## Fractal analysis of self-mixing speckle signal in velocity sensing

Optics Express, Vol. 16, Issue 5, pp. 3204-3211 (2008)

http://dx.doi.org/10.1364/OE.16.003204

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### Abstract

A new method based on fractal theory is proposed to analyze velocity sensing. The waveform of a self-mixing speckle signal is processed as a pattern of a fractal. Fractal boxes are defined as a set of grids used to divide the fractal pattern, and box-counting (BC) is introduced to characterize the statistical property of a speckle signal. A group of simulated speckle signals are analyzed by calculating the BCs corresponding to different velocities of the object. A linear dependence between the BCs of speckle signals and velocities is obtained, the result of which is validated by the analysis of a group of signals obtained from experiments. The performance of the fractal analysis is compared with those of the previous analysis methods. Better linearity and higher measurement sensitivity of the fractal analysis are indicated. The experimental result shows that the fractal method can be used as a valid analysis tool for the self-mixing speckle signal in velocity sensing.

© 2008 Optical Society of America

## 1. Introduction

1. T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, and M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. **45**, 499–503 (1996). [CrossRef]

7. D. A. Zimnyakov and V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. **35**, 4325–4333 (1996). [CrossRef] [PubMed]

## 2. Principle

*f*(

*t*)~

*t*). In this plane, a set of grids with scale

*δ*is used to divide the plane. As a result, a mesh is created. The grid that is crossed by the waveform is called a fractal box, and the BC in scale

*δ*is defined as

*N*(

_{δ}*F*).

*δ*must be an integer and greater than or equal to 1. In fact, the dimension of BC exists only in a special scale range (

*δ*

_{1},

*δ*

_{2}), in which the slopes of log

*N*(

_{δ}*F*) versus log

*δ*approximately maintain a constant, so that the range (

*δ*

_{1},

*δ*

_{2}) is called a scaling range. In the scaling range, the figure

*F*is a fractal and the BCD is unique; thus, the definition of the BCD in Eq. (1) is modified:

*δ*should be calculated, and then the scaling range will be found. Finally, the BCD of the signal will be calculated. The steps of the calculation are listed as follows:

*δ*,

*viz.*, the length of the grid side. Using scale

*δ*to divide

*M*, the total length of the horizontal data, the horizontal axis is divided into

*M*/

*δ*at uniform intervals:

*M*/

*δ*is not an integer, the intervals will be:

*floor*is to round down to the next allowable integer value.

*m*, use scale

*δ*to divide the maximum value of the signal data and then obtain the integer part of the result. The same operation is done with the minimum value in this interval, and another integer is also obtained. The absolute value of the subtraction operation between the two integers gets the BC in the interval

*m*. If the remainder of the value of scale

*δ*dividing the maximum is not equal to zero, the BC will be increased by one.

*δ*, and then the BC N

_{δ}(

*F*) corresponding to

*δ*is obtained.

*δ*. The smaller the value of

*δ*is, the bigger the BC is. However, since the speckle signal we obtained is discrete,

*δ*must be an integer and greater than or equal to 1. When

*δ*=1, the biggest BC will be obtained, which best characterizes the speckle signal in detail. Therefore, in calculating the BC of the speckle signal, we select

*δ*=1 and the calculation is carried out as follows:

*f*(

*t*),

*t*=1,2,…,

*n*., then the BC is calculated by

## 3. Experiments

## 4. Comparison

1. T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, and M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. **45**, 499–503 (1996). [CrossRef]

*, which is defined as the time delay at which the normalized autocorrelation function drops to*

_{c}*1*/

*e*[2

2. S. K. Zdemir, S. Ito, S. Shinohara, H. Yoshida, and M. Sumi, “Correlation-based speckle velocimeter with self-mixing interference in a semiconductor laser diode,” Appl. Opt. **38**, 6859–6865 (1999). [CrossRef]

4. G. G. Romero, E. E. Alanis, and H. J. Rabal, “Statistics of the dynamic speckle produced by a rotating diffuser and its application to the assessment of paint drying,” Opt. Eng. **39**, 1652–1658 (2000). [CrossRef]

