Abstract

This paper describes an extended and improved theory of the displacement of the objective speckle pattern resulting from displacement and/or deformation of a coherently illuminated diffuse object. Using the theory developed by Yamaguchi [Opt. Acta 28, 1359 (1981)], extended expressions are derived that include the influence of surface shape/gradients via the first order approximation of the shape as linear surface gradients. Both the original Yamaguchi expressions and the extended form derived here are shown experimentally to break down as the detector position moves away from the z-axis. As such, improved forms of the expressions are then presented, which remove some of the approximations used by Yamaguchi and can be used to predict the objective speckle displacement over a wide range of detector positions and surface slopes. Finally, these expressions are then verified experimentally for the speckle shifts resulting from object translations.

© 2014 Optical Society of America

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References

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  1. I. Yamaguchi, “Speckle Displacement and Decorrelation in the Diffraction and Image Fields for Small Object Deformation,” Opt. Acta 28(10), 1359–1376 (1981).
    [Crossref]
  2. H. Atcha and R. Tatam, “Heterodyning of fibre optic electronic speckle pattern interferometers using laser diode wavelength modulation,” Meas. Sci. Technol. 5, 704–709 (1994).
  3. D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
    [Crossref]
  4. L. Shirley and G. Hallerman, “Nonconventional 3D Imaging Using Wavelength-Dependent Speckle,” Lincoln Lab. J. 9, 153–186 (1996).
  5. J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996).
    [Crossref] [PubMed]
  6. S. Bianchi, “Vibration detection by observation of speckle patterns,” Appl. Opt. 53(5), 931–936 (2014).
    [Crossref] [PubMed]
  7. I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E. 19, 944–948 (1986).
  8. I. Yamaguchi and T. Fujita, “Laser speckle rotary encoder,” Appl. Opt. 28(20), 4401–4406 (1989).
    [Crossref] [PubMed]
  9. I. Yamaguchi, “Advances in the laser speckle strain gauge,” Opt. Eng. 27(3), 273214 (1988).
    [Crossref]
  10. I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
    [Crossref]
  11. G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
    [Crossref]
  12. P. Smíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to detect an object’s surface slope,” Appl. Opt. 45(27), 6932–6939 (2006).
    [Crossref] [PubMed]
  13. P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
    [Crossref]
  14. T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
    [Crossref]
  15. D. Francis, T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Objective speckle velocimetry for autonomous vehicle odometry,” Appl. Opt. 51(16), 3478–3490 (2012).
    [Crossref] [PubMed]
  16. P. Jacquot and P. K. Rastogi, “Speckle motions induced by rigid-body movements in freespace geometry: an explicit investigation and extension to new cases,” Appl. Opt. 18(12), 2022–2032 (1979).
    [Crossref] [PubMed]
  17. J. Světlík, “Speckle Displacement: Two Related Approaches,” J. Mod. Opt. 39(1), 149–157 (1992).
    [Crossref]
  18. M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
    [Crossref]
  19. P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
    [Crossref]
  20. I. Yamaguchi, “Speckle displacement for general object deformation of a curved surface,” Proc. SPIE 8413, 841307 (2012).
    [Crossref]
  21. K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
    [Crossref]
  22. M. Raffel, C. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer, 2007).

2014 (1)

2012 (2)

2010 (2)

T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
[Crossref]

D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
[Crossref]

2006 (2)

P. Smíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to detect an object’s surface slope,” Appl. Opt. 45(27), 6932–6939 (2006).
[Crossref] [PubMed]

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

2004 (2)

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
[Crossref]

P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
[Crossref]

2001 (1)

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

1999 (1)

M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
[Crossref]

1996 (3)

G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
[Crossref]

L. Shirley and G. Hallerman, “Nonconventional 3D Imaging Using Wavelength-Dependent Speckle,” Lincoln Lab. J. 9, 153–186 (1996).

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996).
[Crossref] [PubMed]

1994 (1)

H. Atcha and R. Tatam, “Heterodyning of fibre optic electronic speckle pattern interferometers using laser diode wavelength modulation,” Meas. Sci. Technol. 5, 704–709 (1994).

1992 (1)

J. Světlík, “Speckle Displacement: Two Related Approaches,” J. Mod. Opt. 39(1), 149–157 (1992).
[Crossref]

1989 (1)

1988 (1)

I. Yamaguchi, “Advances in the laser speckle strain gauge,” Opt. Eng. 27(3), 273214 (1988).
[Crossref]

1986 (1)

I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E. 19, 944–948 (1986).

