## Relation between the contrast in time integrated dynamic speckle patterns and the power spectral density of their temporal intensity fluctuations |

Optics Express, Vol. 18, Issue 21, pp. 21883-21891 (2010)

http://dx.doi.org/10.1364/OE.18.021883

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### Abstract

Scattering fluid flux can be quantified with coherent light, either from the contrast of speckle patterns, or from the moments of the power spectrum of intensity fluctuations. We present a theory connecting these approaches for the general case of mixed static-dynamic patterns of boiling speckles without prior assumptions regarding the particle dynamics. An expression is derived and tested relating the speckle contrast to the intensity power spectrum. Our theory demonstrates that in speckle contrast the concentration of moving particles dominates over the contribution of speed to the particle flux. Our theory provides a basis for comparison of both approaches when used for studying tissue perfusion.

© 2010 Optical Society of America

## 1. Introduction

1. P. Vennemann, R. Lindken, and J. Westerweel, “In vivo whole-field blood velocity measurement techniques,” Exp. Fluids **42**(4), 495–511 (2007). [CrossRef]

2. M. J. Draijer, E. Hondebrink, T. G. van Leeuwen, and W. Steenbergen, “Review of laser speckle contrast techniques for visualizing tissue perfusion,” Lasers Med. Sci. **24**(4), 639–651 (2009). [CrossRef]

3. R. Bonner and R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. **20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

*P*(

*ν*) of intensity fluctuations generated in the dynamic speckle pattern is analyzed in terms of its moments given by : where the zeroth order moment (

*i*= 0) is a measure for the concentration of red blood cells and the first order moment (

*i*= 1) is a measure for the flux or perfusion [3

3. R. Bonner and R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. **20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

3. R. Bonner and R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. **20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

6. D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. **15**(1), 011109 (2010). [CrossRef] [PubMed]

7. J. D. Briers and S. Webster, “Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields,” Opt. Commun. **116**, 36–42 (1995). [CrossRef]

9. A. B. Parthasarathy, W. J. Tom, A. Gopal, X. Zhang, and A. K. Dunn, “Robust flow measurement with multi-exposure speckle imaging,” Opt. Express **16**(3), 1975–1989 (2008). [CrossRef] [PubMed]

9. A. B. Parthasarathy, W. J. Tom, A. Gopal, X. Zhang, and A. K. Dunn, “Robust flow measurement with multi-exposure speckle imaging,” Opt. Express **16**(3), 1975–1989 (2008). [CrossRef] [PubMed]

10. P. Zakharov, A. Völker, A. Buck, B. Weber, and F. Scheffold, “Quantitative modeling of laser speckle imaging,” Opt. Lett. **31**(23), 3465–3467 (2006). [CrossRef] [PubMed]

**20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

11. G. E. Nilsson, T. Tenland, and P. A. Öberg, “A New Instrument for Continuous Measurement of Tissue Blood FlowBy LightBeating Spectroscopy,” IEEE Trans. Biomed. Engin. **27**(1), 12–18 (1980). [CrossRef]

8. J. C. Ramirez-San-Juan, R. Ramos-Garcia, I. Guizar-Iturbide, G. Martinez-Niconoff, and B. Choi, “Impact of velocity distribution assumption on simplified laser speckle imaging equation,” Opt. Express **16**(5), 3197–3203 (2008). [CrossRef] [PubMed]

12. D. D. Duncan and S. J. Kirkpatrick, “Can laser speckle flowmetry be made a quantitative tool?” J. Opt. Soc. Am. A **25**(8), 2088–2094 (2008). [CrossRef]

*C*

^{2}. In this definition, the flux parameter will increase with increasing flow or concentration. Verification of the theory will be done by simulated speckle patterns. Linking speckle contrast flowmetry to a model based on the power spectrum of intensity fluctuations may enable quantification in terms of flux rather than velocity only.

