Abstract

The goal of holographic particle velocimetry is to infer fluid velocity patterns from images reconstructed from doubly exposed holograms of fluid volumes seeded with small particles. The advantages offered by in-line holography in this context usually make it the method of choice, but seeding densities sufficient to achieve high spatial resolution in the sampling of the velocity fields cause serious degradation, through speckle, of the signal-to-noise ratio in the reconstructed images. The in-line method also leads to a great depth of field in paraxial viewing of reconstructed images, making it essentially impossible to estimate particle depth with useful accuracy. We present here an analysis showing that these limitations can be circumvented by variably scaled correlation, or wavelet transformation. The shift variables of the wavelet transform are provided automatically by the optical correlation methodology. The variable scaling of the wavelet transform derives, in this case, directly from the need to accommodate varying particle depths. To provide such scaling, we use a special optical system incorporating prescribed variability in spacings and focal length of lenses to scan through the range of particle depths.

Calculation shows, among other benefits, improvement by approximately two orders of magnitude in depth resolution. A much higher signal-to-noise ratio together with faster data extraction and processing should be attainable.

© 1995 Optical Society of America

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References

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  1. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).
  2. B. J. Thompson, “Diffraction by opaque and transparent objects,” in Photo-Optical Data Reduction, R. J. Gast, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2, 43–46 (1964).
  3. J. D. Trolinger, R. A. Belz, W. M. Farmer, “Holographic techniques for the study of dynamic particle fields,” Appl. Opt. 8, 957–961 (1969).
    [CrossRef] [PubMed]
  4. P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).
  5. Y. J. Lee, J. H. Kim, “A review of holography application in multiphase flow visualization study,” J. Fluids Eng. 108, 279–288 (1986).
    [CrossRef]
  6. L. M. Weinstein, G. R. Beeler, A. M. Lindemann, “High-speed holocinematographic velocimeter for studying turbulent flow control physics,” presented at the Processing American Institute of Aeronautics and Astronautics Shear Flow Control Conference, Boulder, Colo., 12–14 March 1985.
  7. H. Meng, F. Hussain, “Holographic particle velocimetry: a 3-D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
    [CrossRef]
  8. H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
    [CrossRef]
  9. G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
    [CrossRef]
  10. J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).
  11. L. Onural, M. T. Özgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
    [CrossRef]
  12. S. G. Mallet, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
    [CrossRef]
  13. O. Rioul, M. Vetterli, “Wavelets and signal processing,” IEEE Signal Process. Mag. 8(10), 14–38 (1991).
    [CrossRef]
  14. F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9(4), 21–67 (1992).
    [CrossRef]
  15. E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
    [CrossRef]
  16. J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
    [CrossRef]
  17. H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
    [CrossRef] [PubMed]
  18. D. Mendlovic, N. Konforti, “Optical realization of the wavelet transform for two-dimensional objects,” Appl. Opt. 32, 6542–6546 (1993).
    [CrossRef] [PubMed]
  19. A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
    [CrossRef]
  20. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  21. V. Zimin, H. Meng, F. Hussain, “Innovative holographic particle velocimeter: a multibeam technique,” Opt. Lett. 18, 1101–1103 (1993).
    [CrossRef] [PubMed]
  22. H. Meng, F. Hussain, “IROV (in-line recording and off-axis viewing): a novel holographic technique for particle field measurement and holographic particle velocimetry,” Appl. Opt. (to be published).

1993 (3)

1992 (4)

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9(4), 21–67 (1992).
[CrossRef]

L. Onural, M. T. Özgen, “Extraction of three-dimensional object-location information directly from in-line holograms using Wigner analysis,” J. Opt. Soc. Am. A 9, 252–260 (1992).
[CrossRef]

1991 (2)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3-D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

O. Rioul, M. Vetterli, “Wavelets and signal processing,” IEEE Signal Process. Mag. 8(10), 14–38 (1991).
[CrossRef]

1990 (1)

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

1989 (1)

S. G. Mallet, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

1986 (1)

Y. J. Lee, J. H. Kim, “A review of holography application in multiphase flow visualization study,” J. Fluids Eng. 108, 279–288 (1986).
[CrossRef]

1984 (1)

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).

1969 (1)

1964 (1)

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

1960 (2)

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Albersheim, W. J.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

Anderson, W. L.

Argoul, F.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Arneodo, A.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Beeler, G. R.

L. M. Weinstein, G. R. Beeler, A. M. Lindemann, “High-speed holocinematographic velocimeter for studying turbulent flow control physics,” presented at the Processing American Institute of Aeronautics and Astronautics Shear Flow Control Conference, Boulder, Colo., 12–14 March 1985.

Belz, R. A.

Boudreaux-Bartels, G. F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9(4), 21–67 (1992).
[CrossRef]

Caulfield, J.

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

Chen, J.

Darlington, S.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

Farmer, W. M.

