Abstract

In this study, a novel, fast, and accurate in-plane displacement distribution measurement method is proposed that uses a digital camera and arbitrary repeated patterns based on the moiré methodology. The key aspect of this method is the use of phase information of both the fundamental frequency and the high-order frequency components of the moiré fringe before and after deformations. Compared with conventional displacement methods and sensors, the main advantages of the method developed herein are its high resolution, accuracy, speed, low cost, and easy implementation. The effectiveness is confirmed by a simple in-plane displacement measurement experiment, and the experimental results indicate that an accuracy of 1/1000 of the pitch can be achieved for various repeated patterns. This method is useful for various applications ranging from the study of displacement and strain distributions in materials science, the biomimetics field, and mechanical material testing, to secure the integrity of infrastructures.

© 2014 Optical Society of America

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  1. M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
    [CrossRef]
  2. J. J. Lee, M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int 39, 425–432 (2006).
    [CrossRef]
  3. J. M. Burch, C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. 14, 178–185 (1975).
    [CrossRef]
  4. S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
    [CrossRef]
  5. D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994), Chap.4.
    [CrossRef]
  6. B. Han, Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME 117, 185–191 (1995).
    [CrossRef]
  7. S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
    [CrossRef]
  8. H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
    [CrossRef]
  9. S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
    [CrossRef]
  10. M. Tang, H. Xie, J. Zhu, X. Li, Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20, 2942–2955 (2012).
    [CrossRef] [PubMed]
  11. Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
    [CrossRef]
  12. S. Kishimoto, Y. Tanaka, T. Tomimatsu, Y. Kagawa, K. Nagai, “Fabrication of micromodel grid for various moiré methods by femtosecond laser exposure,” Opt. Lett. 34, 112–114 (2009).
    [CrossRef] [PubMed]
  13. S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
    [CrossRef]
  14. M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
    [CrossRef]
  15. S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
    [CrossRef]
  16. S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
    [CrossRef]
  17. Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
    [CrossRef]
  18. K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
    [CrossRef]
  19. Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
    [CrossRef]
  20. S. Ri, T. Muramatsu, “Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis,” Appl. Opt. 51, 3214–3223 (2012).
    [CrossRef] [PubMed]

2014 (1)

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

2013 (3)

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

2012 (3)

2011 (1)

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

2010 (1)

S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

2009 (1)

2007 (1)

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

2006 (1)

J. J. Lee, M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int 39, 425–432 (2006).
[CrossRef]

2004 (1)

S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
[CrossRef]

2003 (1)

M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

1997 (1)

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

1995 (1)

B. Han, Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME 117, 185–191 (1995).
[CrossRef]

1993 (1)

S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
[CrossRef]

1991 (1)

S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
[CrossRef]

1975 (1)

J. M. Burch, C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. 14, 178–185 (1975).
[CrossRef]

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Avril, S.

S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
[CrossRef]

Burch, J. M.

J. M. Burch, C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. 14, 178–185 (1975).
[CrossRef]

Dai, F.

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

Egashira, M.

S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
[CrossRef]

S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
[CrossRef]

Ekielski, M.

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Forno, C.

J. M. Burch, C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. 14, 178–185 (1975).
[CrossRef]

Fujigaki, M.

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

Guo, Y.

B. Han, Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME 117, 185–191 (1995).
[CrossRef]

Han, B.

B. Han, Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME 117, 185–191 (1995).
[CrossRef]

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994), Chap.4.
[CrossRef]

Hÿtch, M. J.

M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

Ifju, P.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994), Chap.4.
[CrossRef]

Jiang, X.

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

Kagawa, Y.

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

S. Kishimoto, Y. Tanaka, T. Tomimatsu, Y. Kagawa, K. Nagai, “Fabrication of micromodel grid for various moiré methods by femtosecond laser exposure,” Opt. Lett. 34, 112–114 (2009).
[CrossRef] [PubMed]

Kazmierczak, P.