## 2. Theory

*T*can be written as : The intensity in an unblurred polarized speckle pattern will have an exponential probability density function [14], so

*C*

^{2}(0) equals unity. Since in the right hand side of equation 3 only

*I*(

_{T}*x, t*) depends on T, substitution of equation 3 in equation 4 gives By using Parsevals’s theorem and the transfer function

*H*(

*T,ν*),

*i*= 0, reduces to since |

*H*(0,

*ν*)| = 1. Equation (8) expresses the speckle contrast in terms of the power spectrum of the local temporal intensity fluctuations in the speckle pattern. For

*T*↓ 0, it holds that

*H*(

*T, ν*) → 1 for all frequencies, reducing Eq. (8) to

*C*

^{2}= 1, which is the required value for a snapshot of the speckle pattern. For

*T*→ ∞,

*H*(

*T, ν*) approaches zero and the last term on the right-hand side cancels out, resulting in

*C*

^{2}(

*T*→ ∞) = 1 –

*M*

_{0}/〈

*I*〉

^{2}. For a completely dynamic speckle pattern, the property of ergodicity leads to

*M*

_{0}/〈

*I*〉

^{2}= 1 so

*C*

^{2}(

*T*→ ∞) → 0. Hence, Eq. (8) shows the required behavior for extreme cases.

*M*

_{0}/〈

*I*〉

^{2}=

*f*(2 –

_{D}*f*) with

_{D}*f*the fraction of Doppler shifted light [15

_{D}15. A. Serov, W. Steenbergen, and F. F. M. de Mul, “Prediction of the photodetector signal generated by Doppler-induced speckle fluctuations: theory and some validations,” J. Opt. Soc. Am. A **18**(3), 622–630 (2001). [CrossRef]

10. P. Zakharov, A. Völker, A. Buck, B. Weber, and F. Scheffold, “Quantitative modeling of laser speckle imaging,” Opt. Lett. **31**(23), 3465–3467 (2006). [CrossRef] [PubMed]

16. D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. **25**(1), 9–15 (2008). [CrossRef]

*T*using Eq. (8).

2. M. J. Draijer, E. Hondebrink, T. G. van Leeuwen, and W. Steenbergen, “Review of laser speckle contrast techniques for visualizing tissue perfusion,” Lasers Med. Sci. **24**(4), 639–651 (2009). [CrossRef]

**20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

*P*(

*ν*) weighed with |

*H*(

*T, ν*)|

^{2}. For the contrast based flux parameter 1 –

*C*

^{2}we can derive from Eq. (8) that : In Fig. 5 the spectral weighting function in Eq. (9) is shown for integration times of 1, 2, 5 and 15 ms, respectively. The weighting function increases nonlinearly from 0 to 1 at

*ν*= 1/

*T*. For

*ν*> 1/

*T*the weighting function is almost constant, showing decaying oscillations between 1 and 0.95. Hence, for integration times 5 ms <

*T*< 15 ms realistic for speckle contrast techniques, frequency dependent weighting is only performed in a frequency interval between 0 Hz and 67 Hz <

*ν*< 200 Hz. For higher frequencies, 1 –

*C*

^{2}mainly provides the zero order moment of the power spectrum. For realistic integration times and for low concentrations of moving particles that are assumed in the theory of Bonner and Nossal [3

**20**(12), 2097–2107 (1981). [CrossRef] [PubMed]

*ν*< 1/

*T*(in Hz). In this interval the frequency weighting is nonlinear, with an approximately 2

*order weighting (∝*

^{nd}*ν*

^{2}) for

*ν*→ 0. The second order weighting can be extended to higher frequencies by reducing integration time

*T*. For instance, assuming that the spectral weighting function associated with 1 –

*C*

^{2}has a second order behavior for 0 <

*ν*< 1/2

*T*, the power spectrum shown in Fig. 3, with a width of 5 kHz which is typical for physiological perfusion, would be obtained for an integration time of approximately 0.1 ms. This will give a contrast value which is only slightly smaller than 1, and therefore will not be very sensitive. Figure 5 also suggests that speckle contrast methods will be particularly sensitive to changes in the power spectrum in the low frequency range, e.g. caused by overall tissue motion or speed variations of particles moving at a low speed. For other flux estimators which provide higher flux or velocity values for lower contrast, such as 1/

*C*, a similar analysis may be made, however this will be mathematically less elegant.