1981 (1)

I. Yamaguchi, “Speckle Displacement and Decorrelation in the Diffraction and Image Fields for Small Object Deformation,” Opt. Acta 28(10), 1359–1376 (1981).
[Crossref]

1979 (1)

Atcha, H.

H. Atcha and R. Tatam, “Heterodyning of fibre optic electronic speckle pattern interferometers using laser diode wavelength modulation,” Meas. Sci. Technol. 5, 704–709 (1994).

Baca, Z.

M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
[Crossref]

Bianchi, S.

Briechle, K.

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

Briers, J. D.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996).
[Crossref] [PubMed]

Charrett, T. O. H.

D. Francis, T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Objective speckle velocimetry for autonomous vehicle odometry,” Appl. Opt. 51(16), 3478–3490 (2012).
[Crossref] [PubMed]

T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
[Crossref]

Francis, D.

D. Francis, T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Objective speckle velocimetry for autonomous vehicle odometry,” Appl. Opt. 51(16), 3478–3490 (2012).
[Crossref] [PubMed]

D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
[Crossref]

Fujita, T.

Groves, R. M.

D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
[Crossref]

Hallerman, G.

L. Shirley and G. Hallerman, “Nonconventional 3D Imaging Using Wavelength-Dependent Speckle,” Lincoln Lab. J. 9, 153–186 (1996).

Hanebeck, U. D.

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

Horváth, P.

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

P. Smíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to detect an object’s surface slope,” Appl. Opt. 45(27), 6932–6939 (2006).
[Crossref] [PubMed]

P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
[Crossref]

M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
[Crossref]

Hrabovský, M.

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

P. Smíd, P. Horváth, and M. Hrabovský, “Speckle correlation method used to detect an object’s surface slope,” Appl. Opt. 45(27), 6932–6939 (2006).
[Crossref] [PubMed]

P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
[Crossref]

M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
[Crossref]

Jacquot, P.

Kobayashi, K.

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
[Crossref]

Paoletty, D.

G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
[Crossref]

Rastogi, P. K.

Shirley, L.

L. Shirley and G. Hallerman, “Nonconventional 3D Imaging Using Wavelength-Dependent Speckle,” Lincoln Lab. J. 9, 153–186 (1996).

Smíd, P.

Šmíd, P.

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
[Crossref]

Spagnolo, G. S.

G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
[Crossref]

Svetlík, J.

J. Světlík, “Speckle Displacement: Two Related Approaches,” J. Mod. Opt. 39(1), 149–157 (1992).
[Crossref]

Tatam, R.

H. Atcha and R. Tatam, “Heterodyning of fibre optic electronic speckle pattern interferometers using laser diode wavelength modulation,” Meas. Sci. Technol. 5, 704–709 (1994).

Tatam, R. P.

D. Francis, T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Objective speckle velocimetry for autonomous vehicle odometry,” Appl. Opt. 51(16), 3478–3490 (2012).
[Crossref] [PubMed]

D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
[Crossref]

T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
[Crossref]

Wagnerova, P.

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

Waugh, L.

D. Francis, T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Objective speckle velocimetry for autonomous vehicle odometry,” Appl. Opt. 51(16), 3478–3490 (2012).
[Crossref] [PubMed]

T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
[Crossref]

Webster, S.

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996).
[Crossref] [PubMed]

Yamaguchi, I.

I. Yamaguchi, “Speckle displacement for general object deformation of a curved surface,” Proc. SPIE 8413, 841307 (2012).
[Crossref]

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
[Crossref]

I. Yamaguchi and T. Fujita, “Laser speckle rotary encoder,” Appl. Opt. 28(20), 4401–4406 (1989).
[Crossref] [PubMed]

I. Yamaguchi, “Advances in the laser speckle strain gauge,” Opt. Eng. 27(3), 273214 (1988).
[Crossref]

I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E. 19, 944–948 (1986).

I. Yamaguchi, “Speckle Displacement and Decorrelation in the Diffraction and Image Fields for Small Object Deformation,” Opt. Acta 28(10), 1359–1376 (1981).
[Crossref]

Yaroslavsky, L.

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
[Crossref]

Zanetta, P.

