Abstract

We present a procedure using continuous wavelet transforms (CWTs) to extract the phase information from moiré interferograms. The relationship between precise ridge detection of the two-dimensional CWT magnitude map and accurate phase extraction is detailed. A cost function is introduced for the adaptive selection of the ridge, and a computationally inexpensive implementation of the cost function ridge detection algorithm is explored with dynamic programming optimization. The results of the proposed ridge detection algorithm on actual interferograms are illustrated. Moreover, the resulting extracted phase is demonstrated to be smooth and accurate. As a result, the sensitivity of the moiré interferometry method is improved to obtain a pixel-by-pixel in-plane strain distribution map.

© 2004 Optical Society of America

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References

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  1. D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, New York, 1994).
    [CrossRef]
  2. B. Han, P. Kunthong, “Micro-mechanical deformation analysis of surface laminar circuit in organic flip-chip package: an experimental study,” J. Electron. Packaging 122, 294–300 (2000).
    [CrossRef]
  3. Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
    [CrossRef]
  4. H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.
  5. W. W. Macy, “Two-dimensional fringe-pattern analysis,” Appl. Opt. 22, 3898–3901 (1983).
    [CrossRef] [PubMed]
  6. K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science B. V., Amsterdam, The Netherlands, 1998), Vol. 26, Chap. 25, pp. 349–393.
    [CrossRef]
  7. M. Cherbuliez, P. Jacquot, X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
    [CrossRef]
  8. P. Tomassini, A. Giulietti, L. A. Gizzi, M. Galimberti, D. Giulietti, M. Borghesi, O. Willi, “Analyzing laser plasma interferograms with a continuous wavelet transform ridge extraction technique: the method,” Appl. Opt. 40, 6561–6568 (2001).
    [CrossRef]
  9. H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
    [CrossRef]
  10. R. A. Carmona, W. L. Hwang, B. Torrésani, “Characterization of signals by the ridges of their wavelet transforms,” IEEE Trans. Signal Process. 45, 2586–2590 (1997).
    [CrossRef]
  11. R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).
  12. A. J. Viterbi, “Error bounds for convolutional codes and a asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theory 13, 260–269 (1967).
    [CrossRef]

2003 (1)

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

2001 (1)

2000 (2)

B. Han, P. Kunthong, “Micro-mechanical deformation analysis of surface laminar circuit in organic flip-chip package: an experimental study,” J. Electron. Packaging 122, 294–300 (2000).
[CrossRef]

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

1997 (1)

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

1983 (1)

1967 (1)

A. J. Viterbi, “Error bounds for convolutional codes and a asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theory 13, 260–269 (1967).
[CrossRef]

Basaran, C.

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.

Bellman, R.

R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).

Borghesi, M.

Carmona, R. A.

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

Cartwright, A. N.

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.

Casey, W.

H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science B. V., Amsterdam, The Netherlands, 1998), Vol. 26, Chap. 25, pp. 349–393.
[CrossRef]

de Lega, X.

M. Cherbuliez, P. Jacquot, X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Dishongh, T.

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

Galimberti, M.

Giulietti, A.

Giulietti, D.

Gizzi, L. A.

Han, B.

B. Han, P. Kunthong, “Micro-mechanical deformation analysis of surface laminar circuit in organic flip-chip package: an experimental study,” J. Electron. Packaging 122, 294–300 (2000).
[CrossRef]

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, New York, 1994).
[CrossRef]

Hwang, W. L.

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

Ifju, P.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, New York, 1994).
[CrossRef]

Jacquot, P.

M. Cherbuliez, P. Jacquot, X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

Kunthong, P.

B. Han, P. Kunthong, “Micro-mechanical deformation analysis of surface laminar circuit in organic flip-chip package: an experimental study,” J. Electron. Packaging 122, 294–300 (2000).
[CrossRef]

Liu, H.

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.

Macy, W. W.

Post, D.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, New York, 1994).
[CrossRef]

Tomassini, P.

Torrésani, B.

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

Viterbi, A. J.

A. J. Viterbi, “Error bounds for convolutional codes and a asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theory 13, 260–269 (1967).
[CrossRef]

Willi, O.

Zhao, Y.

