## Estimation of multiple phases in holographic moiré in presence of harmonics and noise using minimum-norm algorithm

Optics Express, Vol. 13, Issue 11, pp. 4070-4084 (2005)

http://dx.doi.org/10.1364/OPEX.13.004070

Acrobat PDF (4375 KB)

### Abstract

The paper proposes a novel approach for estimating multiple phases in holographic moiré. The need to design such an algorithm is necessitated by the development of optical configurations containing two phase stepping devices, e.g. PZTs, with a view to measure simultaneously two phase distributions. The approach consists of first applying minimum-norm algorithm to extract phase steps imparted to the PZTs. Salient feature of the algorithm lies in its ability to handle nonsinusoidal waveforms and noise. This approach also provides the flexibility of using arbitrary phase steps, a feature most commonly attributed to generalized phase shifting interferometry. Once the phase steps are estimated for each PZT, the Vandermonde system of equations is designed to estimate the phase distributions.

© 2005 Optical Society of America

## 1. Introduction

18. R. Józwicki, M. Kujawinska, and M. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing, Opt. Engg. **31**, 422–433 (1992). [CrossRef]

19. P. de Groot, “Vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A **12**, 354–365 (1995). [CrossRef]

24. K. Kinnnstaetter, A. W. Lohmann, J Schwider, and N. Streibl, “Accuracy of phase shifting interferometry,” Appl. Opt. **27**, 5082–5089 (1988). [CrossRef]

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

29. P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. **2**, 79–103 (1981). [CrossRef]

30. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

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

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

31. R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems **AES-19**, 134–139 (1983). [CrossRef]

32. M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing **ASSP-34**, 331–341 (1986). [CrossRef]

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

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

*N*data frames [31

31. R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems **AES-19**, 134–139 (1983). [CrossRef]

32. M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing **ASSP-34**, 331–341 (1986). [CrossRef]

35. R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoustics, Speech, and Signal Processing **37**, 984–995 (1989). [CrossRef]

*signal*- and

*noise-subspaces*. The phase steps are estimated from the

*noise subspace*. In the following section, we show an important property in which noise subspace is orthogonal to sinusoidals present in the spectrum. Since we draw a parallelism between the frequency present in the spectrum and the phase steps employed by the PZT’s, this property is exploited to extract the latter. Once the phase step values of the two PZT’s are estimated pixelwise, the Vandermonde system of equations are applied to estimate the phase distribution.

## 2. Min-Norm algorithm for holographic moiré

*N*phase shifted moiré images measured at a point (

*x*,

*y*) on the CCD. Usually, an autocovariance matrix is formulated from small overlapping fragments of the data sets using sliding window technique known as spatial smoothing. Equation governing the spatial smoothing technique will be presented in Section 3. Subsequently, an autocovariance matrix is eigen-decomposed to yield the

*signal*- and

*noise subspaces*. The derivation [36–37] in the following section shows that the signal and noise spaces are orthogonal to each other. Therefore, the phase step values of the two PZT’s are estimated pixel-wise from the

*noise subspace*. The following paragraph explains in detail the procedure.

*x*,

*y*) of the

*t*

^{th}frame is given by [25

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

29. P. K. Rastogi, M. Spajer, and J. Monneret, “In-plane deformation measurement using holographic moiré,” Opt. Lasers Eng. **2**, 79–103 (1981). [CrossRef]

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

*ℓ*

_{k}=

*a*

_{k}exp(

*ikφ*

_{1}),

*u*

_{k}=exp(

*ikα*),

*℘*

_{k}=

*b*

_{k}exp(

*ikφ*

_{2}),

*v*

_{k}=exp(

*ikβ*); superscript * denotes the complex conjugate, and

*η*the additive white Gaussian noise with mean zero and variance

*σ*

^{2}. In Eq. (2), 0 (corresponding to

*I*

_{dc}); ±

*α*,±2

*α*…..,±

*κα*; ±

*β*,±2

*β*…..,±

*κβ*; are the frequencies

*α*and

*β*can be identified. Here,

*n*represents the number of frequency components present in the signal. The first step, common to all the methods [31

