Abstract

The Letter proposes a method for phase estimation from a fringe pattern. The proposed method relies on a parametric approach where the phase is locally approximated as a two-dimensional (2D) polynomial, with the ensuing polynomial coefficients as the respective parameters. These coefficients are then estimated using the phase differencing operator. Because of the 2D formulation, the proposed method simultaneously analyzes signal samples along the horizontal and vertical dimensions, which enables robust estimation in the presence of noise. In addition, the method directly provides the desired phase without the requirement of complex unwrapping algorithms. Simulation and experimental results are presented to validate the method’s potential.

© 2012 Optical Society of America

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References

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  1. G. Rajshekhar and P. Rastogi, Opt. Laser Eng. 50, iii (2012).
    [CrossRef]
  2. K. Creath, in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), p. 349.
  3. M. Takeda, H. Ina, and S. Kobayashi, J. Opt. Soc. Am. 72, 156 (1982).
    [CrossRef]
  4. M. Servin, J. L. Marroquin, and F. J. Cuevas, Appl. Opt. 36, 4540 (1997).
    [CrossRef]
  5. L. R. Watkins, S. M. Tan, and T. H. Barnes, Opt. Lett. 24, 905 (1999).
    [CrossRef]
  6. Q. Kemao, Opt. Laser Eng. 45, 304 (2007).
    [CrossRef]
  7. S. S. Gorthi and P. Rastogi, Rev. Sci. Instrum. 80, 073109 (2009).
    [CrossRef]
  8. U. Schnars and W. P. O. Juptner, Meas. Sci. Tech. 13, R85 (2002).
    [CrossRef]
  9. B. Friedlander and J. M. Francos, IEEE Trans. Image Process. 5, 1084 (1996).
    [CrossRef]
  10. J. M. Francos and B. Friedlander, Multidimens. Syst. Signal Process. 9, 173 (1998).
    [CrossRef]
  11. J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
    [CrossRef]
  12. S. Lawrence Marple, IEEE Trans. Signal Process. 47, 2600 (1999).
    [CrossRef]

2012 (1)

G. Rajshekhar and P. Rastogi, Opt. Laser Eng. 50, iii (2012).
[CrossRef]

2009 (1)

S. S. Gorthi and P. Rastogi, Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef]

2007 (1)

Q. Kemao, Opt. Laser Eng. 45, 304 (2007).
[CrossRef]

2002 (1)

U. Schnars and W. P. O. Juptner, Meas. Sci. Tech. 13, R85 (2002).
[CrossRef]

2001 (1)

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

1999 (2)

S. Lawrence Marple, IEEE Trans. Signal Process. 47, 2600 (1999).
[CrossRef]

L. R. Watkins, S. M. Tan, and T. H. Barnes, Opt. Lett. 24, 905 (1999).
[CrossRef]

1998 (1)

J. M. Francos and B. Friedlander, Multidimens. Syst. Signal Process. 9, 173 (1998).
[CrossRef]

1997 (1)

1996 (1)

B. Friedlander and J. M. Francos, IEEE Trans. Image Process. 5, 1084 (1996).
[CrossRef]

1982 (1)

Antonio Gomez-Pedrero, J.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Barnes, T. H.

Creath, K.

K. Creath, in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), p. 349.

Cuevas, F. J.

Francos, J. M.

J. M. Francos and B. Friedlander, Multidimens. Syst. Signal Process. 9, 173 (1998).
[CrossRef]

B. Friedlander and J. M. Francos, IEEE Trans. Image Process. 5, 1084 (1996).
[CrossRef]

Friedlander, B.

J. M. Francos and B. Friedlander, Multidimens. Syst. Signal Process. 9, 173 (1998).
[CrossRef]

B. Friedlander and J. M. Francos, IEEE Trans. Image Process. 5, 1084 (1996).
[CrossRef]

Garcia-Botella, A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef]

Ina, H.

Juptner, W. P. O.

U. Schnars and W. P. O. Juptner, Meas. Sci. Tech. 13, R85 (2002).
[CrossRef]

Kemao, Q.

Q. Kemao, Opt. Laser Eng. 45, 304 (2007).
[CrossRef]

Kobayashi, S.