G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
[Crossref]

Appl. Opt. (5)

J. Biomed. Opt. (1)

J. D. Briers and S. Webster, “Laser speckle contrast analysis (LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. 1(2), 174–179 (1996).
[Crossref] [PubMed]

J. Mod. Opt. (2)

J. Světlík, “Speckle Displacement: Two Related Approaches,” J. Mod. Opt. 39(1), 149–157 (1992).
[Crossref]

P. Horváth, M. Hrabovský, and P. Šmíd, “Full theory of speckle displacement and decorrelation in the image field by wave and geometrical descriptions and its application in mechanics,” J. Mod. Opt. 51(5), 725–742 (2004).
[Crossref]

J. Phys. E. (1)

I. Yamaguchi, “Automatic measurement of in-plane translation by speckle correlation using a linear image sensor,” J. Phys. E. 19, 944–948 (1986).

Lincoln Lab. J. (1)

L. Shirley and G. Hallerman, “Nonconventional 3D Imaging Using Wavelength-Dependent Speckle,” Lincoln Lab. J. 9, 153–186 (1996).

Meas. Sci. Technol. (3)

H. Atcha and R. Tatam, “Heterodyning of fibre optic electronic speckle pattern interferometers using laser diode wavelength modulation,” Meas. Sci. Technol. 5, 704–709 (1994).

D. Francis, R. P. Tatam, and R. M. Groves, “Shearography technology and applications: a review,” Meas. Sci. Technol. 21(10), 102001 (2010).
[Crossref]

T. O. H. Charrett, L. Waugh, and R. P. Tatam, “Speckle velocimetry for high accuracy odometry for a Mars exploration rover,” Meas. Sci. Technol. 21(2), 025301 (2010).
[Crossref]

Opt. Acta (1)

I. Yamaguchi, “Speckle Displacement and Decorrelation in the Diffraction and Image Fields for Small Object Deformation,” Opt. Acta 28(10), 1359–1376 (1981).
[Crossref]

Opt. Commun. (1)

G. S. Spagnolo, D. Paoletty, and P. Zanetta, “Local speckle correlation for vibration analysis,” Opt. Commun. 123(1-3), 41–48 (1996).
[Crossref]

Opt. Eng. (2)

I. Yamaguchi, “Advances in the laser speckle strain gauge,” Opt. Eng. 27(3), 273214 (1988).
[Crossref]

I. Yamaguchi, K. Kobayashi, and L. Yaroslavsky, “Measurement of surface roughness by speckle correlation,” Opt. Eng. 43(11), 2753 (2004).
[Crossref]

Opt. Lasers Eng. (1)

M. Hrabovský, Z. Bača, and P. Horváth, “Theory of speckle displacement and decorrelation and its application in mechanics,” Opt. Lasers Eng. 32(4), 395–403 (1999).
[Crossref]

Proc. SPIE (3)

P. Horváth, P. Šmíd, P. Wagnerova, and M. Hrabovský, “Usage of a speckle correlation for object surface topography,” Proc. SPIE 6034, 603421 (2006).
[Crossref]

I. Yamaguchi, “Speckle displacement for general object deformation of a curved surface,” Proc. SPIE 8413, 841307 (2012).
[Crossref]

K. Briechle and U. D. Hanebeck, “Template matching using fast normalized cross correlation,” Proc. SPIE 4387, 95–102 (2001).
[Crossref]

Other (1)

M. Raffel, C. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: A Practical Guide (Springer, 2007).

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Figures (8)

