Abstract

A novel method is proposed for the direct and simultaneous estimation of multiple phase derivatives corresponding to strain and slope fields from a single moiré fringe pattern in digital holographic moiré. The interference field in a given row/column is a multicomponent complex exponential signal and is represented as a spatially-varying autoregressive (SVAR) process. The spatially-varying coefficients of the SVAR model are computed by approximating them as the linear combination of linearly independent basis functions. Further, the spatially varying poles of the transfer function corresponding to the SVAR model are computed which provide the accurate estimation of the multiple phase derivatives. The simulation and experimental results are provided to substantiate the effectiveness of the proposed method.

© 2014 Optical Society of America

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References

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  1. G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Laser Eng. 50, iii–x (2012).
    [Crossref]
  2. R. Kulkarni, S. S. Gorthi, and P. Rastogi, “Measurement of in-plane and out-of-plane displacements and strains using digital holographic moiré,” J. Mod. Opt. 61, 755–762 (2014).
    [Crossref]
  3. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous measurement of in-plane and out-of-plane displacement derivatives using dual-wavelength digital holographic interferometry,” Appl. Opt. 50, H16–H21 (2011).
    [Crossref] [PubMed]
  4. A. A. Beex and P. Shan, “Time-varying prony method for instantaneous frequency estimation at low SNR,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1999), pp. 5–8.
  5. L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288 (1975).
    [Crossref] [PubMed]
  6. G. Rajshekhar, S. SivaGorthi, and P. Rastogi, “Simultaneous multidimensional deformation measurements using digital holographic moiré,” Appl. Opt. 50, 4189–4197 (2011).
    [Crossref] [PubMed]

2014 (1)

R. Kulkarni, S. S. Gorthi, and P. Rastogi, “Measurement of in-plane and out-of-plane displacements and strains using digital holographic moiré,” J. Mod. Opt. 61, 755–762 (2014).
[Crossref]

2012 (1)

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Laser Eng. 50, iii–x (2012).
[Crossref]

2011 (2)

1975 (1)

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288 (1975).
[Crossref] [PubMed]

Beex, A. A.

A. A. Beex and P. Shan, “Time-varying prony method for instantaneous frequency estimation at low SNR,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1999), pp. 5–8.

Gorthi, S. S.

R. Kulkarni, S. S. Gorthi, and P. Rastogi, “Measurement of in-plane and out-of-plane displacements and strains using digital holographic moiré,” J. Mod. Opt. 61, 755–762 (2014).
[Crossref]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, “Simultaneous measurement of in-plane and out-of-plane displacement derivatives using dual-wavelength digital holographic interferometry,” Appl. Opt. 50, H16–H21 (2011).
[Crossref] [PubMed]

Kulkarni, R.

R. Kulkarni, S. S. Gorthi, and P. Rastogi, “Measurement of in-plane and out-of-plane displacements and strains using digital holographic moiré,” J. Mod. Opt. 61, 755–762 (2014).
[Crossref]

Liporace, L. A.

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288 (1975).
[Crossref] [PubMed]

Rajshekhar, G.

Rastogi, P.

Shan, P.

A. A. Beex and P. Shan, “Time-varying prony method for instantaneous frequency estimation at low SNR,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1999), pp. 5–8.

SivaGorthi, S.

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

L. A. Liporace, “Linear estimation of nonstationary signals,” J. Acoust. Soc. Am. 58, 1288 (1975).
[Crossref] [PubMed]

J. Mod. Opt. (1)

R. Kulkarni, S. S. Gorthi, and P. Rastogi, “Measurement of in-plane and out-of-plane displacements and strains using digital holographic moiré,” J. Mod. Opt. 61, 755–762 (2014).
[Crossref]

Opt. Laser Eng. (1)

G. Rajshekhar and P. Rastogi, “Fringe analysis: premise and perspectives,” Opt. Laser Eng. 50, iii–x (2012).
[Crossref]

Other (1)

A. A. Beex and P. Shan, “Time-varying prony method for instantaneous frequency estimation at low SNR,” in Proceedings of IEEE International Symposium on Circuits and Systems (IEEE, 1999), pp. 5–8.

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Figures (5)

Fig. 1:
Fig. 1: Schematic of the experimental set-up.
Fig. 2:
Fig. 2: (a) Moiré fringe pattern. Pole trajectories in the case of (b) noiseless signal (c) noisy signal for m = 255.
Fig. 3:
Fig. 3: Estimated (a) phase derivative 1 (b) phase derivative 2. Error in the estimation of (c) phase derivative 1 (d) phase derivative 2.
Fig. 4:
Fig. 4: (a) Moiré fringe pattern. (b) linear fringe pattern corresponding to the carrier frequency.
Fig. 5:
Fig. 5: Estimated (a) phase derivative 1 (b) phase derivative 2 (c) Sum of phase derivatives (d) Difference of phase derivatives.

Tables (2)

Tables Icon

Table 1: Errors in the estimation of phase derivatives (in radians/pixel) from a noiseless signal using different basis functions and P = 2

Tables Icon

Table 2: Errors in the estimation of phase derivatives (in radians/pixel) from a noisy signal using different basis functions and model order

Equations (20)

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I [ n , m ] = e j ψ 1 [ n , m ] + e j ψ 2 [ n , m ] + ς [ n , m ] ,
I [ n , m ] = e j ψ 1 [ n , m ] + e j ψ 2 [ n , m ] + ς [ n , m ] ,
I [ n ] = e j ψ 1 [ n ] + e j ψ 2 [ n ] + ς [ n ] .
ψ ˙ 1 [ n ] = ψ 1 [ n ] n ψ ˙ 2 [ n ] = ψ 2 [ n ] n .
I [ n ] = p = 1 P a p [ n ] I [ n p ] + ς [ n ] ,
H ( z ; n ) = 1 1 p = 1 P a p [ n ] z p .
a p [ n ] = k = 0 K a p k β k [ n ] ,
I [ n ] = p = 1 P k = 0 K a p k β k [ n ] I [ n p ] + ς [ n ] .
I ^ [ n ] = p = 1 P k = 0 K a p k β k [ n ] I [ n p ] .
ς [ n ] = I [ n ] I ^ [ n ] .
ς 2 = | ς [ 1 ] | 2 + | ς [ 2 ] | 2 + + | ς [ N ] | 2 .
( x , y ) = n = 1 N x [ n ] y * [ n ] ,
ς 2 = ( ς , ς ) .
ς 2 = ( ς , ς ) = I 2 2 ( I , I ^ ) + I ^ 2 .
w k p = { β k [ n ] I [ n p ] } s k l ( p , q ) = ( w k p w l q ) s 0 l = ( s 0 l ( 0 , 1 ) , , s 0 l ( 0 , P ) ) ,
ς 2 = I 2 2 A s 0 + A S A .
S A = s 0 .
β k [ n ] = ( n N ) k
β k [ n ] = { cos k π n 2 N k even sin ( k + 1 ) π n 2 N k odd
ψ ˙ 2 [ n ] = ψ 2 [ n ] n ω n .

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