31. R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems **AES-19**, 134–139 (1983). [CrossRef]

32. M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing **ASSP-34**, 331–341 (1986). [CrossRef]

35. R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoustics, Speech, and Signal Processing **37**, 984–995 (1989). [CrossRef]

*N*recorded phase shifted sequences. The autocovariance of signal

*I*(

*t*) in Eq. (2) is defined as [38]

*E*[·] represents the expectation operator which averages over the ensemble of realizations. For simplicity, let us consider

*κ*=1 and rewrite Eq. (2) as

*I**(

*t*-

*p*) for

*κ*=1 as

*E*[

*c*

_{1}]=

*E*[

*c*

_{2}]=

*E*[

*c*

_{3}]=

*E*[

*c*

_{4}]=

*E*[

*c*

_{5}]=

*δ*

_{g,h}=1 if

*g*=

*h*; and

*δ*

_{g,h}=0 otherwise. In practice, expectation

*E*in Eq. (3) is computed by averaging over finite number of frames. If a large number of frames is taken for averaging, the exponential terms containing

*t*in

*c*

_{1},

*c*

_{2},

*c*

_{3},

*c*

_{4}, and

*c*

_{5}will oscillate uniformly between 0 and 2π. In this limit, the expectation of

*c*

_{1},

*c*

_{2},

*c*

_{3},

*c*

_{4}, and

*c*

_{5}will approach zero because

*e*

^{iψ}

*d*

*ψ*=0. However, if finite number of frames are taken for averaging, expectation

*c*

_{1},

*c*

_{2},

*c*

_{3},

*c*

_{4}, and

*c*

_{5}will have a small finite value different from zero. Hence, for

*κ*harmonics in the intensity, the final derivation of covariance of

*I*(

*t*) is given by

*I*(

*t*) is assumed to depend only on the lag between the two averaged samples. The autocovariance matrix can thus be written as [35

35. R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoustics, Speech, and Signal Processing **37**, 984–995 (1989). [CrossRef]

*I*(

*t*)=[

*I*(

*t*-1),…..,

*I*(

*t*-

*m*)],

*m*is the autocovariance length, and

*r**(-

*p*)=

*r*(

*p*). An autocovariance matrix

*R*

_{I}can be shown to have the form

**R**

_{s}and

**R**

_{ε}are the signal and noise contributions, A

_{m}×(4

_{κ}+1)=[

**a**(

*ω*

_{0}) . .

**a**(

*ω*

_{4κ})] where for instance element

**a**(

*ω*

_{0})consists of

*m*×1 matrix with unity entries corresponding to

*I*

_{dc};

**a**(

*ω*

_{1})=[1

*e*

^{iα}. .

*eiα*(

*m*-1)]

^{T}; (·)

^{c}and (·)

^{T}is the conjugate transpose and transpose operator, respectively,

**I**is the

*m*×

*m*identity matrix; and

*P*

_{m×m}matrix is

**R**

_{I}is a Hermitian matrix, we have,

**R**

_{I}=

**R***

_{I}∈

**C**

^{m×m}, where

**C**

^{m×m}is a complex matrix. The matrix

**R**

_{I}is said to be positive semidefinite since its eigenvalues are nonnegative. The singular value decomposition of

**R**

_{I}yields a

**P**matrix whose diagonal elements (also known as eigenvalues) can be ordered as

*A*

_{4κ}

^{2}≥….

**S**

_{m×n}=[s

_{1},s

_{2},…..s

_{n}] be the orthonormal eigenvectors associated with

*G*

_{m×(m-n)}=[

**g**

_{1},

**g**

_{2},…..

