}(

*P*) - φ

_{2}(

*P*) = (2

_{χ}+ 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

_{+}.

*nʹ*,

*jʹ*) on the CCD be written as

*n*

_{0},

*j*

_{0}) is the origin for the intensity image of

*n*×

*j*pixels corresponding to phase in Eq. (5a). In Eq. (5),

*n*

_{0}is identical in both equations and the centers of the concentric fringes are offset by

*p*

_{0}in y-direction only. Here, Φ

_{ran1}and Φ

_{ran2}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 Φ

_{ran1}= Φ

_{ran2}= 0 and Φ

_{ran1}= Φ

_{ran2}≠ 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, φ

_{ran1}- φ

_{ran2}= (2

_{χ}+ 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 φ

_{1}(

*P*) - φ

_{2}(

*P*) = (2

_{χ}+ 1)π, there is a jump of ±2π, from basic trigonometry we get cos [φ

_{1}(

*P*) - φ

_{2}(

*P*) ± [2

_{χ}+1)π] = cos [φ

_{1}(

*P*) - φ

_{2}(

*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.

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

_{-}in Figs. 2(a)– 2(b) are shown in Figs. 6(a)– 6(b), respectively. The plot in Fig. 7 drawn across a central row shows that whenever φ

_{1}(

*P*)+ φ

_{2}(

*P*) = (2

_{χ}+ 1)π, the discontinuity (shown by R2) is ±2π in Φ

_{-}. This discontinuity is removed in a similar way as explained in the previous paragraph. Figure 7 shows the continuous function

*g*(

*x*) obtained from

*g*(

*x*) = cos(Φ

_{-}). Applying phase shifts of 0, π/2, π, and 3π/2 to

*g*(

*x*) and solving the four phase shifted images results in the estimation of wrapped difference of phases. Figures 8(a)–8(b) show the wrapped difference of phases obtained for fringes in Figs. 2(a)–2(b), respectively.

**31**, 1680–1681 (1992). [CrossRef] [PubMed]

**32**, 3669–3675 (1993). [CrossRef] [PubMed]

### 2.2. Flexible least-squares method

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

**32**, 3669–3675 (1993). [CrossRef] [PubMed]

*I*

_{dc}*V*cosφ

_{1}(

*P*), δ = -

*I*

_{dc}*V*sinφ

_{1}(

*P*), μ =

*I*

_{dc}*V*cosφ

_{2}(

*P*), and

*v*= -

*I*

_{dc}*V*sinφ

_{2}(

*P*). Here,

*N*is the number of data frames. Since Eq. (6) is linear with respect to unknown coefficients

*I*, γ, δ, μ, and

_{dc}*v*, we can use least-squares technique to minimize

*E*(

*P*), defined as

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]

*E*(

*P*)). This is done by setting the first derivative of

*E*(

*P*) with respect to the unknown coefficients

*I*, γ, δ, μ, and

_{dc}*v*, written as

*∂E*/

*∂I*,

_{dc}*∂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,

**B**= [

*I*γ δ

_{dc}*μ*

*v*]

^{T}, and

**X**= [∑

*I*∑

_{n}*I*cos α

_{n}_{n}∑

*I*sin α

_{n}_{n}∑

*I*cos β

_{n}_{n}∑

*I*sinβ

_{n}_{n}]

^{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

*I*, γ, δ,

_{dc}*μ*, and

*v*, and subsequently, in the determination of φ

_{1}and φ

_{2}. The sum and difference of phases can then be obtained using

#### 2.2.1. Constraints in flexible least-squares method

**A**is non-singular. It has also been observed that the use of sequential phase steps such as

*n*α and

*n*β in matrix

**A**results in the determinant to be zero. In practical situations although the successive phase steps cannot be exact multiples of

*n*α and

*n*β, 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.

**7**, 368–370 (1982). [CrossRef] [PubMed]

*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

*n*α and

*n*β. Hence, the matrix for determining phases φ

_{1}and φ

_{2}can be written in the form

_{0},β

_{0}), (α

_{1},β

_{1}),..,and (α

_{N-1},β

_{N-1}) are phase steps for frames

*I*

_{0},

*I*

_{1},

*I*

_{2},…,and

*I*

_{N-1}, respectively. The phase distribution φ

_{1}and φ

_{2}are subsequently computed from the argument of ℓ and ℘, where ℓ = 0.5

*I*

_{dc}*V*exp(

*j*(φ

_{1}) and ℘ = 0.5

*I*

_{dc}*V*exp(φ

_{2}). Here, ∗ denotes the complex conjugate.

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

*et al*. [17

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]

### 2.3. Signal processing approach

*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]

*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

*I*in Eq. (2) into a complex frequency domain by taking its Z-transform. Let the Z-transform of

_{n}*I*be denoted by

_{n}**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*).

