Abstract

We present a technique to measure velocity, acceleration, and higher order derivatives of motion using periodic and nonperiodic spatial filters. The technique can be applied using a single detector or an array of detectors. In one configuration, the velocity distribution of an object such as a fluid can be measured by imaging the object onto an array of detectors. In another configuration, multiple projections of an object are used to reconstruct a cross-sectional velocity distribution using a tomography algorithm. The advantages and disadvantages of our technique applied to uniform and spatially varying motions are described.

© 2009 Optical Society of America

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References

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  1. J. T. Ator, “Image-velocity sensing with parallel-slit reticles,” J. Opt. Soc. Am. 53, 1416-1419 (1963).
    [CrossRef]
  2. J. T. Ator, “Image velocity sensing by optical correlation,” Appl. Opt. 5, 1325-1331 (1966).
    [CrossRef] [PubMed]
  3. Y. Aizu and T. Asakura, “Principles and development of spatial filtering velocimetry,” Appl. Phys. B 43, 209-224 (1987).
    [CrossRef]
  4. Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications (Springer, 2006).
  5. K. Christofori and K. Michel, “Velocimetry with spatial filters based on sensor arrays,” Flow Meas. Instrum. 7, 265-272(1996).
    [CrossRef]
  6. M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
    [CrossRef] [PubMed]
  7. M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
    [CrossRef]
  8. D. V. Semenov, E. Nippolainen, and A. A. Kamshilin, “Accuracy and resolution of a dynamic-speckle profilometer,” Appl. Opt. 45, 411-418 (2006).
    [CrossRef] [PubMed]
  9. T. Ushizaka and T. Asakura, “Measurements of flow velocity in a microscopic region using a transmission grating,” Appl. Opt. 22, 1870-1874 (1983).
    [CrossRef] [PubMed]
  10. O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
    [CrossRef]
  11. M. S. Uddin, H. Inaba, Y. Itakura, Y. Yoshida, and M. Kasahara, “Adaptive computer-based spatial-filtering method for more accurate estimation of the surface velocity of debris flow,” Appl. Opt. 38, 6714-6721 (1999).
    [CrossRef]
  12. P. Reinicke and J. Meyer-ter-Vehn, “The point explosion with heat conduction,” Phys. Fluids A 3, 1807-1818 (1991).
    [CrossRef]
  13. S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44, 623-631 (2008).
    [CrossRef]
  14. M. P. Wernet, “Symmetric phase only filtering: a new paradigm for DPIV data processing,” Meas. Sci. Technol. 16, 601-618 (2005).
    [CrossRef]
  15. F. Kato and I. Shimizu, “Optical processing of particle tracking velocimetry under deformed double exposure,” Meas. Sci. Technol. 11, 646-654 (2000).
    [CrossRef]
  16. K. T. Christensen and R. J. Adrian, “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy,” Exp. Fluids 33, 759-769(2002).
  17. X. Liu and J. Katz, “Instantaneous pressure and material acceleration measurements using a four-exposure PIV system,” Exp. Fluids 41, 227-240 (2006).
    [CrossRef]
  18. S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
    [CrossRef]
  19. H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. 14, 307-310 (1975).
  20. S. Bergeler and H. Krambeer, “Novel optical spatial filtering methods based on two-dimensional photodetector arrays,” Meas. Sci. Technol. 15, 1309-1315 (2004).
    [CrossRef]
  21. M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
    [CrossRef]
  22. A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
    [CrossRef]
  23. W. F. Hughes and J. A. Brighton, Fluidic Dynamics, 3rd ed. (McGraw Hill, 1999), p. 137.
  24. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).
  25. H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 217-286 (2006).
    [CrossRef]
  26. M. Kachelriess and W. A. Kalender, “Presampling, algorithm factors, and noise: consideration for CT in particular and for medical imaging in general,” Med. Phys. 32, 1321-1334 (2005).
    [CrossRef] [PubMed]

2008

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44, 623-631 (2008).
[CrossRef]