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Kishimoto, S.

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

S. Kishimoto, Y. Tanaka, T. Tomimatsu, Y. Kagawa, K. Nagai, “Fabrication of micromodel grid for various moiré methods by femtosecond laser exposure,” Opt. Lett. 34, 112–114 (2009).
[CrossRef] [PubMed]

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
[CrossRef]

S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
[CrossRef]

Kobayashi, D.

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Lee, J. J.

J. J. Lee, M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int 39, 425–432 (2006).
[CrossRef]

Li, X.

Li, Y.

Masaya, A.

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

Morimoto, Y.

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

Muramatsu, T.

S. Ri, T. Muramatsu, “Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis,” Appl. Opt. 51, 3214–3223 (2012).
[CrossRef] [PubMed]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Nagai, K.

Nanbara, K.

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Patorski, K.

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Pénisson, J-M.

M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

Post, D.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994), Chap.4.
[CrossRef]

Putaux, J-L.

M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

Ri, S.

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

S. Ri, T. Muramatsu, “Theoretical error analysis of the sampling moiré method and phase compensation methodology for single-shot phase analysis,” Appl. Opt. 51, 3214–3223 (2012).
[CrossRef] [PubMed]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

Saka, M.

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

Shimo, K.

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

Shinozuka, M.

J. J. Lee, M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int 39, 425–432 (2006).
[CrossRef]

Shinya, N.

S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
[CrossRef]

S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
[CrossRef]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Surrel, Y.

S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
[CrossRef]

Tanaka, Y.

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

S. Kishimoto, Y. Tanaka, T. Tomimatsu, Y. Kagawa, K. Nagai, “Fabrication of micromodel grid for various moiré methods by femtosecond laser exposure,” Opt. Lett. 34, 112–114 (2009).
[CrossRef] [PubMed]

Tang, M.

Tomimatsu, T.

Vautrin, A.

S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
[CrossRef]

Wang, Q.

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

Wielgus, M.

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Xie, H.

M. Tang, H. Xie, J. Zhu, X. Li, Y. Li, “Study of moiré grating fabrication on metal samples using nanoimprint lithography,” Opt. Express 20, 2942–2955 (2012).
[CrossRef] [PubMed]

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Yamauchi, Y.

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

Zhu, J.

Appl. Opt. (1)

Chem. Eur. J. (1)

Q. Wang, S. Kishimoto, X. Jiang, Y. Yamauchi, “Spot moiré fringes: determination of domain boundaries and structural parameters in ordered nanoporous structures,” Chem. Eur. J. 20, 2179–2183 (2014).
[CrossRef]

Exp. Mech. (4)

S. Ri, M. Fujigaki, Y. Morimoto, “Sampling moiré method for accurate small deformation distribution measurement,” Exp. Mech. 50, 501–508 (2010).
[CrossRef]

S. Ri, T. Muramatsu, M. Saka, K. Nanbara, D. Kobayashi, “Accuracy of the sampling moiré method and its application to deflection measurements of large-scale structures,” Exp. Mech. 52, 331–340 (2012).
[CrossRef]

S. Ri, M. Saka, K. Nanbara, D. Kobayashi, “Dynamic thermal deformation measurement of large-scale, high-temperature piping in thermal power plants utilizing the sampling moiré method and grating magnets,” Exp. Mech. 53, 1635–1646 (2013).
[CrossRef]

S. Avril, A. Vautrin, Y. Surrel, “Grid method: application to the characterization of cracks,” Exp. Mech. 44, 37–43 (2004).
[CrossRef]

J. Appl. Phys. (1)

H. Xie, Q. Wang, S. Kishimoto, F. Dai, “Characterization of planar periodic structure using inverse laser scanning confocal microscopy moiré method and its application in the structure of butterfly wing,” J. Appl. Phys. 101,103511 (2007).
[CrossRef]