Freysz, E.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

Hlawatsch, F.

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9(4), 21–67 (1992).
[CrossRef]

Hussain, F.

H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

V. Zimin, H. Meng, F. Hussain, “Innovative holographic particle velocimeter: a multibeam technique,” Opt. Lett. 18, 1101–1103 (1993).
[CrossRef] [PubMed]

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3-D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

H. Meng, F. Hussain, “IROV (in-line recording and off-axis viewing): a novel holographic technique for particle field measurement and holographic particle velocimetry,” Appl. Opt. (to be published).

Kim, J. H.

Y. J. Lee, J. H. Kim, “A review of holography application in multiphase flow visualization study,” J. Fluids Eng. 108, 279–288 (1986).
[CrossRef]

Klauder, J. R.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

Konforti, N.

Lee, Y. J.

Y. J. Lee, J. H. Kim, “A review of holography application in multiphase flow visualization study,” J. Fluids Eng. 108, 279–288 (1986).
[CrossRef]

Lindemann, A. M.

L. M. Weinstein, G. R. Beeler, A. M. Lindemann, “High-speed holocinematographic velocimeter for studying turbulent flow control physics,” presented at the Processing American Institute of Aeronautics and Astronautics Shear Flow Control Conference, Boulder, Colo., 12–14 March 1985.

Liu, D. D.

Mallet, S. G.

S. G. Mallet, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

Malyak, P. H.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).

Mendlovic, D.

Meng, H.

V. Zimin, H. Meng, F. Hussain, “Innovative holographic particle velocimeter: a multibeam technique,” Opt. Lett. 18, 1101–1103 (1993).
[CrossRef] [PubMed]

H. Meng, W. L. Anderson, F. Hussain, D. D. Liu, “Intrinsic speckle noise in in-line holography,” J. Opt. Soc. Am. A 10, 2046–2058 (1993).
[CrossRef]

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3-D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

H. Meng, F. Hussain, “IROV (in-line recording and off-axis viewing): a novel holographic technique for particle field measurement and holographic particle velocimetry,” Appl. Opt. (to be published).

Onural, L.

Özgen, M. T.

Pouligny, B.

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Price, A. C.

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

Rioul, O.

O. Rioul, M. Vetterli, “Wavelets and signal processing,” IEEE Signal Process. Mag. 8(10), 14–38 (1991).
[CrossRef]

Sheng, Y.

Szu, H.

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

H. Szu, Y. Sheng, J. Chen, “Wavelet transform as a bank of matched filters,” Appl. Opt. 31, 3267–3277 (1992).
[CrossRef] [PubMed]

Thompson, B. J.

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).

B. J. Thompson, “Diffraction by opaque and transparent objects,” in Photo-Optical Data Reduction, R. J. Gast, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2, 43–46 (1964).

Trolinger, J. D.

Turin, G. L.

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

VanderLugt, A. B.

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

Vetterli, M.

O. Rioul, M. Vetterli, “Wavelets and signal processing,” IEEE Signal Process. Mag. 8(10), 14–38 (1991).
[CrossRef]

Weinstein, L. M.

L. M. Weinstein, G. R. Beeler, A. M. Lindemann, “High-speed holocinematographic velocimeter for studying turbulent flow control physics,” presented at the Processing American Institute of Aeronautics and Astronautics Shear Flow Control Conference, Boulder, Colo., 12–14 March 1985.

Zimin, V.

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

J. R. Klauder, A. C. Price, S. Darlington, W. J. Albersheim, “The theory and design of chirp radars,” Bell Syst. Tech. J. 39, 745–806 (1960).

Fluid Dyn. Res. (1)

H. Meng, F. Hussain, “Holographic particle velocimetry: a 3-D measurement technique for vortex interactions, coherent structures and turbulence,” Fluid Dyn. Res. 8, 33–52 (1991).
[CrossRef]

IEEE Signal Process. Mag. (2)

O. Rioul, M. Vetterli, “Wavelets and signal processing,” IEEE Signal Process. Mag. 8(10), 14–38 (1991).
[CrossRef]

F. Hlawatsch, G. F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations,” IEEE Signal Process. Mag. 9(4), 21–67 (1992).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. B. VanderLugt, “Signal detection by complex spatial filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (1)

S. G. Mallet, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell. 11, 674–693 (1989).
[CrossRef]

IRE Trans. Inf. Theory (1)

G. L. Turin, “An introduction to matched filters,” IRE Trans. Inf. Theory 6, 311–329 (1960).
[CrossRef]

J. Fluids Eng. (1)

Y. J. Lee, J. H. Kim, “A review of holography application in multiphase flow visualization study,” J. Fluids Eng. 108, 279–288 (1986).
[CrossRef]

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 93, 429–457 (1946).

J. Opt. Soc. Am. A (2)

Opt. Eng. (2)

P. H. Malyak, B. J. Thompson, “Particle displacement and velocity measurement using holography,” Opt. Eng. 23, 567–576 (1984).