**g**

_{m-n}] be the set of orthonormal eigenvectors associated with eigen values

**APA**

^{c}∈

**C**

^{m×m}has rank

*n*(

*n*<

*m*), it has

*n*eigen values and the remaining

*m*-

*n*eigen values are zero. If we further suppose that (

*A*

^{2},

**r**) is an eigenpair of

**L**∈

**C**

^{m×m}and

**W**=

**L**+

*ρ*

**I**with

*ρ*∈

**C**, then (

*A*

^{2}+

*ρ*,

**r**) is an eigenpair of

**W**. In consequence we obtain

*σ*

^{2}, where

*t*spans from 1,2,3,…,

*m*. We observe that

*σ*

^{2}and

*σ*

^{2}. Following this corollary and from Eq. (11) we get

**a**(

*ω*

_{k})

_{=0}are orthogonal to

*noise subspace*. This also indicates that {

**g**

*k*

**G**belong to the null space of

**A**

^{c}(can be written as

**g**

*k*∈

**N**(

**A**

^{c}). The dimension of

**N**(

**A**

^{c}) is equal to

*m*-

*n*which is also the dimension of

*range space*of

**G**, written as R(

**G**). This fact combined to the observation that

**A**

^{c}

**G**=0 leads us to

**G**)=

**N**(

**S**

^{c}); we have,

**N**(

**S**

^{c})=

**N**(

**A**

^{c}). We further deduce that since R(

**S**) and R(

**A**) are orthogonal complements to

**N**(

**S**

^{c}) and

**N**(

**A**

^{c}), respectively, it follows that

**S**) and R(

**G**) are called

*signal subspace*and

*noise subspace*, respectively.

*R̂*

_{I}of

*R*

_{I}is available, only the estimate

**Ĝ**of

**G**can be determined. MUSIC uses

*m*-

*n*linearly independent vectors in R(

**Ĝ**) to obtain the frequency estimates. Using Eq. (14), in

*root*MUSIC, the frequencies are estimated as angular positions of

*n*roots of equation [39]

**a**(

*z*) is obtained from

**a**(

*ω*), and by replacing

*e*

^{iω}by

*z*we get

**a**(

*z*)=[1

*z*

^{-1}.

*z*

^{-(m-1)}]

^{T}. Since the minimum possible value of m is

*n*+1, it can be observed from Eq. (11) that we need

*N*data frames that have at least twice the number of sinusoidal components in the signal. For example, for

*κ*=2, we have

*n*=9 (since

*n*=2H

_{κ}+1, where H is the number of PZT’s); we need at least eighteen data frames (the dc component is also counted as dc frequency). Hence, the minimum number of data frames required for phase extraction is 2

*n*. If only one PZT is employed for phase shifting in an optical setup, then number of data frames required is 4

*κ*+2.

**Ĝ**) is orthogonal to {

**a**(

*ω*

_{k})}

*n*

_{k=0}, therefore, without sacrificing the accuracy too much, only one such vector can be used. This would result in substantial computational savings. The statistical accuracy of the min-norm algorithm is similar to that obtained using MUSIC. Hence, the performance of MUSIC algorithm is achieved at reduced computational cost [32

**ASSP-34**, 331–341 (1986). [CrossRef]

*n*roots of the polynomial

**Ĝ**) with first element equal to unity, that has minimum Euclidean norm. The following explains the rational behind this specific selection [30

30. 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), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-20-4681. [CrossRef] [PubMed]

**AES-19**, 134–139 (1983). [CrossRef]

**R̂**

_{I}of

**R**

_{I}is available, only the estimate

**Ŝ**of

**S**can be determined. Let the matrix

**Ŝ**be partitioned as

**I**can be written as

**I**-

**χχ**

^{c}is equal to 1-‖

**χ**‖2 and the remaining

*n*-1 eigenvalues of

**I**-

**χχ**

^{c}are equal to unity, the inverse in Eq. (23) exists if and only if

**ĝ**]

^{T}in R(

**Ĝ**). Condition in Eq. (25) is equivalent to rank(

**S̄**

^{c}

**S̄**)=

*n*which, in turn, holds if and only if

**S**made from more than

*n*consecutive rows should have rank equal to

*n*. Therefore, Eq. (26) is valid as long as

*N*is sufficiently large. This completes the derivation of min-norm frequency estimator defined in Eq. (19). In addition, it has been shown that by using the min-norm vector R(

**Ĝ**) as defined in Eq. (19), the spurious frequency estimates can be reduced, a problem sometimes associated with the MUSIC method.

