Abstract

The phase unwrapping algorithm plays a very important role in many noncontact optical profilometries based on triangular measurement theory. Here we focus on discussing how to diminish the phase error caused by incorrect unwrapping path in wavelet transform profilometry. We employ the amplitude value map of wavelet transform coefficients at the wavelet-ridge position to identify the reliability of the phase data and the path of phase unwrapping. This means that the wrapped phase located at the pixel with the highest amplitude value will be selected as the starting point of the phase unwrapping, and that pixels with higher amplitude value will be unwrapped earlier. So the path of phase unwrapping is always in the direction of the pixel with highest amplitude value to the one with lowest amplitude value. Making full use of the amplitude information of wavelet coefficients at the wavelet-ridge position keeps the phase unwrapping error limited to local minimum areas even in the worst case. Computer simulations and experiments verify our theory.

© 2008 Optical Society of America

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References

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  1. M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  2. X. Su and W. Chen, “Fourier transform profilometry review,” Opt. Lasers Eng. 35, 263-284 (2001).
    [CrossRef]
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    [CrossRef]
  4. K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
    [CrossRef]
  5. W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
    [CrossRef]
  6. J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895-899(2004).
    [CrossRef]
  7. A. Z. Abid, M. A. Gdeisat, and D. R. Burton, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120-6126 (2007).
    [CrossRef] [PubMed]
  8. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722-8732 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  17. R. Cusack, J. M. Huntley, and H. T. Goldrein, “Improved noise-immune phase-unwrapping algorithm,” Appl. Opt. 34, 781-789 (1995).
    [CrossRef] [PubMed]
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    [CrossRef]
  19. J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2-9 (1987).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  25. X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
    [CrossRef]
  26. J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
    [CrossRef]
  27. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2007 (2)

2006 (1)

2005 (2)

W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
[CrossRef]

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560-2562 (2005).
[CrossRef] [PubMed]

2004 (6)

J. Zhong and J. Weng, “Spatial carrier-fringe pattern analysis by means of wavelet transform: wavelet transform profilometry,” Appl. Opt. 43, 4993-4998 (2004).
[CrossRef] [PubMed]

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895-899(2004).
[CrossRef]

K. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

A. Durson, S. Ozder, and N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

2002 (1)

M. Afifi, A. Fassi-Fihri, M. Marjane, , “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

2001 (2)

X. Su and W. Chen, “Fourier transform profilometry review,” Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

1996 (1)

1995 (1)

1993 (1)

X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

1992 (2)

J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
[CrossRef]

N. H. Ching, D. V. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

1989 (1)

1988 (1)

R. M. Goldstern, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry two dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

1987 (1)

J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2-9 (1987).

1984 (1)

1983 (1)

Abe, T.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345-2351 (l996).
[CrossRef]

Abid, A. Z.

Afifi, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, , “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Asundi, A.

Bally, G. V.

X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Barnes, T. H.

Braun, M.

N. H. Ching, D. V. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Bryanston-Cross, P. J.

J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
[CrossRef]

Burton, D. R.

Cao, Y.

W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
[CrossRef]

Carmona, R. A.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Chen, W.

J. Sun, W. Chen, and X. Su, “Study the measurement range of wavelet transform profilometry,” Acta Optica Sin. 27, 647-653 (2007).

W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry review,” Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

Ching, N. H.

N. H. Ching, D. V. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Cusack, R.

Durson, A.

A. Durson, S. Ozder, and N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Ecevit, N.

A. Durson, S. Ozder, and N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Fassi-Fihri, A.

M. Afifi, A. Fassi-Fihri, M. Marjane, , “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Gdeisat, M. A.

Ghiglia, D. C.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A. 4, 267-280 (l987).
[CrossRef]

D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

Gierloff, J. J.

J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2-9 (1987).

Goldrein, H. T.

Goldstern, R. M.

R. M. Goldstern, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry two dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Halioua, M.

Huntley, J. M.