J. Caulfield, H. Szu, “Parallel discrete and continuous wavelet transforms,” Opt. Eng. 31, 1835–1839 (1992).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

E. Freysz, B. Pouligny, F. Argoul, A. Arneodo, “Optical wavelet transform of fractal aggregates,” Phys. Rev. Lett. 64, 7745–7748 (1990).
[CrossRef]

Other (4)

H. Meng, F. Hussain, “IROV (in-line recording and off-axis viewing): a novel holographic technique for particle field measurement and holographic particle velocimetry,” Appl. Opt. (to be published).

B. J. Thompson, “Diffraction by opaque and transparent objects,” in Photo-Optical Data Reduction, R. J. Gast, ed., Proc. Soc. Photo-Opt. Instrum. Eng. 2, 43–46 (1964).

L. M. Weinstein, G. R. Beeler, A. M. Lindemann, “High-speed holocinematographic velocimeter for studying turbulent flow control physics,” presented at the Processing American Institute of Aeronautics and Astronautics Shear Flow Control Conference, Boulder, Colo., 12–14 March 1985.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

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Figures (2)

Fig. 1
Fig. 1

In-line hologram recording of small particle scattering.

Fig. 2
Fig. 2

Diverging illumination of a hologram for wavelet transformation by signal scaling. The scaling involves varying the distance d k and concomitant adjustments of z 1 and z 2. H, hologram; L’s, lenses; F, Fourier plane; I, correlation plane.

Equations (20)

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I H ( x , y ) = U H U H * = 1 + O + O * = 1 + k = 1 N I k ( x , y , x k , y k , z k ) ,
I k ( x , y , x k , y k , z k ) = sin ( k r k 2 2 z k ) J 1 ( k b r k z k ) ( k b r k z k ) ,
WT s ( a , ξ , η ) = 1 a s ( x - ξ a , y - η a ) h * ( x , y ) d x d y .
WT s ( a , ξ , η ) = 1 a s h * ,
WT s ( c , ξ , η ) = 1 c s ( x - ξ , y - η ) h * ( x c , y c ) d x d y ,
WT s ( c , ξ , η ) = 1 c s h * ,
S WT ( α , ξ , η ) ( f x , f y ) = S ( a , ξ , η ) ( f x , f y ) H * ( f x , f y ) ,
S ( a , ξ , η ) ( f x , f y ) = s ( a , ξ , η ) ( x , y ) exp [ - j 2 π ( x f x + y f y ) ] d x d y = a exp [ - j 2 π ( ξ f x + η f y ) ] S ( a f x , a f y ) .
I ( x , y , 0 ) = 1 + 1 α 2 d k 2 + 2 α λ d k cos { k 2 d k [ ( x - x k ) 2 + ( y - y k ) 2 ] } ,
t H ( x ) = C + 2 D cos [ k 2 d k ( x - x k ) 2 ] = C + D { exp [ j k 2 d k ( x - x k ) 2 ] + exp [ - j k 2 d k ( x - x k ) 2 ] } .
U ( x ) = C exp [ j k 2 ( x 2 d k ) ] + D exp { j k 2 [ x 2 d k + ( x - x k ) 2 d k ] } + D exp ( - j k x k 2 2 d k ) exp ( j k x x k d k ) .
U 0 ( x ) = exp [ j k 2 ( x 2 d k ) ] equals the zero - order or direct ( undiffracted ) wave , U 1 ( x ) = exp { j k 2 [ x 2 d k + ( x - x k ) 2 d k ] equals the virtual image wave ( first - order diffraction ) , U 2 ( x ) = exp ( - j k x k 2 2 d k ) exp ( j k x x k d k ) equals the real image wave ( first - order diffraction ) .
U 0 ( x 2 ) = exp [ j k 2 z 2 ( 1 + 1 M ) x 2 2 ] exp ( j k x 2 2 2 M 2 d k )
U 0 ( x 2 , y 2 ) = exp [ j k ( x 2 2 + y 2 2 2 M 2 d k ] ,
U 1 ( x 2 , y 2 ) = exp [ j k ( x k 2 + y k 2 ) 2 d k ] exp [ j k ( x 2 2 + y 2 2 ) M 2 d k ] × exp [ j k ( x k x 2 + y k y 2 ) M d k ] ,
U 2 ( x 2 , y 2 ) = exp [ - j k ( x k 2 + y k 2 ) 2 d k ] exp [ - j k ( x k x 2 + y k y 2 ) M d k ] .
U 2 ( a f x , a f y ) = exp [ - j 2 π ( a f x x k + a f y y k ) ] ,
S ( a f x , a f x ) exp [ - j 2 π ( a f x x k + a f y y k ) ] ,
f x = x 2 λ d ,             f y = y 2 λ d ,
a = 1 / M = d k / d .

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