Abstract

Presented herein is a performance analysis of a maximum likelihood estimator for calculating small speckle motions. Such estimators are important in a variety of speckle techniques used in non-destructive evaluation. The analysis characterizes the performance (bias and RMS deviation) of the estimator as a function of the signal-to-noise ratio. This SNR parameter is a convenient surrogate for decorrelation of sequential speckle patterns such as are seen in biological tissues. Although the particular estimator is predicated on speckle motions that are a small fraction of a pixel, accurate performance is demonstrated for instantaneous motions of up to ±0.8 pixel/record. Beyond this point, more conventional approaches have been shown to perform well.

© 2002 Optical Society of America

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References

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  1. D. D. Duncan, F. F. Mark, and L. W. Hunter, ???A new speckle technique for noncontact measurement of small creep rates,??? Opt. Eng. 31, 1583-1589 (1992).
    [CrossRef]
  2. D. D. Duncan, S. J. Kirkpatrick, F. F. Mark, and L. W. Hunter, ???Transform method of processing for speckle strain-rate measurements,??? Appl. Opt. 33, 5177-5186 (1994).
    [CrossRef] [PubMed]
  3. D. D. Duncan and S. J. Kirkpatrick, ???Processing algorithms for tracking speckle shifts in optical elastography of biological tissues,??? J. Biomed Opt. 6, 418-426 (2001).
    [CrossRef] [PubMed]
  4. S. J. Kirkpatrick and D. D. Duncan, ???Optical Assessment of Tissue Mechanics??? in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE Press, Bellingham, WA. 2002).
  5. E. E. Konofagou, T. Varghese, J. Ophir, and S. K. Alam, ???Power spectral strain estimators in elastography,??? Ultrasound Med. Biol. 27, 1115-1129 (1999).
    [CrossRef]
  6. A. Oulamara, G. Tribillon, and J. Duvernoy, ???Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,??? J. Mod. Opt. 36, 165-179 (1989).
    [CrossRef]
  7. I. Yamaguchi, ???Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,??? Opt. Acta. 28, 1359-1376 (1981).
    [CrossRef]
  8. S. J. Kirkpatrick and D. D. Duncan, ???Noncontact microstrain measurements in orthodontic wires,??? J. Biomed. Mater. Res. 29, 1437-1442 (1995).
    [CrossRef] [PubMed]
  9. S. J. Kirkpatrick and B. W. Brooks, ???Micromechanical behavior of cortical bone as inferred from laser speckle data,??? J. Biomed. Mater. Res. 39, 373-379 (1998).
    [CrossRef] [PubMed]
  10. S. J. Kirkpatrick and M. J. Cipolla, ???High resolution imaged laser speckle strain gauge for vascular applications,??? J. Biomed Opt. 5, 62-71 (2000).
    [CrossRef] [PubMed]
  11. J. W. Goodman, ???Statistical properties of laser speckles??? in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, Second Enlarged Edition, J. C. Dainty ed. (Springer Verlag, New York 1984).
  12. B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC Press, Boca Raton 1997).
  13. R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd edition (John Wiley & Sons, Inc. New York 2001).
  14. C. F. Gerald, Applied Numerical Analysis (Addison-Wesley Publishing Company, Reading 1973).
  15. T. Takemori, K. Fujita, and I. Yamaguchi, ???Resolution improvement in speckle displacement and strain sensor by correlation interpolation,??? Laser Interferometry IV: Computer Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 137-148 (1990).

Appl. Opt. (1)

J. Biomed Opt. (2)

D. D. Duncan and S. J. Kirkpatrick, ???Processing algorithms for tracking speckle shifts in optical elastography of biological tissues,??? J. Biomed Opt. 6, 418-426 (2001).
[CrossRef] [PubMed]

S. J. Kirkpatrick and M. J. Cipolla, ???High resolution imaged laser speckle strain gauge for vascular applications,??? J. Biomed Opt. 5, 62-71 (2000).
[CrossRef] [PubMed]

J. Biomed. Mater. Res. (2)

S. J. Kirkpatrick and D. D. Duncan, ???Noncontact microstrain measurements in orthodontic wires,??? J. Biomed. Mater. Res. 29, 1437-1442 (1995).
[CrossRef] [PubMed]

S. J. Kirkpatrick and B. W. Brooks, ???Micromechanical behavior of cortical bone as inferred from laser speckle data,??? J. Biomed. Mater. Res. 39, 373-379 (1998).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

A. Oulamara, G. Tribillon, and J. Duvernoy, ???Biological activity measurement on botanical specimen surfaces using a temporal decorrelation effect of laser speckle,??? J. Mod. Opt. 36, 165-179 (1989).
[CrossRef]

Opt. Acta. (1)

I. Yamaguchi, ???Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,??? Opt. Acta. 28, 1359-1376 (1981).
[CrossRef]

Opt. Eng. (1)

D. D. Duncan, F. F. Mark, and L. W. Hunter, ???A new speckle technique for noncontact measurement of small creep rates,??? Opt. Eng. 31, 1583-1589 (1992).
[CrossRef]

Proc. SPIE (1)

T. Takemori, K. Fujita, and I. Yamaguchi, ???Resolution improvement in speckle displacement and strain sensor by correlation interpolation,??? Laser Interferometry IV: Computer Aided Interferometry, R. J. Pryputniewicz, ed., Proc. SPIE 1553, 137-148 (1990).