Hwang, W. L.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Judge, J. T.

J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
[CrossRef]

Lalor, M. J.

Liu, H. C.

Marjane, M.

M. Afifi, A. Fassi-Fihri, M. Marjane, , “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Mastin, G. A.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A. 4, 267-280 (l987).
[CrossRef]

Mutoh, K.

Ozder, S.

A. Durson, S. Ozder, and N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Qian, K.

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

K. Qian, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

Quan, C.

J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
[CrossRef]

Romero, L. A.

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A. 4, 267-280 (l987).
[CrossRef]

D. C. Ghiglia and L. A. Romero, “Minimum Lp-norm two-dimensional phase unwrapping,” J. Opt. Soc. Am. A 13, 1999-2013 (1996).
[CrossRef]

Rosenfeld, D. V.

N. H. Ching, D. V. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

Srinivasan, V.

Su, X.

J. Sun, W. Chen, and X. Su, “Study the measurement range of wavelet transform profilometry,” Acta Optica Sin. 27, 647-653 (2007).

W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
[CrossRef]

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry review,” Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Sun, J.

J. Sun, W. Chen, and X. Su, “Study the measurement range of wavelet transform profilometry,” Acta Optica Sin. 27, 647-653 (2007).

Takeda, M.

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345-2351 (l996).
[CrossRef]

M. Takeda and K. Mutoh, “Fourier transform profilometry for the automatic measurement of 3-D object shapes,” Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

Tan, S. M.

Torresani, B.

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

Vukicevic, D.

X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

Watkins, L. R.

Weng, J.

Werner, C. L.

R. M. Goldstern, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry two dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Xue, L.

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

Zebker, H. A.

R. M. Goldstern, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry two dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

Zhong, J.

Zhou, W.

Acta Optica Sin. (1)

J. Sun, W. Chen, and X. Su, “Study the measurement range of wavelet transform profilometry,” Acta Optica Sin. 27, 647-653 (2007).

Appl. Opt. (9)

IEEE Trans. Image Process. (1)

N. H. Ching, D. V. Rosenfeld, and M. Braun, “Two-dimensional phase unwrapping using a minimum spanning tree algorithm,” IEEE Trans. Image Process. 1, 355 (1992).
[CrossRef]

IEEE Trans. Signal Process. (1)

R. A. Carmona, W. L. Hwang, and B. Torresani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586-2590 (1997).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. A. (1)

D. C. Ghiglia, G. A. Mastin, and L. A. Romero, “Cellular-automata method for phase unwrapping,” J. Opt. Soc. Am. A. 4, 267-280 (l987).
[CrossRef]

Meas. Sci. Technol. (1)

A. Durson, S. Ozder, and N. Ecevit, “Continuous wavelet transform analysis of projected fringe patterns,” Meas. Sci. Technol. 15, 1768-1772 (2004).
[CrossRef]

Opt. Commun. (2)

X. Su, G. V. Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150(1993).
[CrossRef]

M. Afifi, A. Fassi-Fihri, M. Marjane, , “Paul wavelet-based algorithm for optical phase distribution evaluation,” Opt. Commun. 211, 47-51 (2002).
[CrossRef]

Opt. Eng. (5)

K. Qian, “Windowed Fourier transform method for demodulation of carrier fringes,” Opt. Eng. 43, 1472-1473 (2004).
[CrossRef]

J. Zhong and J. Weng, “Dilating Gabor transform for the fringe analysis of 3-D shape measurement,” Opt. Eng. 43, 895-899(2004).
[CrossRef]

M. Takeda and T. Abe, “Phase unwrapping by a maximum cross-amplitude spanning tree algorithm: a comparative study,” Opt. Eng. 35, 2345-2351 (l996).
[CrossRef]

J. T. Judge, C. Quan, and P. J. Bryanston-Cross, “Holographic deformation measurements by Fourier transform technique with automatic phase unwrapping,” Opt. Eng. 31, 533-543(1992).
[CrossRef]