## 3. Conclusion

*C*, the flux parameter 1 –

*C*

^{2}provides a weighted average of the power spectrum of photocurrent fluctuations similar to that in laser Doppler flowmetry, however with nonlinear rather than linear weighting. For very small integration times 1 –

*C*

^{2}would imply a second order weighting. For realistic integration times

*T*> 5 ms, speckle contrast mainly provides information regarding the concentration of particles moving within a static matrix, with speed information only present as far as represented by the power spectrum for frequencies between zero and 1/

*T*Hz. The presented theory will enable further research into the use of speckle contrast as an estimator of tissue perfusion.

## References and links

1. | P. Vennemann, R. Lindken, and J. Westerweel, “In vivo whole-field blood velocity measurement techniques,” Exp. Fluids |

2. | M. J. Draijer, E. Hondebrink, T. G. van Leeuwen, and W. Steenbergen, “Review of laser speckle contrast techniques for visualizing tissue perfusion,” Lasers Med. Sci. |

3. | R. Bonner and R. Nossal, “Model for Laser Doppler Measurements of Blood Flow in Tissue,” Appl. Opt. |

4. | J. D. Briers and S. Webster, “Laser speckle contrast analysis LASCA): a nonscanning, full-field technique for monitoring capillary blood flow,” J. Biomed. Opt. |

5. | H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, “Modified laser speckle imaging method with improved spatial resolution,” J. Biomed. Opt. |

6. | D. A. Boas and A. K. Dunn, “Laser speckle contrast imaging in biomedical optics,” J. Biomed. Opt. |

7. | J. D. Briers and S. Webster, “Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields,” Opt. Commun. |

8. | J. C. Ramirez-San-Juan, R. Ramos-Garcia, I. Guizar-Iturbide, G. Martinez-Niconoff, and B. Choi, “Impact of velocity distribution assumption on simplified laser speckle imaging equation,” Opt. Express |

9. | A. B. Parthasarathy, W. J. Tom, A. Gopal, X. Zhang, and A. K. Dunn, “Robust flow measurement with multi-exposure speckle imaging,” Opt. Express |

10. | P. Zakharov, A. Völker, A. Buck, B. Weber, and F. Scheffold, “Quantitative modeling of laser speckle imaging,” Opt. Lett. |

11. | G. E. Nilsson, T. Tenland, and P. A. Öberg, “A New Instrument for Continuous Measurement of Tissue Blood FlowBy LightBeating Spectroscopy,” IEEE Trans. Biomed. Engin. |

12. | D. D. Duncan and S. J. Kirkpatrick, “Can laser speckle flowmetry be made a quantitative tool?” J. Opt. Soc. Am. A |

13. | P. A. Lynn, |

14. | J. W. Goodman and G. Parry, |

15. | A. Serov, W. Steenbergen, and F. F. M. de Mul, “Prediction of the photodetector signal generated by Doppler-induced speckle fluctuations: theory and some validations,” J. Opt. Soc. Am. A |

16. | D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, “Statistics of local speckle contrast,” J. Opt. Soc. Am. A Opt. Image Sci. Vis. |

**OCIS Codes**

(170.1650) Medical optics and biotechnology : Coherence imaging

(170.3340) Medical optics and biotechnology : Laser Doppler velocimetry

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.4580) Medical optics and biotechnology : Optical diagnostics for medicine