2006

X. Liu and J. Katz, “Instantaneous pressure and material acceleration measurements using a four-exposure PIV system,” Exp. Fluids 41, 227-240 (2006).
[CrossRef]

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 217-286 (2006).
[CrossRef]

D. V. Semenov, E. Nippolainen, and A. A. Kamshilin, “Accuracy and resolution of a dynamic-speckle profilometer,” Appl. Opt. 45, 411-418 (2006).
[CrossRef] [PubMed]

2005

M. Kachelriess and W. A. Kalender, “Presampling, algorithm factors, and noise: consideration for CT in particular and for medical imaging in general,” Med. Phys. 32, 1321-1334 (2005).
[CrossRef] [PubMed]

M. P. Wernet, “Symmetric phase only filtering: a new paradigm for DPIV data processing,” Meas. Sci. Technol. 16, 601-618 (2005).
[CrossRef]

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
[CrossRef]

2004

S. Bergeler and H. Krambeer, “Novel optical spatial filtering methods based on two-dimensional photodetector arrays,” Meas. Sci. Technol. 15, 1309-1315 (2004).
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
[CrossRef] [PubMed]

2002

K. T. Christensen and R. J. Adrian, “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy,” Exp. Fluids 33, 759-769(2002).

2001

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

2000

F. Kato and I. Shimizu, “Optical processing of particle tracking velocimetry under deformed double exposure,” Meas. Sci. Technol. 11, 646-654 (2000).
[CrossRef]

S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
[CrossRef]

1999

1997

M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
[CrossRef]

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

1996

K. Christofori and K. Michel, “Velocimetry with spatial filters based on sensor arrays,” Flow Meas. Instrum. 7, 265-272(1996).
[CrossRef]

1991

P. Reinicke and J. Meyer-ter-Vehn, “The point explosion with heat conduction,” Phys. Fluids A 3, 1807-1818 (1991).
[CrossRef]

1987

Y. Aizu and T. Asakura, “Principles and development of spatial filtering velocimetry,” Appl. Phys. B 43, 209-224 (1987).
[CrossRef]

1983

1975

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. 14, 307-310 (1975).

1966

1963

Adrian, R. J.

K. T. Christensen and R. J. Adrian, “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy,” Exp. Fluids 33, 759-769(2002).

Aizu, Y.

Y. Aizu and T. Asakura, “Principles and development of spatial filtering velocimetry,” Appl. Phys. B 43, 209-224 (1987).
[CrossRef]

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications (Springer, 2006).

Asakura, T.

Y. Aizu and T. Asakura, “Principles and development of spatial filtering velocimetry,” Appl. Phys. B 43, 209-224 (1987).
[CrossRef]

T. Ushizaka and T. Asakura, “Measurements of flow velocity in a microscopic region using a transmission grating,” Appl. Opt. 22, 1870-1874 (1983).
[CrossRef] [PubMed]

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications (Springer, 2006).

Ator, J. T.

Barrett, H. H.

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 217-286 (2006).
[CrossRef]

Bergeler, S.

S. Bergeler and H. Krambeer, “Novel optical spatial filtering methods based on two-dimensional photodetector arrays,” Meas. Sci. Technol. 15, 1309-1315 (2004).
[CrossRef]

Brighton, J. A.

W. F. Hughes and J. A. Brighton, Fluidic Dynamics, 3rd ed. (McGraw Hill, 1999), p. 137.

Christensen, K. T.

K. T. Christensen and R. J. Adrian, “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy,” Exp. Fluids 33, 759-769(2002).

Christofori, K.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

K. Christofori and K. Michel, “Velocimetry with spatial filters based on sensor arrays,” Flow Meas. Instrum. 7, 265-272(1996).
[CrossRef]

Coupland, J.

S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
[CrossRef]

Fiedler, O.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

Gray, C.

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Greated, C. A.

M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
[CrossRef]

Grue, J.

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Hanson, S. G.

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
[CrossRef] [PubMed]

Harvey, D.

M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
[CrossRef]

Hocknell, A.

S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
[CrossRef]

Hughes, W. F.