^{th}order harmonics can be written as

*a*and

_{k}*b*are the complex Fourier coefficients of the κ

_{k}^{th}order harmonic;

*j*= √-1. Let us rewrite Eq. (11) in the following form

_{k}=

*a*exp(

_{k}*jk*φ

_{1}),

*u*= exp(

_{k}*jk*α), ℘

_{k}=

*b*exp(

_{k}*jk*φ

_{2}), and ϑ

_{k}= exp(

*jk*α). The task now reduces to designing an annihilation filter of the form

**P**(

*z*), represented as

**P**

_{n}, with the intensity signal, represented as

**I**

_{n}, vanishes identically, and can be written as

**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

**12**, 4681–4697 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

#### 2.3.2. Constraints in signal processing approach

## 3. Optimizing phase shifts by signal processing approach

*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 φ

_{1}and φ

_{2}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 φ

_{1}and φ

_{2}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 φ

_{1}and φ

_{2}can be reasonably estimated. Hence, there is a need to set up guidelines for the selection of the phase steps.

### 3.1. Cramér-Rao bound for holographic moiré

**I**ʹ =

**I**+ η) denote the vector consisting of measured intensities in the presence of noise η, where

**I**= [

*I*

_{0}

*I*

_{1}

*I*

_{2}∙

*I*

_{N-1}]

^{T}. The vector

**I**ʹ is characterized by the probability density function

*p*(

**I**ʹ;Ψ) =

*p*(

**I**ʹ), where Φ is the set of unknown parameters in the moiré fringes. In the present case, Φ = (

*I*

_{dc1}

*I*

_{dc2},

*V*

_{1},

*V*

_{2},φ

_{1},φ

_{2},φ

_{1},φ

_{2},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}=

*I*and

_{dc}*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

*E*{

^{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]

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

_{r}(any rth element of

*r*) is an unbiased estimator of deterministic ψ

^{th}_{r}, based on

**I**ʹ, whose CRB is given by

**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 σ

^{2}, the probability density function

*p*(

**I**ʹ;Ψ) is defined by the mean and variance of the noise. Thus, the joint probability of the vector

**I**ʹ is given by

**J**

^{-1}. Simplifying Eq. (18) by taking the logarithmic function (note that log

*p*is asymptotic to

*p*(

**I**ι; Ψ)), we obtain

_{s}. Therefore Eq. (15) for the

*r*and

^{th}*s*element in Ψ can be written as

^{th}_{r}) ≥

**J**

^{-1},

_{r,r}. In the present example the typical Fisher Information matrix will be

*N*, let us first look at the following simple calculation. Suppose that, we have a variable

*q*= Ψ

^{T}

**U**, which represents the difference between the phase steps α and β, then the variance of q is given by

**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}.

### 3.2. Results of CRB analysis

_{1}and φ

_{2}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.

*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 rad

^{2}using the annihilation method.

*N*= 20. Figure 10 shows that as the separation between the phase steps α and β increases, their values can be reliably estimated at lower SNRs. Figures 10(e)–10(f) show that as the separation between the phase steps increases the bounds obtained by annihilation filter method reaches the theoretical bounds given by the CRB at much lower SNR as compared to that obtained in Figs. 10(a)–10(d). This analysis once again reemphasizes the fact that larger the separation between the phase steps, more reliably is the phase obtained at lower SNRs. Similar analysis can also be performed for other algorithms.

## 4. Conclusion

## Acknowledgments

## References and links

1. | P. K. Rastogi, “Phase shifting applied to four-wave holographic interferometers,” Appl. Opt. |

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 |

3. | P. K. Rastogi, “Phase shifting four wave holographic interferometry,” J. Mod. Opt. |

4. | C. J. Morgan, “Least squares estimation in phase-measurement interferometry,” Opt. Lett. |

5. | J. E. Grievenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. |

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 |

7. | A. Patil and P. Rastogi, “Subspace-based method for phase retrieval in interferometry,” Opt. Express |

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 |

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 |

10. | A. Patil and P. Rastogi, “Maximum-likelihood estimator for dual phase extraction in holographic moiré,” Opt. Lett. |

11. | D. C. Rife and R. R. Boorstyn, “Single-tone parameter estimation from discrete-time observations,”IEEE Transactions on Information Theory |

12. | M. Marcus and H. Minc, “Vandermonde Matrix,”in |

13. | |

14. | |

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 |

17. | Abhijit Patil, Benny Raphael, and Pramod Rastogi, “Generalized phase-shifting interferometry by use of a direct stochastic algorithm for global search,” Opt. Lett. |

**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

- P. K. Rastogi, "Phase shifting applied to four-wave holographic interferometers," Appl. Opt. 31, 1680-1681 (1992). [CrossRef] [PubMed]
- 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]
- P. K. Rastogi, "Phase shifting four wave holographic interferometry," J. Mod. Opt. 39, 677-680 (1992). [CrossRef]
- J. E. Grievenkamp, "Generalized data reduction for heterodyne interferometry," Opt. Eng. 23, 350-352 (1984).
- 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]
- 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]
- 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]
- 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]
- A. Patil and P. Rastogi, "Maximum-likelihood estimator for dual phase extraction in holographic moiré," Opt. Lett. 17, 2227-2229 (2005). [CrossRef]
- M. Marcus and H. Minc, "Vandermonde Matrix," in A Survey of Matrix Theory and Matrix Inequalities (Dover, New York, 1992), pp. 15-16.
- <a href= http://www.mathworks.com/>http://www.mathworks.com/</a>.
- <a href= http://mathworld.wolfram.com/VandermondeMatrix.html>http://mathworld.wolfram.com/VandermondeMatrix.html</a>.
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