Abstract

In this research work, we introduce a novel approach for phase estimation from noisy reconstructed interference fields in digital holographic interferometry using an unscented Kalman filter. Unlike conventionally used unwrapping algorithms and piecewise polynomial approximation approaches, this paper proposes, for the first time to the best of our knowledge, a signal tracking approach for phase estimation. The state space model derived in this approach is inspired from the Taylor series expansion of the phase function as the process model, and polar to Cartesian conversion as the measurement model. We have characterized our approach by simulations and validated the performance on experimental data (holograms) recorded under various practical conditions. Our study reveals that the proposed approach, when compared with various phase estimation methods available in the literature, outperforms at lower SNR values (i.e., especially in the range 0–20 dB). It is demonstrated with experimental data as well that the proposed approach is a better choice for estimating rapidly varying phase with high dynamic range and noise.

© 2014 Optical Society of America

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References

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  1. U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
    [CrossRef]
  2. S. S. Gorthi and P. Rastogi, “Piece-wise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A Pure Appl. Opt. 11, 065405 (2009).
    [CrossRef]
  3. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).
  4. F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
    [CrossRef]
  5. M. J. Huang and W. Sheu, “Histogram-data-orientated filter for inconsistency removal of interferometric phase maps,” Opt. Eng. 44, 045602 (2005).
    [CrossRef]
  6. Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
    [CrossRef]
  7. E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46, 106–116 (2008).
  8. S. S. Gorthi and P. Rastogi, “Estimation of phase derivatives using discrete chirp-Fourier-transform-based method,” Opt. Lett. 34, 2396–2398 (2009).
    [CrossRef]
  9. S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
    [CrossRef]
  10. G. Rajshekhar and P. Rastogi, “Phase estimation using a state-space approach based method,” Opt. Laser Eng. 51, 1004–1007 (2013).
  11. S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
    [CrossRef]
  12. G. R. K. S. Subrahmanyam, “Recursive image estimation and inpainting in noise using non-Gaussian MRF prior,” Ph.D. thesis (Indian Institute of Technology, Madras, 2008).
  13. J. Gal, A. Campeanu, and I. Nafornita, “Estimation of chirp signals in Gaussian noise by Kalman filtering,” in International Symposium on Signals, Circuits and Systems (IEEE, 2007), Vol. 1, pp. 1–4.
  14. J. Gal, A. Campeanu, and I. Nafornita, “Identification of polynomial phase signals by extended Kalman filtering,” in Proceedings of the European Signal Processing Conference (EURASIP, 2008).

2013 (1)

G. Rajshekhar and P. Rastogi, “Phase estimation using a state-space approach based method,” Opt. Laser Eng. 51, 1004–1007 (2013).

2010 (1)

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).

2009 (3)

2008 (2)

Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
[CrossRef]

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46, 106–116 (2008).

2005 (1)

M. J. Huang and W. Sheu, “Histogram-data-orientated filter for inconsistency removal of interferometric phase maps,” Opt. Eng. 44, 045602 (2005).
[CrossRef]

2004 (2)

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
[CrossRef]

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

2002 (1)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Busca, G.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46, 106–116 (2008).

Campeanu, A.

J. Gal, A. Campeanu, and I. Nafornita, “Estimation of chirp signals in Gaussian noise by Kalman filtering,” in International Symposium on Signals, Circuits and Systems (IEEE, 2007), Vol. 1, pp. 1–4.

J. Gal, A. Campeanu, and I. Nafornita, “Identification of polynomial phase signals by extended Kalman filtering,” in Proceedings of the European Signal Processing Conference (EURASIP, 2008).

Gal, J.

J. Gal, A. Campeanu, and I. Nafornita, “Estimation of chirp signals in Gaussian noise by Kalman filtering,” in International Symposium on Signals, Circuits and Systems (IEEE, 2007), Vol. 1, pp. 1–4.

J. Gal, A. Campeanu, and I. Nafornita, “Identification of polynomial phase signals by extended Kalman filtering,” in Proceedings of the European Signal Processing Conference (EURASIP, 2008).

Gonalves, E.

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).

S. S. Gorthi and P. Rastogi, “Piece-wise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Estimation of phase derivatives using discrete chirp-Fourier-transform-based method,” Opt. Lett. 34, 2396–2398 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
[CrossRef]

Huang, M. J.

M. J. Huang and W. Sheu, “Histogram-data-orientated filter for inconsistency removal of interferometric phase maps,” Opt. Eng. 44, 045602 (2005).
[CrossRef]

Julier, S. J.