W. F. Hughes and J. A. Brighton, Fluidic Dynamics, 3rd ed. (McGraw Hill, 1999), p. 137.

Inaba, H.

Itakura, Y.

Jakobsen, M. L.

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
[CrossRef]

M. L. Jakobsen and S. G. Hanson, “Lenticular array for spatial filtering velocimetry of laser speckles from solid surfaces,” Appl. Opt. 43, 4643-4651 (2004).
[CrossRef] [PubMed]

M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
[CrossRef]

Jensen, A.

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Kachelriess, M.

M. Kachelriess and W. A. Kalender, “Presampling, algorithm factors, and noise: consideration for CT in particular and for medical imaging in general,” Med. Phys. 32, 1321-1334 (2005).
[CrossRef] [PubMed]

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Kalender, W. A.

M. Kachelriess and W. A. Kalender, “Presampling, algorithm factors, and noise: consideration for CT in particular and for medical imaging in general,” Med. Phys. 32, 1321-1334 (2005).
[CrossRef] [PubMed]

Kamshilin, A. A.

Kasahara, M.

Kato, F.

F. Kato and I. Shimizu, “Optical processing of particle tracking velocimetry under deformed double exposure,” Meas. Sci. Technol. 11, 646-654 (2000).
[CrossRef]

Katz, J.

X. Liu and J. Katz, “Instantaneous pressure and material acceleration measurements using a four-exposure PIV system,” Exp. Fluids 41, 227-240 (2006).
[CrossRef]

Kim, S.

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44, 623-631 (2008).
[CrossRef]

Krambeer, H.

S. Bergeler and H. Krambeer, “Novel optical spatial filtering methods based on two-dimensional photodetector arrays,” Meas. Sci. Technol. 15, 1309-1315 (2004).
[CrossRef]

Kumpart, J.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

Labahn, N.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

Larsen, H. E.

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
[CrossRef]

Lee, S. J.

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44, 623-631 (2008).
[CrossRef]

Liu, X.

X. Liu and J. Katz, “Instantaneous pressure and material acceleration measurements using a four-exposure PIV system,” Exp. Fluids 41, 227-240 (2006).
[CrossRef]

Meyer-ter-Vehn, J.

P. Reinicke and J. Meyer-ter-Vehn, “The point explosion with heat conduction,” Phys. Fluids A 3, 1807-1818 (1991).
[CrossRef]

Michel, K.

K. Christofori and K. Michel, “Velocimetry with spatial filters based on sensor arrays,” Flow Meas. Instrum. 7, 265-272(1996).
[CrossRef]

Nippolainen, E.

Ogiwara, H.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. 14, 307-310 (1975).

Reinicke, P.

P. Reinicke and J. Meyer-ter-Vehn, “The point explosion with heat conduction,” Phys. Fluids A 3, 1807-1818 (1991).
[CrossRef]

Richon, J. B.

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Rothberg, S.

S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
[CrossRef]

Semenov, D. V.

Shimizu, I.

F. Kato and I. Shimizu, “Optical processing of particle tracking velocimetry under deformed double exposure,” Meas. Sci. Technol. 11, 646-654 (2000).
[CrossRef]

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

Sveen, J. K.

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Uddin, M. S.

Ukita, H.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. 14, 307-310 (1975).

Ushizaka, T.

Wernet, M. P.

M. P. Wernet, “Symmetric phase only filtering: a new paradigm for DPIV data processing,” Meas. Sci. Technol. 16, 601-618 (2005).
[CrossRef]

Werther, J.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

Yoshida, Y.

Appl. Opt.

Appl. Phys. B

Y. Aizu and T. Asakura, “Principles and development of spatial filtering velocimetry,” Appl. Phys. B 43, 209-224 (1987).
[CrossRef]

Exp. Fluids

S. Kim and S. J. Lee, “Effect of particle number density in in-line digital holographic particle velocimetry,” Exp. Fluids 44, 623-631 (2008).
[CrossRef]

K. T. Christensen and R. J. Adrian, “Measurement of instantaneous Eulerian acceleration fields by particle image accelerometry: method and accuracy,” Exp. Fluids 33, 759-769(2002).