*n*is determined. The details on selection of

*m*and

*N*will be explained in next Section. Determining the number of harmonics is a typical problem in signal processing and more details can be obtained from any standard signal processing text book. One way [40

40. J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing **36**, 1846–1853 (1988). [CrossRef]

**R**

_{I}matrix in Eq. (10).

## 3. Simulation of the min-norm algorithm to holographic moiré

*κ*=2 and noise

*η*as

*π*/4,

*β*=7

*π*/18, and phases

*φ*

_{1},

*φ*

_{2}at (

*x*,

*y*) on the intensity map are defined by

*p′*,

*p′*) is the center for circular fringe pattern corresponding to phase

*φ*

_{1}, (

*p″*,

*p′*) is the center for circular fringe pattern corresponding to phase

*φ*

_{2}and,

*φ*

_{R}

_{1}and

*φ*

_{R}

_{2}are the random phases. Figure 2 shows the fringe pattern corresponding to Eq. (1) for two different values of

*κ*for different noise levels. Figures 2(a) and (b) show fringe maps for

*κ*=1 ; and, no noise and 10dB SNR, respectively. Figures 2(c) and (d) show fringe maps for

*κ*=2 ; and, no noise and 10dB SNR, respectively.

**R**

_{I}is formulated from small overlapping fragments of data sets using sliding window technique known as spatial smoothing. Equation governing the spatial smoothing technique for the design of an autocovariance matrix is given by [35

**37**, 984–995 (1989). [CrossRef]

**R̂**

_{I}is the estimate of

**R**

_{I}. The approach which obtains frequency estimates from

**R̂**

_{I}in Eq. (30) is also called the

*forward approach*. We study the retrieval of phase steps

*α*and

*β*at a pixel (

*x*,

*y*) using this approach in the presence of additive white Gaussian noise with SNR between 0 and 70 dB. Let us assume that the number of frequencies

*n*present in the signal is determined to be nine using the method suggested in Section 2. Hence, the number of harmonics present in the moiré fringes is

*κ*=2, which sets lower limit on the number of data frames as eighteen. Once

*n*is identified, an appropriate value of

*m*must be selected such that

*m*>

*n*. The minimum value of

*m*is

*n*+1 and it is observed that though

*m*>

*n*increases the accuracy of frequency estimates, this is at a higher computational cost. On the other hand,

*m*too close to

*N*does not yield an autocovariance matrix

**R̂**

_{I}similar to

**R**

_{I}. This in turn results in spurious frequency estimates. Performing eigendecomposition of

**R̂**

_{I}gives estimates for eigenvectors

**Ĝ**. Phase steps

*α*and

*β*are then estimated using Eq. (19).

*N*=18 and

*m*=12 is selected. The phase step values

*α*and

*β*at any arbitrary pixel location on the data frame cannot be estimated from this plot. In the second case, we choose the number of data frames to be

*N*=22. Figures 3(b)–(d) show typical plots for phase steps

*α*and

*β*for

*m*=12, 15, and 19, respectively. From these plots it can be observed that even if the number of frames is increased the phase steps can be estimated only when appropriate value for

*m*is selected. Figure 3(b) shows that phase steps

*α*and

*β*can be estimated reliably from 35 dB onwards. In the third case we choose

*N*=26, and plots for

*m*=18 and 23 are shown in Figs. 3(e) and 3(f), respectively. Figure 3(e) shows that phase steps can be estimated reliably for values of SNR 35 dB and above. This shows that with small increase in data frames, a substantial improvement in estimating the phase steps is not obtained. On the other hand, Fig. 3(f) shows that value of

*m*too close to

*N*does not yield phase step estimates.

41. B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing **41**, 788–803 (1993). [CrossRef]

**R̂**

_{I}in Eq. (31) is called the

*forward-backward approach*. It has been shown that the accuracy of frequency estimates can be enhanced if this sample autocovariance matrix is applied.