Ultrasound Med. Biol. (1)

E. E. Konofagou, T. Varghese, J. Ophir, and S. K. Alam, ???Power spectral strain estimators in elastography,??? Ultrasound Med. Biol. 27, 1115-1129 (1999).
[CrossRef]

Other (5)

S. J. Kirkpatrick and D. D. Duncan, ???Optical Assessment of Tissue Mechanics??? in Handbook of Optical Biomedical Diagnostics, V. V. Tuchin, ed. (SPIE Press, Bellingham, WA. 2002).

J. W. Goodman, ???Statistical properties of laser speckles??? in Topics in Applied Physics, Vol. 9: Laser Speckle and Related Phenomena, Second Enlarged Edition, J. C. Dainty ed. (Springer Verlag, New York 1984).

B. Jähne, Practical Handbook on Image Processing for Scientific Applications (CRC Press, Boca Raton 1997).

R. O. Duda, P. E. Hart, and D. G. Stork, Pattern Classification, 2nd edition (John Wiley & Sons, Inc. New York 2001).

C. F. Gerald, Applied Numerical Analysis (Addison-Wesley Publishing Company, Reading 1973).

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Figures (20)

Fig. 1.
Fig. 1.

Illustration of frozen speckle concept

Fig. 2.
Fig. 2.

Simulated 1-dimensional speckle pattern

Fig. 3.
Fig. 3.

Instantaneous speckle shift used in simulation

Fig. 4.
Fig. 4.

Typical simulated speckle history without additive noise

Fig. 5.
Fig. 5.

Instantaneous speckle shifts using spatial derivative operator [-1, 0, 1]/2

Fig. 6.
Fig. 6.

Instantaneous speckle shifts using spatial derivative operator [1, -8, 0, 8, -1]/12

Fig. 7.
Fig. 7.

Instantaneous speckle shifts using spatial derivative operator [3, -32, 168, -672, 0, 672, -168, 32, -3]/840 (4th order central difference)

Fig. 8.
Fig. 8.

Instantaneous speckle shifts using spatial derivative operator [0.12019, - 0.74038, 0, 0.74038, - 0.12019] (MMSE optimized derivative operator [12])

Fig. 9.
Fig. 9.

Bias error as a function of SNR for instantaneous speckle motion of ±0.1 pixel

Fig. 10.
Fig. 10.

RMS error as a function of SNR for instantaneous speckle motion of ±0.1 pixel

Fig. 11.
Fig. 11.

Bias as a function of SNR for an instantaneous speckle shift of ±0.2 pixel

Fig. 12.
Fig. 12.

Bias as a function of SNR for an instantaneous speckle shift of ±0.3 pixel

Fig. 13.
Fig. 13.

Typical noiseless speckle history for instantaneous speckle shift of ±0.3 pixel

Fig. 14.
Fig. 14.

Alignment of speckle history based on first estimate of instantaneous speckle shift

Fig. 15.
Fig. 15.

Effect of iterative refinement algorithm for larger speckle shifts

Fig 16.
Fig 16.

Bias performance of iterative refinement algorithm for instantaneous speckle motion of ±0.3 pixel, using temporal derivative kernel [-1, 0, 1]/2 . Note lower SNR threshold.

Fig. 17.
Fig. 17.

RMS noise performance of iterative refinement algorithm for instantaneous speckle motion of ±0.3 pixel, using temporal derivative kernel [-1, 0, 1]/2

Fig. 18
Fig. 18

Instantaneous speckle shift estimates for a ±0.3 pixel shift using temporal derivative kernel [1, - 8, 0, 8, - 1]

Fig. 19
Fig. 19

High-SNR bias levels for iterative and non-iterative approaches

Fig. 20.
Fig. 20.

High-SNR RMS noise asymptotes for iterative and non-iterative approaches

Equations (13)

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

ε = β ( + θ s ) β ( θ s ) 2 L o sin θ s ,
g j + 1 ( x i ) = g j ( x i β ) ,
g j + 1 ( x i ) g j ( x i ) β g j ( x i ) .
[ g j + 1 ( x i ) , g j 1 ( x i ) ] .
ε j 2 = i = 1 N [ g j + 1 ( x i + β ) g j 1 ( x i β ) ] 2 ,
β ̂ j = i = 1 N [ g j + 1 ( x i ) g j 1 ( x i ) ] [ g j + 1 ( x i ) + g j 1 ( x i ) ] i = 1 N [ g j + 1 ( x i ) g j 1 ( x i ) ] 2 .
[ 0.12019 , 0.74038,0 , 0.74038 , 0.12019 ]
[ 0.0178608 , 0.0949175,0.298974 , 0.88464,0 ,
0.88464 , 0.298974,0.0949175 , 0.0178608 ] .
I = F { exp ( ix ) } 2 ,
s ( SNR ) = 1 exp { 2 α ( SNR SNR ) } 1 + exp { 2 α ( SNR SNR ) } ,
bias ( SNR ) = ( b + 1 2 ) s ( SNR ) + ( b 1 2 ) ,
RMS ( SNR ) = ( h l 2 ) s ( SNR ) + ( h + l 2 ) ,

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