Lawrence Marple, S.

S. Lawrence Marple, IEEE Trans. Signal Process. 47, 2600 (1999).
[CrossRef]

Marroquin, J. L.

Quiroga, J. A.

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Rajshekhar, G.

G. Rajshekhar and P. Rastogi, Opt. Laser Eng. 50, iii (2012).
[CrossRef]

Rastogi, P.

G. Rajshekhar and P. Rastogi, Opt. Laser Eng. 50, iii (2012).
[CrossRef]

S. S. Gorthi and P. Rastogi, Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Juptner, Meas. Sci. Tech. 13, R85 (2002).
[CrossRef]

Servin, M.

Takeda, M.

Tan, S. M.

Watkins, L. R.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

B. Friedlander and J. M. Francos, IEEE Trans. Image Process. 5, 1084 (1996).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Lawrence Marple, IEEE Trans. Signal Process. 47, 2600 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

Meas. Sci. Tech. (1)

U. Schnars and W. P. O. Juptner, Meas. Sci. Tech. 13, R85 (2002).
[CrossRef]

Multidimens. Syst. Signal Process. (1)

J. M. Francos and B. Friedlander, Multidimens. Syst. Signal Process. 9, 173 (1998).
[CrossRef]

Opt. Commun. (1)

J. A. Quiroga, J. Antonio Gomez-Pedrero, and A. Garcia-Botella, Opt. Commun. 197, 43 (2001).
[CrossRef]

Opt. Laser Eng. (2)

G. Rajshekhar and P. Rastogi, Opt. Laser Eng. 50, iii (2012).
[CrossRef]

Q. Kemao, Opt. Laser Eng. 45, 304 (2007).
[CrossRef]

Opt. Lett. (1)

Rev. Sci. Instrum. (1)

S. S. Gorthi and P. Rastogi, Rev. Sci. Instrum. 80, 073109 (2009).
[CrossRef]

Other (1)

K. Creath, in Progress in Optics, E. Wolf, ed. (North-Holland, 1988), p. 349.

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Figures (3)

Fig. 1.
Fig. 1.

2D block representation.

Fig. 2.
Fig. 2.

(a) Simulated fringe pattern. (b) Estimated phase in radians. Estimation error in radians for (c) proposed method, (d) 1D polynomial phase approximation method, and (e) WFT method.

Fig. 3.
Fig. 3.

(a) Experimental fringe pattern. (b) Estimated phase in radians.

Tables (1)

Tables Icon

Table 1. Performance at Various Noise Levels

Equations (14)

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Γ(x,y)=a(x,y)exp[jϕ(x,y)]+η(x,y),
Γmn(x,y)=amn(x,y)exp[jϕmn(x,y)]+ηmn(x,y),
ϕmn(x,y)=(k,l)Ic(k,l)xkyl,
ϕmn(x,y)={c(0,0)}layer 1+{c(0,1)y+c(1,0)x}layer 2++{c(0,d)yd+c(1,d1)xyd1c(d1,1)xd1y+c(d,0)xd}layerd+1,
PDx(P),y(dP1)[Γmn(x,y)]=q=0dP1{p=0P{Γmn((p+q))(x+pτx,y+qτy)}(Pp)}(dP1q),
Γmn((p+q))={Γmn,p+qevenΓmn*,p+qodd,
PDx(P),y(dP1)[Γmn(x,y)]exp[j(ωxx+ωyy)],
c(P,dP)=ωy(1)d1P!(dP)!τxPτydP1,
c(P+1,dP1)=ωx(1)d1(P+1)!(dP1)!τxPτydP1.
ω^x,ω^y=argmaxω1,ω2|yxPDx(P),y(dP1)[Γmn(x,y)]exp[j(ω1x+ω2y)]|2,
Γmnd1(x,y)=Γmn(x,y)exp[jP=0dc(P,dP)xPydP].
c(0,0)=angle{1LxLyy=0Ly1x=0Lx1Γmn0(x,y)}.
ϕoffset=1Lyy=0Ly1(ϕmn(Lx,y)ϕm+1n(0,y)),
ϕ^m+1n(x,y)=ϕm+1n(x,y)+ϕoffset.

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