*N*=18 and

*m*=12. Phase step values

*α*and

*β*can be estimated from this plot for values of SNR of 40dB and above. In the second case, we choose the number of data frames

*N*to be 22. Figures 4(b)–(d) show typical plots for phase steps

*α*and

*β*with

*m*=12, 15, and 19, respectively. Figures 4(b)–(c) shows that the values of

*α*and

*β*can be estimated at much lower SNR (25 dB and above). Figure 4(d) shows that value of

*m*is critical and its value too close to

*N*does not yield result. In the third case, we choose

*N*=26 ; and corresponding plots for

*m*=18 and 23 are shown in Figs. 4(e) and 4(f), respectively. From these plots, we observe that phase steps

*α*and

*β*can be reliably estimated from Fig. 4(e) as compared to Fig. 4(f). We also observe from Fig. 4(e) that phase steps can be estimated at much lower SNR (20dB and above) as compared to that obtained using other plots. Again, Fig. 4(f) shows that the values of

*m*too close to

*N*do not yield reliable phase step estimates. From these three cases we can conclude that phase step values

*α*and

*β*can be reliably estimated at lower SNR’s with increase in data frames

*N*and

*m*not too close to

*N*and

*n*.

## 4. Phase extraction

*φ*

_{1}and

*φ*

_{2}can be obtained by solving the linear Vandermonde system of equations. To design the equations, the number of harmonics

*κ*present in the signal and values of phase steps

*α*and

*β*pixel-wise must be known. Once, these values are known, the parameters

*ℓ*

_{k}and

*℘*

_{k}can be solved using the linear Vandermonde system of equations obtained from Eq. (2). Phase distributions

*φ*

_{1}and

*φ*

_{2}are subsequently computed from the arguments of

*ℓ*

_{1}and

*℘*

_{1}. The matrix thus obtained can be written as

_{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 advantage of Vandermonde system of equations is that the matrix shown in Eq. (32) can always be inverted as long as different values of

*α*and

*β*are used in the design of equations. Figures 5(a) and 5(b) show typical errors in the computation of phase values

*φ*

_{1}and

*φ*

_{2}under the assumption that SNR=30,

*N*=22, and

*m*=15. For obtaining the errors in typical phase maps, the results obtained in Fig. 4(c) using the forward-backward approach is considered. Figure 6 shows wrapped phase maps for

*φ*

_{1}and

*φ*

_{2}, all the other parameters remaining the same.

## 5. Conclusion

## Acknowledgment

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30. | A. Patil, R. Langoju, and P Rastogi, “An integral approach to phase shifting interferometry using a super-resolution frequency estimation method,” Opt. Express |

31. | R. Kumaresan and D. W. Tufts, “Estimating the angles of arrival of multiple plane waves,” IEEE Transactions on Aerospace and Electronic Systems |

32. | M. Kaveh and A. J Barabell, “The statistical performance of the MUSIC and the Minimum-Norm algorithms in resolving plane waves in noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing |

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

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

35. | R. Roy and T. Kailath, “ESPRIT-Estimation of signal parameters via rotational invariance techniques,” IEEE Trans. Acoustics, Speech, and Signal Processing |

36. | R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” in |

37. | G. Bienvenu, “Influence of the spatial coherence of the background noise on high resolution passive methods,” in |

38. | T. Söderström and P. Stoica, “Accuracy of high-order Yule-Walker methods for frequency estimation of complex sine waves,” IEEE Proceedings-F |

39. | A. J. Barabell, “Improving the resolution performance of eigenstructure-based direction-finding algorithms,” in |

40. | J. J. Fuchs, “Estimating the number of sinusoids in additive white noise,” IEEE Transactions on Acoustics, Speech, and Signal Processing |

41. | B. D. Rao and K. V. S. Hari, “Weighted subspace methods and spatial smoothing: analysis and comparison”, IEEE Trans. Signal Processing |

**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:**

Research Papers

**History**

Original Manuscript: March 24, 2005

Revised Manuscript: May 13, 2005

Published: May 30, 2005

**Citation**

Abhijit Patil and Pramod 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.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-11-4070

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### References

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