Abstract

This Letter proposes a new method for the estimation of the first- and second-order phase derivatives corresponding to strain and curvature from a single fringe pattern in digital holographic interferometry. The method is based on a discrete energy separation algorithm, which provides a biased phase derivative estimate in a noisy environment. Subsequently, the least-squares spline approximation with optimal number of knots selection technique is used to obtain the accurate estimation of phase derivatives. The accuracy and computational efficiency of the proposed method is validated with simulation and experimental results.

© 2014 Optical Society of America

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References

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  1. Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
    [CrossRef]
  2. C. Liu, Opt. Eng. 42, 3443 (2003).
    [CrossRef]
  3. G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
    [CrossRef]
  4. G. Rajshekhar and P. Rastogi, Opt. Lett. 36, 3738 (2011).
    [CrossRef]
  5. J. F. Kaiser, in International Conference on Acoustics, Speech and Signal Processing (1990), Vol. 1, p. 381.
  6. P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
    [CrossRef]
  7. C. DeBoor, A Practical Guide To Splines (Springer, 1978).
  8. T. C. M. Lee, J. Stat. Comput. Simul. 72, 647 (2002).

2011

2009

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef]

2003

C. Liu, Opt. Eng. 42, 3443 (2003).
[CrossRef]

2002

T. C. M. Lee, J. Stat. Comput. Simul. 72, 647 (2002).

1994

Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
[CrossRef]

1993

P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
[CrossRef]

DeBoor, C.

C. DeBoor, A Practical Guide To Splines (Springer, 1978).

Gorthi, S. S.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef]

Kaiser, J. F.

P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
[CrossRef]

J. F. Kaiser, in International Conference on Acoustics, Speech and Signal Processing (1990), Vol. 1, p. 381.

Lee, T. C. M.

T. C. M. Lee, J. Stat. Comput. Simul. 72, 647 (2002).

Liu, C.

C. Liu, Opt. Eng. 42, 3443 (2003).
[CrossRef]

Maragos, P.

P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
[CrossRef]

Pedrini, G.

Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
[CrossRef]

Quatieri, T. F.

P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, Opt. Lett. 36, 3738 (2011).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef]

Rastogi, P.

G. Rajshekhar and P. Rastogi, Opt. Lett. 36, 3738 (2011).
[CrossRef]

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef]

Tiziani, H.

Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
[CrossRef]

Zou, Y.

Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
[CrossRef]

IEEE Trans. Signal Process.

P. Maragos, J. F. Kaiser, and T. F. Quatieri, IEEE Trans. Signal Process. 41, 3024 (1993).
[CrossRef]

J. Stat. Comput. Simul.

T. C. M. Lee, J. Stat. Comput. Simul. 72, 647 (2002).

Opt. Commun.

Y. Zou, G. Pedrini, and H. Tiziani, Opt. Commun. 111, 427 (1994).
[CrossRef]

Opt. Eng.

C. Liu, Opt. Eng. 42, 3443 (2003).
[CrossRef]

Opt. Lett.

Rev. Sci. Instrum.

G. Rajshekhar, S. S. Gorthi, and P. Rastogi, Rev. Sci. Instrum. 80, 093107 (2009).
[CrossRef]

Other

J. F. Kaiser, in International Conference on Acoustics, Speech and Signal Processing (1990), Vol. 1, p. 381.

C. DeBoor, A Practical Guide To Splines (Springer, 1978).

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Figures (4)

Fig. 1.
Fig. 1.

Phase derivative estimation of 1D signal. LSSA, least-squares spline approximation.

Fig. 2.
Fig. 2.

(a) Interference phase ϕ(m,n) in radians. (b) Fringe pattern.

Fig. 3.
Fig. 3.

(a) Estimate of ϕ˙n(m,n) in radians/pixel. (b) Estimate of ϕ¨n(m,n) in radians/pixel2. (c) Error in estimation of ϕ˙n(m,n). (d) Error in estimation of ϕ¨n(m,n).

Fig. 4.
Fig. 4.

(a) Object image. (b) Experimental fringe pattern. (c) First-order phase derivative estimate in radians/pixel. (d) Second-order phase derivative estimate in radians/pixel2.

Equations (13)

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Γ(m,n)=e[jϕ(m,n)]+ϵ(m,n),
ϕ˙n(m,n)=(ϕ(m,n)/n),ϕ¨n(m,n)=2ϕ(m,n)n2.
Γ(n)=e[jϕ(n)]+ϵ(n).
Γc(n)=Γ(n)e[jωcn].
r(n)=R{Γc(n)}=cos[ωcn+ϕ(n)]+ϵr(n)=cos[Ω(n)]+ϵr(n),
Ψ[r(n)]r2(n)r(n1)r(n+1).
Ω˙(n)=arccos(1Ψ[q(n)]+Ψ[q(n+1)]4Ψ[r(n)]),
ϕ˙(n)=Ω˙(n)ωc.
fd,κ(n)=i=1m1αiBi,d,κ(n),
Bi,0(n)={1ifkin<ki+10otherwiseBi,j(n)=nkiki+jkiBi,j1(n)+ki+j+1nki+j+1ki+1Bi+1,j1(n)
O:=n[ϕ˙(n)iαiBi,d,κ(n)]2.
GCV(m)=1Nn=1N(ϕ˙(n)f^(n))2(1g(m)/N)2,
Γ1(n)=e[jf^(n)].

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