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

Appl. Opt. (2)

IEEE Trans. Inf. Theory (1)

A. J. Viterbi, “Error bounds for convolutional codes and a asymptotically optimum decoding algorithm,” IEEE Trans. Inf. Theory 13, 260–269 (1967).
[CrossRef]

IEEE Trans. Signal Process. (1)

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

J. Electron. Packaging (1)

B. Han, P. Kunthong, “Micro-mechanical deformation analysis of surface laminar circuit in organic flip-chip package: an experimental study,” J. Electron. Packaging 122, 294–300 (2000).
[CrossRef]

Mech. Mater. (1)

Y. Zhao, C. Basaran, A. N. Cartwright, T. Dishongh, “Thermomechanical behavior of micron scale solder joints under dynamic loads,” Mech. Mater. 32, 161–173 (2000).
[CrossRef]

Opt. Eng. (1)

H. Liu, A. N. Cartwright, C. Basaran, “Sensitivity improvement in phase shifted moiré interferometry using 1D continuous wavelet transform image processing,” Opt. Eng. 42, 2646–2652 (2003).
[CrossRef]

Other (5)

R. Bellman, Dynamic Programming (Princeton University, Princeton, N.J., 1957).

H. Liu, A. N. Cartwright, C. Basaran, W. Casey, “Moiré interferometry for microelectronic packaging interface fatigue reliability,” in International Conference on Compound Semiconductor Manufacturing Technology Digest (GaAs MANTECH, St. Louis, Mo., 2002), pp. 165–168.

K. Creath, “Phase-measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier Science B. V., Amsterdam, The Netherlands, 1998), Vol. 26, Chap. 25, pp. 349–393.
[CrossRef]

M. Cherbuliez, P. Jacquot, X. de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
[CrossRef]

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, New York, 1994).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Simulation of the direct ridge detection method: (a) original signal—a truncated chirped cosine signal, (b) the CWT magnitude, (c) the ridge index, (d) the real part of the detected ridge, (e) the phase extracted from the ridge.

Fig. 2
Fig. 2

Original noisy interferogram.

Fig. 3
Fig. 3

Typical magnitude curve (b = 393, fifth image row). Point 2 is the global maximum of this curve; points 1, 2, and 3 are three local maximums of the curve.

Fig. 4
Fig. 4

Diagram of the cost function ridge detection algorithm for a typical image row. The black dots of each column represent the candidate ridge points for the kth pixel of that image row. The arrows are the paths from one candidate of the kth pixel to one candidate of the (k + 1)th pixel.

Fig. 5
Fig. 5

Comparison of two ridge detection algorithms on an example interference signal (fifth row of Fig. 2). (a) Two-dimensional CWT magnitude map; the horizontal axis is the shifting parameter b and the vertical axis is the scaling parameter a. (b) The ridge selection by use of the direct maximum ridge detection and cost function ridge detection. (c) The reconstructed ridge of the direct maximum ridge detection and cost function ridge detection, as well as the original noisy interference signal.

Fig. 6
Fig. 6

Comparison of the direct maximum ridge detection and the cost ridge detection: (a) filtering with direct maximum ridge detection, (b) filtering with cost function ridge detection.

Fig. 7
Fig. 7

(a) Phase reconstruction, (b) first derivative of the phase map in the horizontal direction, (c) the normal x strain of the image in Fig. 2.

Equations (13)

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I=I01+γ cos φ,
εx=12πfφux,εy=12πfφvy,γxy=12πfφuy+φvx,
Sa, b=1a- stMt-badb,
Mx=exp-x22expjω0x,
Sa0, b=12 I0γ2πexpjωsb.
ridgebi=max|Saj, bi|,
cos tϕb, b=-C0b |Sϕb, b|2db+C1bϕbb2db,
cost=k=2L-|Sϕk, k|2+|ϕk-ϕk-1|2,
cost=k=2j-1-|Sϕk, k|2+|ϕk-ϕk-1|2+-|Sm, j|2+|m-ϕj-1|2+k=j+1L-|Sϕk, k|2+|ϕk-ϕk-1|2.
costm, j=minϕk=2j-1-|Sϕk, k|2+|ϕk-ϕk-1|2+-|Sm, j|2+|m-ϕj-1|2,
costn, j+1=minmcostm, j-|Sϕn, j+1|2+|n-m|2.
cost=k=3L-|Sϕk-1, k-1|2+|ϕk-1-ϕk-2|2+|2ϕk-1-ϕk-ϕk-2|2,
costp, j+2=minm,ncostm, j-|Sn, j+1|2+|n-m|2+|2m-n-ϕj-1|2-|Sp, j+2|2+|p-n|2+|2n-m-p|2.

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