X. Su and L. Xue, “Phase unwrapping algorithm based on fringe frequency analysis in Fourier-transform profilometry,” Opt. Eng. 40, 637-643 (2001).
[CrossRef]

Opt. Lasers Eng. (3)

X. Su and W. Chen, “Reliability-guided phase unwrapping algorithm: a review,” Opt. Lasers Eng. 42, 245-261 (2004).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry review,” Opt. Lasers Eng. 35, 263-284 (2001).
[CrossRef]

W. Chen, X. Su, and Y. Cao, , “Method for eliminating zero spectrum in Fourier transform profilometry,” Opt. Lasers Eng. 43, 1267-1276 (2005).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (1)

J. J. Gierloff, “Phase unwrapping by regions,” Proc. SPIE 818, 2-9 (1987).

Radio Sci. (1)

R. M. Goldstern, H. A. Zebker, and C. L. Werner, “Satellite radar interferometry two dimensional phase unwrapping,” Radio Sci. 23, 713-720 (1988).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

(a) Deformed fringe pattern. (b) Amplitude of wavelet coefficients of the deformed fringe pattern at the wavelet-ridge position.

Fig. 3
Fig. 3

Amplitude-guide phase unwrapping algorithm in WTP.

Fig. 4
Fig. 4

(a) Simulated high object with shadow. (b) Deformed fringe pattern. (c) Restored object using traditional 1D phase unwrapping algorithm. (d) Error distribution.

Fig. 5
Fig. 5

(a) Residues map. (b) Object restored using the cut-line phase unwrapping algorithm. (c) Error distribution.

Fig. 6
Fig. 6

(a) Amplitude map of coefficient at the wavelet-ridge position. (b) Object restored using the amplitude-guided phase unwrapping algorithm. (c) Error distribution.

Fig. 7
Fig. 7

(a) Simulated object with slot. (b) Deformed fringe pattern with break zones. (c) Restored object using traditional 1D phase unwrapping algorithm. (d) Error distribution.

Fig. 8
Fig. 8

(a) Residues distribution map. (b) Object restored using the cut-line phase unwrapping algorithm. (c) Error distribution.

Fig. 9
Fig. 9

(a) Amplitude map of coefficients at the wavelet-ridge position. (b) Object restored using the amplitude-guided phase unwrapping algorithm. (c) Error distribution.

Fig. 10
Fig. 10

Experiment: (a) deformed fringe pattern captured by CCD; (b) amplitude of wavelet coefficients of the deformed fringe pattern at the wavelet-ridge position; (c) residues distribution map; (d) object restored using the traditional 1D phase unwrapping algorithm; (e) object restored using the cut-line phase unwrapping algorithm; (f) object restored using the amplitude-guided phase unwrapping algorithm.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

g ( x , y ) = I ( x , y ) + B ( x , y ) cos [ 2 π f 0 x + ϕ ( x , y ) ] ,
W f ( a , b ) = 1 a r f ( x ) Ψ ¯ ( t b a ) d t = 1 a f ( x ) , ψ ¯ a , b ( x ) ,
A ( a , b ) = { Im [ W f ( a , b ) ] } 2 + { Re [ W f ( a , b ) ] } 2 ,
ϕ ( a , b ) = arctan { Im [ W f ( a , b ) ] / Re [ W f ( a , b ) ] } ,
φ ( b ) = ϕ ( a r b , b ) .
ϕ ( x , y ) = Δ ϕ ( x , y ) = 2 π f 0 d h ( x , y ) / l .
M ( x ) = 1 π 4 2 π γ exp [ ( 2 π / γ ) 2 x 2 2 + j 2 π x ] ,
peaks ( x , y ) = 3 * ( 1 x ) 2 exp ( x 2 ( y + 1 ) 2 ) 10 * ( x 5 x 3 y 5 ) * exp ( x 2 y 2 ) 1 3 * exp ( ( x + 1 ) 2 y 2 ) .

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