Fig. 1
Fig. 1 General representation of the geometry used in the development of the speckle shift theory via cross-correlation of speckle intensities.
Fig. 2
Fig. 2 The 3D printed test objects used in this experiment. From left to right mx,y = 1.0, 0.8, 0.6, 0.4, 0.2 and 0.0 and the stage mounting plate. The ramps have a constant working height of 20mm above the stage at the centre point and can be rotated in 45° increments in the mounting plate to enable the surface gradient direction to be rotated in a controlled manner.
Fig. 3
Fig. 3 Comparison of measured and theoretical translational scaling factors for a source and detector positioned at S = (0,-150,265) mm and D = (Dx,0,300) mm, for three surface gradients (a) mx = my = 0; (b) mx = + 1, my = 0 and (c) mx = −1, my = 0. The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimental results; (✕) Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az, and the dashed lines shows the values predicted by the Yamaguchi’s equations (for mx = my = 0) and the extended equations presented in section 2. The solid lines (green) are the values predicted by the improved equations presented in section 4. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.
Fig. 4
Fig. 4 Comparison of measured and theoretical translational scaling factors for a source at S = (0,-150,265) mm, and the detector positioned on the z-axis at D = (0,0,300) mm, for varying surface gradients; (a) for varying mx; (b) for varying my and (c) for varying directions of a fixed magnitude gradient as it is rotated about the z-axis (mx = cosϕ, my = -sinϕ). The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimentally measured results; (✕) Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az, and the solid lines show the values predicted by both the extended equations (section 2) and the improved equations (section 4), which are identical when the detector is located on the z-axis. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.
Fig. 5
Fig. 5 Comparison of measured and theoretical translational scaling factors for a source and detector positioned at S = (0,-150,265) mm and D = (−80,0,300) mm for varying surface gradients; (a) for varying mx; (b) for varying my and (c) for varying directions of a fixed magnitude gradient as it is rotated about the z-axis (mx = cosϕ, my = -sinϕ). The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimentally measured results; (✕)Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az and the solid lines show the values predicted by the improved equations, presented in section 4. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.
Fig. 6
Fig. 6 Comparison of measured and theoretical translational scaling factors for a source and detector positioned at S = (0,-150,265) mm and D = ( + 80,0,300) mm for varying surface gradients; (a) for varying mx; (b) for varying my and (c) for varying directions of a fixed magnitude gradient as it is rotated about the z-axis (mx = cosϕ, my = -sinϕ). The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimentally measured results; (✕)Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az and the solid lines show the values predicted by the improved equations, presented in section 4. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.
Fig. 7
Fig. 7 Comparison of measured and theoretical translational scaling factors for a source and detector positioned at S = (0,-150,265) mm and D = (0,-80,300) mm for varying surface gradients; (a) for varying mx; (b) for varying my and (c) for varying directions of a fixed magnitude gradient as it is rotated about the z-axis (mx = cosϕ, my = -sinϕ). The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimentally measured results; (✕)Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az and the solid lines show the values predicted by the improved equations, presented in section 4. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.
Fig. 8
Fig. 8 Comparison of measured and theoretical translational scaling factors for a source and detector positioned at S = (0,-150,265) mm and D = (−20,-80,300) mm for varying surface gradients; (a) for varying mx; (b) for varying my and (c) for varying directions of a fixed magnitude gradient as it is rotated about the z-axis (mx = cosϕ, my = -sinϕ). The top row shows the scaling factors for the speckle shift in the x direction (Ax) resulting from the object translations (ax, ay and az) while the bottom row shows scaling factors for the speckle shift in the y direction (Ay). The data points are the experimentally measured results; (✕)Ax,y/ax, (●) Ax,y/ay and (★) Ax,y/az and the solid lines show the values predicted by the improved equations, presented in section 4. Here the thickness of the line denotes the minimum and maximum bounds predicted assuming an error in source and detector positions of ± 1mm in all directions.

Equations (29)