X. Liu and J. Katz, “Instantaneous pressure and material acceleration measurements using a four-exposure PIV system,” Exp. Fluids 41, 227-240 (2006).
[CrossRef]

A. Jensen, J. K. Sveen, J. Grue, J. B. Richon, and C. Gray, “Accelerations in water waves by extended particle image velocimetry,” Exp. Fluids 30, 500-510 (2001).
[CrossRef]

Flow Meas. Instrum.

K. Christofori and K. Michel, “Velocimetry with spatial filters based on sensor arrays,” Flow Meas. Instrum. 7, 265-272(1996).
[CrossRef]

J. Opt. A: Pure Appl. Opt.

M. L. Jakobsen, H. E. Larsen, and S. G. Hanson, “Optical spatial filtering velocimetry sensor for sub-micron, in-plane vibration measurements,” J. Opt. A: Pure Appl. Opt. 7, S303-S307 (2005).
[CrossRef]

J. Opt. Soc. Am.

Jpn. J. Appl. Phys.

H. Ogiwara and H. Ukita, “A speckle pattern velocimeter using a periodical differential detector,” Jpn. J. Appl. Phys. 14, 307-310 (1975).

Meas. Sci. Technol.

S. Bergeler and H. Krambeer, “Novel optical spatial filtering methods based on two-dimensional photodetector arrays,” Meas. Sci. Technol. 15, 1309-1315 (2004).
[CrossRef]

M. L. Jakobsen, D. Harvey, and C. A. Greated, “Particle image velocimetry for predictions of acceleration fields and force within fluid flows,” Meas. Sci. Technol. 8, 1502-15161502 (1997).
[CrossRef]

M. P. Wernet, “Symmetric phase only filtering: a new paradigm for DPIV data processing,” Meas. Sci. Technol. 16, 601-618 (2005).
[CrossRef]

F. Kato and I. Shimizu, “Optical processing of particle tracking velocimetry under deformed double exposure,” Meas. Sci. Technol. 11, 646-654 (2000).
[CrossRef]

Med. Phys.

M. Kachelriess and W. A. Kalender, “Presampling, algorithm factors, and noise: consideration for CT in particular and for medical imaging in general,” Med. Phys. 32, 1321-1334 (2005).
[CrossRef] [PubMed]

Opt. Lasers Eng.

S. Rothberg, A. Hocknell, and J. Coupland, “Developments in laser Doppler accelerometry (LDAc) and comparison with laser Doppler velocimetry,” Opt. Lasers Eng. 32, 549-564 (2000).
[CrossRef]

Phys. Fluids A

P. Reinicke and J. Meyer-ter-Vehn, “The point explosion with heat conduction,” Phys. Fluids A 3, 1807-1818 (1991).
[CrossRef]

Powder Technol.

O. Fiedler, J. Werther, N. Labahn, J. Kumpart, and K. Christofori, “Measurement of local particle velocities and velocity distributions in gas-solid flows by means of the spatial filter method,” Powder Technol. 94, 51-57 (1997).
[CrossRef]

Prog. Opt.

H. H. Barrett, “The Radon transform and its applications,” Prog. Opt. 21, 217-286 (2006).
[CrossRef]

Other

Y. Aizu and T. Asakura, Spatial Filtering Velocimetry, Fundamentals and Applications (Springer, 2006).

W. F. Hughes and J. A. Brighton, Fluidic Dynamics, 3rd ed. (McGraw Hill, 1999), p. 137.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE, 1988).

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Figures (8)

Fig. 1
Fig. 1

Schematic of an experimental setup for spatial filtering velocimetry. The moving object is imaged onto the spatial filter.

Fig. 2
Fig. 2

(a) Simulated signal as a function of time for an object moving with parameters a = 1.5 , v 0 = 0.3 , and Δ = 2 , and a grating with period p = 8 and M = 1 . (b) Same signal after a change of variable to u, assuming t 0 = 0.2 . (c) Phase of the Fourier transform of the signal. (d) Log of the Fourier power spectrum of the signal in the new coordinate u.