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
[CrossRef]

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Li, Y.

Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
[CrossRef]

Nafornita, I.

J. Gal, A. Campeanu, and I. Nafornita, “Estimation of chirp signals in Gaussian noise by Kalman filtering,” in International Symposium on Signals, Circuits and Systems (IEEE, 2007), Vol. 1, pp. 1–4.

J. Gal, A. Campeanu, and I. Nafornita, “Identification of polynomial phase signals by extended Kalman filtering,” in Proceedings of the European Signal Processing Conference (EURASIP, 2008).

Palacios, F.

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, “Phase estimation using a state-space approach based method,” Opt. Laser Eng. 51, 1004–1007 (2013).

Rastogi, P.

G. Rajshekhar and P. Rastogi, “Phase estimation using a state-space approach based method,” Opt. Laser Eng. 51, 1004–1007 (2013).

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).

S. S. Gorthi and P. Rastogi, “Piece-wise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Estimation of phase derivatives using discrete chirp-Fourier-transform-based method,” Opt. Lett. 34, 2396–2398 (2009).
[CrossRef]

S. S. Gorthi and P. Rastogi, “Improved high-order ambiguity-function method for the estimation of phase from interferometric fringes,” Opt. Lett. 34, 2575–2577 (2009).
[CrossRef]

Ricardo, J.

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Shen, W.

Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
[CrossRef]

Sheu, W.

M. J. Huang and W. Sheu, “Histogram-data-orientated filter for inconsistency removal of interferometric phase maps,” Opt. Eng. 44, 045602 (2005).
[CrossRef]

Subrahmanyam, G. R. K. S.

G. R. K. S. Subrahmanyam, “Recursive image estimation and inpainting in noise using non-Gaussian MRF prior,” Ph.D. thesis (Indian Institute of Technology, Madras, 2008).

Uhlmann, J. K.

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
[CrossRef]

Valin, J. L.

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Zappa, E.

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46, 106–116 (2008).

Zhu, J.

Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
[CrossRef]

J. Opt. A Pure Appl. Opt. (1)

S. S. Gorthi and P. Rastogi, “Piece-wise polynomial phase approximation approach for the analysis of reconstructed interference fields in digital holographic interferometry,” J. Opt. A Pure Appl. Opt. 11, 065405 (2009).
[CrossRef]

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Jüptner, “Digital recording and numerical reconstruction of holograms,” Meas. Sci. Technol. 13, R85–R101 (2002).
[CrossRef]

Opt. Commun. (1)

F. Palacios, E. Gonalves, J. Ricardo, and J. L. Valin, “Adaptive filter to improve the performance of phase-unwrapping in digital holography,” Opt. Commun. 238, 245–251 (2004).
[CrossRef]

Opt. Eng. (1)

M. J. Huang and W. Sheu, “Histogram-data-orientated filter for inconsistency removal of interferometric phase maps,” Opt. Eng. 44, 045602 (2005).
[CrossRef]

Opt. Laser Eng. (1)

G. Rajshekhar and P. Rastogi, “Phase estimation using a state-space approach based method,” Opt. Laser Eng. 51, 1004–1007 (2013).

Opt. Lasers Eng. (2)

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?” Opt. Lasers Eng. 48, 133–140 (2010).

E. Zappa and G. Busca, “Comparison of eight unwrapping algorithms applied to Fourier-transform profilometry,” Opt. Lasers Eng. 46, 106–116 (2008).

Opt. Lett. (2)

Optik (1)

Y. Li, J. Zhu, and W. Shen, “Phase unwrapping algorithms, respectively, based on path-following and discrete cosine transform,” Optik 119, 545–547 (2008).
[CrossRef]

Proc. IEEE (1)

S. J. Julier and J. K. Uhlmann, “Unscented filtering and nonlinear estimation,” Proc. IEEE 92, 401–422 (2004).
[CrossRef]

Other (3)

G. R. K. S. Subrahmanyam, “Recursive image estimation and inpainting in noise using non-Gaussian MRF prior,” Ph.D. thesis (Indian Institute of Technology, Madras, 2008).

J. Gal, A. Campeanu, and I. Nafornita, “Estimation of chirp signals in Gaussian noise by Kalman filtering,” in International Symposium on Signals, Circuits and Systems (IEEE, 2007), Vol. 1, pp. 1–4.

J. Gal, A. Campeanu, and I. Nafornita, “Identification of polynomial phase signals by extended Kalman filtering,” in Proceedings of the European Signal Processing Conference (EURASIP, 2008).