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

U n ( D n ) = I 0 ( R + α T n ( R ) ) ζ ( R ) exp ( i ϕ ) × exp ( i k [ | L S ( R ) a n ( R ) | + | L D ( R , D n ) a n ( R ) | ] ) d x d y
| L S ( R ) a n ( R ) | L S ( R ) a n ( R ) · s ( R ) | L D ( R , D n ) a n ( R ) | L D ( R , D n ) a n ( R ) · d ( R , D n )
U n ( D n ) = I 0 ( R + α T n ( R ) ) exp ( i ϕ ) × exp ( i k [ L S ( R ) + L D ( R , D n ) a n ( R ) · p ( R , D n ) ] ) d x d y
I 1 ( D 1 ) I 2 * ( D 2 ) = | U 1 ( D 1 ) | 2 | U 2 ( D 2 ) | 2 + | U 1 ( D 1 ) U 2 * ( D 2 ) | 2
| U 1 ( D 1 ) U 2 * ( D 2 ) | = I 0 ( R ) I 0 ( R + α T 2 ( R ) ) × exp ( i k [ L D ( R , D ) L D ( R , D + A ) p ( R , D ) · a 1 ( R ) + p ( R , D + A ) · a 2 ( R ) ] ) d x d y
p ( R , D ) · a 1 R + p ( R , D + A ) · a 2 ( R ) p ( R , D ) · ( a 2 ( R ) a 1 ( R ) ) = p ( R , D ) · a ( R )
p ( R , D ) · a ( R ) = p ( 0 , D ) · a ( 0 ) + [ p ( R , D ) · a ( R ) ] 0 · R
L D ( R , D ) = L 0 + | R D | 2 2 L 0 L D ( R , D + A ) = L 0 +   | R D A | 2 2 L 0
| U 1 ( D 1 ) U 2 * ( D 2 ) | = exp ( i k [ D A L 0 p ( 0 , D ) · a 1 ( 0 ) ] ) × I 0 ( R ) I 0 ( R + α T 2 ( R ) ) × exp ( i k [ A + L 0 [ p ( R , D ) · a ( R ) ] 0 ] R L 0 ) d x d y
A = L 0 [ p ( R , D ) · a ( R ) ] 0
δ a x δ x = ε x x δ a y δ y = ε y y 1 2 ( δ a x δ y + δ a y δ x ) = ε x y = ε y x δ a z δ y = Ω x δ a z δ x = Ω y 1 2 ( δ a y δ x δ a x δ y ) = Ω z
[ A x A y ] = [ T ] [ a x a y a z ] + [ R ] [ Ω x Ω y Ω z ] + [ D ] [ ε x x ε x y ε y y ] [ T ] = L 0 [ δ ( s x + d x ) δ x δ ( s y + d y ) δ x δ ( s z + d z ) δ x δ ( s x + d x ) δ y δ ( s y + d y ) δ y δ ( s z + d z ) δ y ] [ R ] = L 0 [ 0 ( s z + d z ) ( s y + d y ) ( s z + d z ) 0 ( s x + d x ) ] [ D ] = L 0 [ ( s x + d x ) ( s y + d y ) 0 0 ( s x + d x ) ( s y + d y ) ]
δ s δ x = 1 L S ( s x 2 1 , s x s y , s x s z ) δ s δ y = 1 L S ( s x s y , s y 2 1 , s y s z ) δ d δ x = 1 L D ( d x 2 1 , d x d y , d x d z ) δ d δ y = 1 L D ( d x d y , d y 2 1 , d y d z )
[ T ] = L 0 [ ( s x 2 1 ) L S + ( d x 2 1 ) L 0 ( s x s y ) L S + ( d x d y ) L 0 ( s x s z ) L S + ( d x d z ) L 0 ( s x s y ) L S + ( d x d y ) L 0 ( s y 2 1 ) L S + ( d y 2 1 ) L 0 ( s y s z ) L S + ( d y d z ) L 0 ]
δ s δ x = 1 L S ( m x s x s z + s x 2 1 , m x s y s z + s x s y , m x ( s z 2 1 ) + s x s z ) δ s δ y = 1 L S ( m y s x s z + s x s y , m y s y s z + s y 2 1 , m y ( s z 2 1 ) + s y s z ) δ d δ x = 1 L D ( m x d x d z + d x 2 1 , m x d y d z + d x d y , m x ( d z 2 1 ) + d x d z ) δ d δ y = 1 L D ( m y d x d z + d x d y , m y d y d z + d y 2 1 , m y ( d z 2 1 ) + d y d z )
[ A x A y ] = ( [ T ] + [ T ' ] ) [ a x a y a z ] + [ R ] [ Ω x Ω y Ω z ] + [ D ] [ ε x x ε x y ε y y ]
[ T ' ] = L 0 [ m x ( s x s z L S + d x d z L D ) m x ( s y s z L S + d y d z L D ) m x ( ( s z 2 1 ) L S + ( d z 2 1 ) L D ) m y ( s x s z L S + d x d z L D ) m y ( s y s z L S + d y d z L D ) m y ( ( s z 2 1 ) L S + ( d z 2 1 ) L D ) ]