Fig. 3
Fig. 3

Sample signal as a function of time. The local period of the oscillation is shown.

Fig. 4
Fig. 4

Fourier transform of a simulated signal after a change of variable using different values of t 0 . The parameters are a = 1.5 , v 0 = 3.7 , Δ = 2 , p = 8 , and M = 1 . From the figure, we see that the sideband location for t 0 = 2.5 is f a = 0.187 .

Fig. 5
Fig. 5

(a) Spatial filter on top of a light detector with a microlens and (b) a set of spatial filters with periodicity p. The four filters can be put in front of four pixels of an array detector to measure the velocity and acceleration of a localized region of an image.

Fig. 6
Fig. 6

(a) Schematic cross section of a laminar flow through a parallel plate, (b) equal contours for magnitude of the velocity, and (c) a nonperiodic spatial filter that matches the velocity flow.

Fig. 7
Fig. 7

Schematic showing the projection of a 3D object at angle θ. An array of detector with spatial filter is also shown. Tomography can be applied to the projection data to obtain the density of the object with pixel resolution of Δ x by Δ y by Δ z as a function of time.

Fig. 8
Fig. 8

Procedure to calculate the velocity distribution of an object from its projection data.

Equations (27)

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

v 0 = p f v M .
S ( f ) = A δ ( f ) + 2 π B 2 f a cos [ π f 2 2 f a + π 4 ] ,
a = p M f a .
x ( t ) = x 0 + v 0 t + a t 2 2 + O ( t 3 ) .
x ( t ) = a 2 ( t + t 0 ) 2 + x 0 a t 0 2 2 ,
T ( t = 1 2 ( t n + 1 + t n ) ) = 2 ( t n + 1 t n ) ,
s ( t ) = A + B cos [ 2 π M p x ( t ) ] = A + B cos [ 2 π M p ( v 0 + a t 2 ) t ] = A + B cos ( ω ( t ) t ) ,
ω ( t ) = 2 π M p ( v 0 + a t 2 ) = 2 π T ( t )
x ( T d + Δ T ) x ( T d ) = p ,
Δ T = ( T d + t 0 ) 2 + 2 p / a ( T d + t 0 ) .
S ( f a ) > S noise ( f a ) ,
v ¯ = ( v x ( x , y ) , v y ( x , y ) ) = ( v 0 g x ( x , y ) , v 0 g y ( x , y ) ) .
f v = M p v p ,
h ( x , y ) = 1 + cos [ 2 π p ( v ¯ | v ¯ | r ¯ ) ] = 1 + cos [ 2 π f v M v 0 ( g x ( x , y ) x + g y ( x , y ) y g x 2 + g y 2 ) ]
v ¯ = ( V c ( y / δ ) 1 / 7 , 0 ) ,
P ( θ , s , t ) = d x d y f ( x , y , t ) δ ( x cos θ + y sin θ s ) .
S ( θ , ω , t ) = d s P ( θ , s , t ) e j 2 π ω s .
F ( u , v , t ) = d x d y f ( x , y , t ) e j 2 π ( u x + v y ) .
F ( u , 0 , t ) = S ( θ = 0 , ω , t ) .
T acq < Δ z / v z .
D D t = t + v x x + v y y ,
f ( x , y ) = intensity distribution of the object , h ( x , y ) = intensity transmittance of the spatial filter.
s ( t ) = g ( x r ( t ) , y r ( t ) ) = f ( x r x , y r y ) h ( x , y ) d x d y ,
G ( ν M v x ) = 1 | M v x | F ( ν M v x ) H ( ν M v x ) .
f ( x , y ) = i = 0 N exp [ ( x x i ) 2 + ( y y i ) 2 2 Δ ] .
h ( x , y ) = 1 + cos [ 2 π y p ] ,
s ( t ) = d x d y f ( x ( t ) , y ) h ( x ( t ) , y ) ,

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