J. Electron. Packaging, Trans. ASME (1)

B. Han, Y. Guo, “Thermal deformation analysis of various electronic packaging products by moiré and microscopic moiré interferometry,” J. Electron. Packaging, Trans. ASME 117, 185–191 (1995).
[CrossRef]

J. Mod. Opt. (1)

Y. Arai, S. Yokozeki, K. Shiraki, T. Yamada, “High precision two-dimensional spatial fringe analysis method,” J. Mod. Opt. 44, 739–751 (1997).
[CrossRef]

J. Soc. Mat. Sci. (1)

S. Kishimoto, M. Egashira, N. Shinya, “Observation of micro-deformation by moiré method using a scanning electron microscope,” J. Soc. Mat. Sci. 40, 637–641 (1991).
[CrossRef]

Meas. Sci. Technol. (1)

K. Patorski, M. Wielgus, M. Ekielski, P. Kaźmierczak, “AFM nanomoiré technique with phase multiplication,” Meas. Sci. Technol. 24,035402 (2013).
[CrossRef]

Nature (1)

M. J. Hÿtch, J-L. Putaux, J-M. Pénisson, “Measurement of the displacement field of dislocations to 0.03Å by electron microscopy,” Nature 423, 270–273 (2003).
[CrossRef]

NDT&E Int (1)

J. J. Lee, M. Shinozuka, “A vision-based system for remote sensing of bridge displacement,” NDT&E Int 39, 425–432 (2006).
[CrossRef]

Opt. Eng. (3)

J. M. Burch, C. Forno, “A high sensitivity moiré grid technique for studying deformation in large objects,” Opt. Eng. 14, 178–185 (1975).
[CrossRef]

S. Kishimoto, M. Egashira, N. Shinya, “Microcreep deformation measurements by a moiré method using electron beam lithography and electron beam scan,” Opt. Eng. 32, 522–526 (1993).
[CrossRef]

M. Fujigaki, K. Shimo, A. Masaya, Y. Morimoto, “Dynamic shape and strain measurements of rotating tire using a sampling moiré method,” Opt. Eng. 50,101506 (2011).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Optics and Lasers in Engineering (1)

Q. Wang, S. Kishimoto, Y. Tanaka, Y. Kagawa, “Micro/submicro grating fabrication on metals for deformation measurement based on ultraviolet nanoimprint lithography,” Optics and Lasers in Engineering 51, 944–948 (2013).
[CrossRef]

Other (1)

D. Post, B. Han, P. Ifju, High Sensitivity Moiré: Experimental Analysis for Mechanics and Materials (Springer-Verlag, 1994), Chap.4.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Basic principle of the sampling moiré method for the single-shot phase analysis of fringe pattern: (a) optical setup, (b) down-sampling and intensity interpolation to generate multiple phase-shifted moiré fringes from a single grating image.

Fig. 2
Fig. 2

Schematic representation of a periodic repeated pattern as the sum of cosine functions in a Fourier series: (a) arbitrary original repeated pattern, (b) moiré fringe of (a) after performing down-sampling and intensity interpolation.

Fig. 3
Fig. 3

Comparison of the conventional sampling moiré method and the proposed method. In the proposed method, both the fundamental frequency component and the high-order frequency components of the moiré fringe are used to improve the measurement accuracy for any repeated pattern.

Fig. 4
Fig. 4

Experimental setup.

Fig. 5
Fig. 5

Experimental results: (a) the recorded image (260×280 pixels), where the pattern pitch is approximately 20 pixels; (b) displacement error distribution by the conventional method; (c) displacement error distribution by the proposed method after a 0.5 mm displacement; (d) displacement error in the y directional section in blue line of (b); and (e) the displacement error in the y directional section of the red line of (c).

Fig. 6
Fig. 6

Experimental results of the RMS measurement error by the conventional method (closed circle •) and the proposed method (open circle ○) in the case of (a) rectangles, (b) the number “3”, and (c) Chinese characters of one of the author’s first name, and (d) the character “A”.