| L S ( R ) a 1 ( R ) | L S ( R ) a 1 ( R ) · s ( R ) | L S ( R ) a 2 ( R ) | L S ( R ) a 2 ( R ) · s ( R ) | L D ( R , D ) a 1 ( R ) | L D ( R , D ) a 1 ( R ) · d ( R , D ) | L D ( R , D + A ) a 2 ( R ) | L D ( R , D ) a 2 ( R ) · d ( R , D ) + A · d ( R , D )
U 1 ( R , D ) = I 0 ( R + α T 1 ( R ) ) exp ( i ϕ ) × exp ( i k [ L S ( R ) + L D ( R , D ) a 1 ( R ) · p ( R , D ) ] ) d x d y U 2 ( R , D + A ) = I 0 ( R + α T 2 ( R ) ) exp ( i ϕ ) × exp ( i k [ L S ( R ) + L D ( R , D ) a 2 ( R ) · p ( R , D ) A · d ( R , D ) ] ) d x d y
| U 1 ( R , D ) U 2 * ( R , D + A ) | = I 0 ( R ) I 0 ( R + α T 2 ( R ) ) × exp ( i k [ p ( R , D ) · a ( R ) - A · d ( R , D ) ] ) d x d y
A · d ( R , D ) = A x d x ( R , D ) A y d y ( R , D ) A x ( d x ( 0 , D ) + [ d x ( R , D ) ] 0 · R ) A y ( d y ( 0 , D ) + [ d y ( R , D ) ] 0 · R ) = - A · d ( 0 , D ) A x [ d x ( R , D ) ] 0 · R A y [ d y ( R , D ) ] 0 · R
| U 1 ( R , D ) U 2 * ( R , D + A ) | = exp ( i k [ A · d ( 0 , D ) p ( 0 , D ) · a ( 0 ) ] ) × I 0 ( R ) I 0 ( R + α T 2 ( R ) ) × exp ( i k [ A x [ d x ( R , D ) ] 0 A y [ d y ( R , D ) ] 0 + [ p ( R , D ) · a ( R ) ] 0 ] · R ) d x d y
[ A x [ d x ( R , D ) ] 0 A y [ d y ( R , D ) ] 0 + [ p ( R , D ) · a ( R ) ] 0 ] = 0
A x = δ d y δ y ( δ p · a δ x ) δ d y δ x ( δ p · a δ y ) ( δ d x δ x δ d y δ y δ d x δ y δ d y δ x ) A y = δ d x δ x ( δ p · a δ y ) δ d x δ y ( δ p · a δ x ) ( δ d x δ x δ d y δ y δ d x δ y δ d y δ x )
[ A x A y ] = [ P ] ( [ T ] [ a x a y a z ] + [ R ] [ Ω x Ω y Ω z ] + [ D ] [ ε x x ε x y ε y y ] )
[ P ] = 1 ( δ d x δ x δ d y δ y δ d x δ y δ d y δ x ) [ δ d y δ y δ d y δ x δ d x δ y δ d x δ x ] [ T ] = [ δ ( s x + d x ) δ x δ ( s y + d y ) δ x δ ( s z + d z ) δ x δ ( s x + d x ) δ y δ ( s y + d y ) δ y δ ( s z + d z ) δ y ] [ R ] = [ 0 ( s z + d z ) ( s y + d y ) ( s z + d z ) 0 ( s x + d x ) ] [ D ] = [ ( s x + d x ) ( s y + d y ) 0 0 ( s x + d x ) ( s y + d y ) ]
[ P ] = L D d z 2 [ ( d y 2 1 ) ( d x d y ) ( d x d y ) ( d x 2 1 ) ] [ T ] = [ ( s x 2 1 ) L S + ( d x 2 1 ) L D ( s x s y ) L S + ( d x d y ) L D ( s x s z ) L S + ( d x d z ) L D ( s x s y ) L S + ( d x d y ) L D ( s y 2 1 ) L S + ( d y 2 1 ) L D ( s y s z ) L S + ( d y d z ) L D ]
[ A x A y ] = [ P ] ( ( [ T ] + [ T ' ] ) [ a x a y a z ] + [ R ] [ Ω x Ω y Ω z ] + [ D ] [ ε x x ε x y ε y y ] )
[ P ] = L D ( d z 2 m x d x d z m y d y d z ) [ ( m y d y d z + d y 2 1 ) ( m x d y d z + d y d x ) ( m x d x d z + d y d x ) ( m x d x d z + d x 2 1 ) ] [ T ] = [ ( s x 2 1 ) L S + ( d x 2 1 ) L D ( s x s y ) L S + ( d x d y ) L D ( s x s z ) L S + ( d x d z ) L D ( s x s y ) L S + ( d x d y ) L D ( s y 2 1 ) L S + ( d y 2 1 ) L D ( s y s z ) L S + ( d y d z ) L D ] [ T ' ] = [ m x ( s x s z L S + d x d z L D ) m x ( s y s z L S + d y d z L D ) m x ( ( s z 2 1 ) L S + ( d z 2 1 ) L D ) m y ( s x s z L S + d x d z L D ) m y ( s y s z L S + d y d z L D ) m y ( ( s z 2 1 ) L S + ( d z 2 1 ) L D ) ]

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