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Figures (10)

Fig. 1.
Fig. 1.

3D mesh plot of the original phase and the corresponding fringe pattern at a SNR of 20 dB.

Fig. 2.
Fig. 2.

(a)–(d) Error in phase estimation with different methods at a SNR of 20 dB. (e) RMSE versus SNR and (f) divergence rate versus SNR.

Fig. 3.
Fig. 3.

3D mesh plot of the original phase (20×peaks) and the corresponding fringe pattern with a SNR of 20 dB.

Fig. 4.
Fig. 4.

Estimated phase comparison along the middle column of the phase map at a SNR of 20 dB.

Fig. 5.
Fig. 5.

Phase estimation error for the phase map generated by 20×peaks showing relatively poor performance of polynomial-approximation-based methods at a larger dynamic range of the phase map.

Fig. 6.
Fig. 6.

Comparison of EKFSig and UKFSig for signal-tracking-based phase estimation for rapidly varying phase maps at 20 dB. (a) and (b) The 3D mesh plot of the true phase map and corresponding 2D image of the fringe pattern. (c) and (d) The error in phase estimation using EKFSig and UKFSig. (e) and (f) Comparison of the performance of EKFSig and UKFSig for different values of SNR in terms of RMSE and divergence rate.

Fig. 7.
Fig. 7.

Experimental setup.

Fig. 8.
Fig. 8.

Intensity image of the reconstructed hologram.

Fig. 9.
Fig. 9.

Comparison of EKFPara and UKFSig for experimental data. (a) and (b) The noisy fringe pattern of the reconstructed interference field. (c) and (d) The fringes corresponding to phase estimated by EKFPara whereas (e) and (f) are that of proposed UKFSig.

Fig. 10.
Fig. 10.

3D mesh plot of the estimated phase using the proposed method.

Tables (1)

Tables Icon

Table 1. Comparison of Computational Time

Equations (27)

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f(m,n)=a(m,n)eiϕ(m,n)+η(m,n),
f(n)=a(n)eiϕ(n)+η(n).
ϕ(n+1)=ϕ(n)+11!ϕ(n)+12!ϕ(n)+w0(n).
ϕ(n+1)=ϕ(n)+11!ϕ(n)+w1(n),
ϕ(n+1)=ϕ(n)+w2(n).
a(n+1)=a(n)+wa(n).
[a(n+1)ϕ(n+1)ϕ(n+1)ϕ(n+1)]=[10000111!12!00111!0001][a(n)ϕ(n)ϕ(n)ϕ(n)]+[wa(n)w0(n)w1(n)w2(n)],
x(n+1)=Fx(n)+w(n).
R[f(n)]=a(n)cos(ϕ(n)),
I[f(n)]=a(n)sin(ϕ(n)).
h(x)=[x1cos(x2)x1sin(x2)],
ν(y)=[R[η(y)]I[η(y)]].
R=kRσv2I,
z(n)=h(x(n))+ν(n).
χ0=x¯,
χi=x¯+((L+λ)Px)i,i=1,,L,χi=x¯((L+λ)Px)i,i=L+1,,2L,
w0(μ)=λ(L+λ),
w0(c)=λ(L+λ)+(1α2+β),
wi(μ)=wi(c)=12(L+λ),i=1,,2L,
Zi=h(χi),i=0,,2L.
z¯=i=02Lwi(μ)Zi,
Pz=i=02Lwi(c)[(Ziz¯)(Ziz¯)T].
x^0=E{x0},P0=E{(x0x^0)(x0x^0)T},x^0a=E{xa}=[x^0T00]T,P0a=E{(x0ax^0a)(x0ax^0a)T}=[P0000Pu000Pv],
χk1a=[x^k1ax^k1a+(L+λ)Pk1ax^k1a(L+λ)Pk1a].
χk|k1x=f(χk1x,χk1u),x^k=i=02Lwi(μ)χi,k|k1x,Pk=i=02Lwi(c)[χi,k|k1xx^k][χi,k|k1xx^k]T,Zk|k1=h(χk|k1x,χk1v),z^k=i=02Lwi(μ)Zi,k|k1.
Pzz=i=02Lwi(c)[Zi,k|k1z^k][Zi,k|k1z^k]T,Pxz=i=02Lwi(c)[χi,k|k1x^k][Zi,k|k1z^k]T,K=PxzPzz1,x^k=x^k+K(zkz^k),Pk=PkKPzzKT.
θ(n):=CFnx(n),

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