## 5. Conclusion

## Acknowledgement

1. | T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, and M. Sumi, “Laser speckle velocimeter using self-mixing laser diode,” IEEE Trans. Instrum. Meas. |

2. | S. K. Zdemir, S. Ito, S. Shinohara, H. Yoshida, and M. Sumi, “Correlation-based speckle velocimeter with self-mixing interference in a semiconductor laser diode,” Appl. Opt. |

3. | O. K. Sahin, I. Satoshi, T. Sotetsu, I. Satoshi, T. Sotetsu, S. Shigenobu, Y. Hirofumi, and S. Masao, “Velocity measurement by a self-mixing laser diode using speckle correlation,” |

4. | G. G. Romero, E. E. Alanis, and H. J. Rabal, “Statistics of the dynamic speckle produced by a rotating diffuser and its application to the assessment of paint drying,” Opt. Eng. |

5. | M. Wang, M. Lu, H. Hao, and J. Zhou, “Statistics of the self-mixing speckle interference in a laser diode and its application to the measurement of flow velocity,” Opt. Commun. |

6. | D. Han, M. Wang, and J. Zhou, “Self-mixing speckle interference in DFB lasers,” Opt. Express |

7. | D. A. Zimnyakov and V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. |

8. | J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. |

9. | Z. Li, H. Li, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE |

10. | H. Funamizu and J. Uozumi, “Generation of fractal speckles by means of a spatial light modulator,” Opt Express |

11. | B.B. Mandelbrot, |

**OCIS Codes**

(030.6140) Coherence and statistical optics : Speckle

(280.7250) Remote sensing and sensors : Velocimetry

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

**ToC Category:**

Coherence and Statistical Optics

**History**

Original Manuscript: September 14, 2007

Revised Manuscript: November 27, 2007

Manuscript Accepted: January 17, 2008

Published: February 22, 2008

**Citation**

Daofu Han, Ming Wang, and Junping Zhou, "Fractal analysis of self-mixing speckle signal in velocity sensing," Opt. Express **16**, 3204-3211 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3204

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### References

- T. Shibata, S. Shinohara, H. Ikeda, H. Yoshida, T. Sawaki, and M. Sumi, "Laser speckle velocimeter using self-mixing laser diode," IEEE Trans. Instrum. Meas. 45, 499-503 (1996). [CrossRef]
- S. K. Zdemir, S. Ito, S. Shinohara, H. Yoshida, and M. Sumi, "Correlation-based speckle velocimeter with self-mixing interference in a semiconductor laser diode," Appl. Opt. 38, 6859-6865 (1999). [CrossRef]
- O. K. Sahin, I. Satoshi, T. Sotetsu, I. Satoshi, T. Sotetsu, S. Shigenobu, Y. Hirofumi, and S. Masao, "Velocity measurement by a self-mixing laser diode using speckle correlation," in Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference (IEEE 1999), pp. 1756-1760.
- G. G. Romero, E. E. Alanis, and H. J. Rabal, "Statistics of the dynamic speckle produced by a rotating diffuser and its application to the assessment of paint drying," Opt. Eng. 39, 1652-1658 (2000). [CrossRef]
- M. Wang, M. Lu, H. Hao, and J. Zhou, "Statistics of the self-mixing speckle interference in a laser diode and its application to the measurement of flow velocity," Opt. Commun. 60, 242-247 (2006). [CrossRef]
- D. Han, M. Wang, and J. Zhou, "Self-mixing speckle interference in DFB lasers," Opt. Express 14, 3312-3317, (2006). [CrossRef] [PubMed]
- D. A. Zimnyakov and V. V. Tuchin, "Fractality of speckle intensity fluctuations," Appl. Opt. 35, 4325-4333 (1996). [CrossRef] [PubMed]
- J. Uozumi, M. Ibrahim, and T. Asakura, "Fractal speckles," Opt. Commun. 156, 350-358 (1998). [CrossRef]
- Z. Li, H. Li, and Y. Qiu, "Fractal analysis of laser speckle for measuring roughness," Proc. SPIE 6027, 470-476 (2006).
- H. Funamizu and J. Uozumi, "Generation of fractal speckles by means of a spatial light modulator," Opt Express 15, 7415-7422 (2007). [CrossRef] [PubMed]
- B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

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