Fig. 7
Fig. 7

Relation between the highest frequency order and the RMS displacement error for three types of repeated patterns after (a) 0.1 mm, and (b) 1.0 mm displacements in the x direction.

Fig. 8
Fig. 8

Analyzed Fourier spectrum for the case of repeated pattern for (a) the number “3”, and (b) the character “A” at the central and lower points, respectively.

Fig. 9
Fig. 9

Fourier spectra of the repeated pattern for the number “3” at the central point when the order of the intensity interpolation is changed from 1st-order (linear) to 3rd-order in case of (a) T = 20 pixels, and (b) T = 18pixels.

Fig. 10
Fig. 10

The relation between the order of the intensity interpolation and the RMS displacement error obtained by the sampling moiré method and the proposed method for the cases of (a) T = 20 pixels, and (b) T = 18 pixels. The RMS displacement error is obtained in the central 20×20 pixels evaluation area (400 points) for the repeated pattern “A”.

Equations (16)

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f ( x , y ) = a ( x , y ) cos { 2 π x P + φ 0 } + b ( x , y ) = a ( x , y ) cos { φ ( x , y ) } + b ( x , y )
f m ( x , y ; k ) = a ( x , y ) cos { 2 π ( 1 P 1 T ) x + φ 0 + 2 π k T } + b ( x , y ) = a ( x , y ) cos { φ m ( x , y ) + 2 π k T } + b ( x , y ) , ( k = 0 , 1 , , T 1 )
φ m ( x , y ) = arctan k = 0 T 1 f m ( x , y ; k ) sin ( 2 π k / T ) k = 0 T 1 f m ( x , y ; k ) cos ( 2 π k / T )
δ ( x , y ) = p 2 π Δ φ m ( x , y )
g ( x , y ) = w = 1 W a ( w ) ( x , y ) cos w { 2 π x P + φ 0 } + b ( x , y ) = w = 1 W a ( w ) ( x , y ) cos w { φ ( x , y ) } + b ( x , y )
g m ( x , y ; k ) = w = 1 W a ( w ) ( x , y ) cos w { 2 π ( 1 P 1 T ) x + φ 0 + 2 π k T } + b ( x , y ) = w = 1 W a ( w ) ( x , y ) cos w { φ m ( x , y ) + 2 π k T } + b ( x , y ) , ( k = 0 , 1 , , T 1 )
a ( w ) = 2 T [ k = 0 T 1 g m ( x , y ; k ) cos w ( 2 π k T ) ] 2 + [ k = 0 T 1 g m ( x , y ; k ) sin w ( 2 π k T ) ] 2
φ m ( w ) ( x , y ) = arctan k = 0 T 1 g m ( x , y ; k ) sin w ( 2 π k / T ) k = 0 T 1 g m ( x , y ; k ) cos w ( 2 π k / T )
δ ( w ) ( x , y ) = p 2 π w Δ φ m ( w ) ( x , y )
δ ^ ( x , y ) = w W h a ^ ( w ) ( x , y ) δ ( w ) ( x , y )
a ^ ( w ) ( x , y ) = a ( w ) ( x , y ) w = 1 W h a ( w ) ( x , y )
φ m ( w ) ( x , y ) x = 2 π ( 1 P 1 T ) φ m ( w ) ( x + 1 , y ) φ m ( w ) ( x 1 , y ) 2
P ( w ) ( x , y ) = 4 π T 4 π + [ φ m ( w ) ( x + 1 , y ) φ m ( w ) ( x 1 , y ) ] T
P ^ ( x , y ) = w = 1 W h a ^ ( w ) ( x , y ) P ( w ) ( x , y )
N = { ( O + 1 ) T ( when O is even and T is odd number ) ( O + 1 ) T 1 ( otherwise )
σ φ n = 2 N σ n a

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