**ToC Category:**

Medical Optics and Biotechnology

**History**

Original Manuscript: July 6, 2010

Revised Manuscript: September 9, 2010

Manuscript Accepted: September 14, 2010

Published: September 30, 2010

**Virtual Issues**

Vol. 5, Iss. 14 *Virtual Journal for Biomedical Optics*

**Citation**

Matthijs J. Draijer, Erwin Hondebrink, Marcus Larsson, Ton G. van Leeuwen, and Wiendelt Steenbergen, "Relation between the contrast in time integrated dynamic speckle patterns an the
power spectral density of their temporal intensity fluctuations," Opt. Express **18**, 21883-21891 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-21-21883

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### References

- P. Vennemann, R. Lindken, and J. Westerweel, "In vivo whole-field blood velocity measurement techniques," Exp. Fluids 42(4), 495-511 (2007). [CrossRef]
- M. J. Draijer, E. Hondebrink, T. G. van Leeuwen, and W. Steenbergen, "Review of laser speckle contrast techniques for visualizing tissue perfusion," Lasers Med. Sci. 24(4), 639-651 (2009). [CrossRef]
- R. Bonner and R. Nossal, "Model for Laser Doppler Measurements of Blood Flow in Tissue," Appl. Opt. 20(12), 2097-2107 (1981). [CrossRef] [PubMed]
- 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, 174-179 (1996). [CrossRef]
- H. Cheng, Q. Luo, S. Zeng, S. Chen, J. Cen, and H. Gong, "Modified laser speckle imaging method with improved spatial resolution," J. Biomed. Opt. 8(3), 559-564 (2003). [CrossRef] [PubMed]
- D. A. Boas and A. K. Dunn, "Laser speckle contrast imaging in biomedical optics," J. Biomed. Opt. 15(1), 011109 (2010). [CrossRef] [PubMed]
- J. D. Briers and S. Webster, "Quasi real-time digital version of single-exposure speckle photography for full-field monitoring of velocity or flow fields," Opt. Commun. 116, 36-42 (1995). [CrossRef]
- J. C. Ramirez-San-Juan, R. Ramos-Garcia, I. Guizar-Iturbide, G. Martinez-Niconoff, and B. Choi, "Impact of velocity distribution assumption on simplified laser speckle imaging equation," Opt. Express 16(5), 3197-3203 (2008). [CrossRef] [PubMed]
- A. B. Parthasarathy, W. J. Tom, A. Gopal, X. Zhang, and A. K. Dunn, "Robust flow measurement with multiexposure speckle imaging," Opt. Express 16(3), 1975-1989 (2008). [CrossRef] [PubMed]
- P. Zakharov, A. V¨olker, A. Buck, B. Weber, and F. Scheffold, "Quantitative modeling of laser speckle imaging," Opt. Lett. 31(23), 3465-3467 (2006). [CrossRef] [PubMed]
- G. E. Nilsson, T. Tenland, and P. A. Oberg, "A New Instrument for Continuous Measurement of Tissue Blood Flow by Light Beating Spectroscopy," IEEE Trans. Biomed. Eng. 27(1), 12-18 (1980). [CrossRef]
- D. D. Duncan and S. J. Kirkpatrick, "Can laser speckle flowmetry be made a quantitative tool?" J. Opt. Soc. Am. A 25(8), 2088-2094 (2008). [CrossRef]
- P. A. Lynn, Electronic signals and systems (Macmillan, London, 1986).
- J. W. Goodman and G. Parry, Laser Speckle and Related Phenomena (Springer-Verlag, New York, 1975).
- A. Serov, W. Steenbergen, and F. F. M. de Mul, "Prediction of the photodetector signal generated by Dopplerinduced speckle fluctuations: theory and some validations," J. Opt. Soc. Am. A 18(3), 622-630 (2001). [CrossRef]
- D. D. Duncan, S. J. Kirkpatrick, and R. K. Wang, "Statistics of local speckle contrast,"J. Opt. Soc. Am. A: Opt. Image Sci. Vis. 25(1), 9-15 (2008). [CrossRef]

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