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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3204–3211
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Fractal analysis of self-mixing speckle signal in velocity sensing

Daofu Han, Ming Wang, and Junping Zhou  »View Author Affiliations


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

In self-mixing speckle experiments, signals that represent light intensity changes along a time axis are generated in a laser cavity by illuminating a random rough surface. These are temporal sequences that have fractal properties for each pixel of the speckle signals. The purpose of this paper is to propose a new approach with which to analyze self-mixing speckle signals. The proposition is owing to two factors: fractal analysis is an effective approach to study the temporal or spatial-ordered non-linear events that exist in nature, and it has rarely been used to study the dynamic speckle signal, though it was applied to a wide variety of scientific problems, including spatial speckle patterns [7-10

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

]. In this paper, the analysis is carried out by calculating the BCs of a speckle waveform. The validity of this method is demonstrated by the analysis of speckle signals obtained from numerical simulation and experiments. The performance of fractal analysis is compared with those of pulse counting, the autocorrelation function, and FFT.

2. Principle

In Fig. 1, the curve is a section of the waveform of a self-mixing speckle signal. From the method proposed by Mandelbrot [11

11. B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

], the signal waveform obtained from SMSI is considered as a fractal pattern F in the plane of intensity versus time(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 Nδ(F).

Fig. 1. Fractal boxes: δ=5, Nδ(F)=23 (gray squares).

Equation (1) mathematically defines the dimension of BC (BCD) as a limit:

Dim=limδ0logNδ(F)logδ,
(1)

Because the speckle signal is discrete, scale 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 logNδ(F) versus logδ 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:

Dim=ΔlogNδ(F)Δlogδδ(δ1,δ2),
(2)

where Δ represents the change of value.

In order to calculate the BCD of a speckle signal, first the BC corresponding to each scale 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:

(1) Select scale 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:

[(m1)δ+1,mδ],m=1,2,...,Mδ.
(3)

When M/δ is not an integer, the intervals will be:

[(m1)δ+1,mδ],m=1,2,...,floor(Mδ).
[(floor(Mδ)·δ+1),M]thelastinterval
(4)

(2) In the interval of 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 scaleδ dividing the maximum is not equal to zero, the BC will be increased by one.

Nδ(m)=max[f((m1)δ+1),f(mδ)]δ
min[f((m1)δ+1),f(mδ)]δ+1
(max[f((m1)δ+1),f(mδ)]δ0).
(5)

(3) Add the BC of all the intervals, and the total BC of the signal waveform will be obtained.

Nδ(F)=m=1MδNδ(m).
(6)

(4) Processes (1)–(3) are repeated for different scales of δ, and then the BC Nδ(F) corresponding to δ is obtained.

The BCD can be used as one measure to determine whether a signal is a fractal or not. When a speckle signal is a fractal, we can introduce the BC as a parameter to characterize the statistical property of the signal. For a fixed signal, the BC is determined by scale 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:

(1) Multiply a common number to make all the speckle waveform data become integers. This operation does not change the waveform of the speckle signal but enlarges the amplitude. (2) Provided that the enlarged speckle waveform is expressed with the function f (t),t=1,2,…,n., then the BC is calculated by

N=i=1nf(i)f(i+1)δ,i=1,2,3...(n1).
(7)

The BC of a speckle signal relates to the amplitude and frequency of the waveform containing the velocity information of the object being tested. Calculating the BCs of the speckle signals corresponding to different velocities, one can find a relationship between the BC and the velocity. The relationship can be use in velocity sensing.

Fig. 2. Plot of log Nδ(F)versus log δ.
Fig. 3. Plot of slope versus scale δ.

3. Experiments

To demonstrate the validity of the fractal method, we first used a group of speckle signals produced by numerical simulation to do the fractal analysis experimentally. The signals are shown in Fig. 4 representing a group of velocities that are 98mm/s, 167mm/s, 242mm/s, 340mm/s, 431mm/s, and 580mm/s. Using Eq. (7), the BC of each speckle signal is calculated. We plot each BC versus the corresponding velocity in Fig. 5; a linear relationship between the BC and the velocity is indicated. The linear relation can be expressed as Y=1710+41.73X, and the linearity reaches to 100%. The fractal analysis of the simulated speckle signals obtained a numerical result that is a linear relationship between the BC and the velocity.

Fig. 4. Group of self-mixing speckle signals produced by numerical simulation.
Fig. 5. Linear relationship between BC and velocity, the result to the fractal analysis of the simulated speckle signal.

The numerical result is in agreement with that of the experiment, which indicates that fractal analysis is valid for determining the self-mixing speckle signal in velocity sensing.

Fig. 6. Group of self-mixing speckle signals produced by experiments.
Fig. 7. Linear relationship between BC and velocity, the result of the fractal analysis of simulated speckle signal.

4. Comparison

Fig. 8. Analysis results of pulse-counting, autocorrelation, and FFT; (a), (b), and (c) show that mean pulse frequency, autocorrelation time, and mean spectrum frequency are approximately linearly changed with velocities, respectively.
Fig. 9. Comparing results using normalized parameters for each analysis.

A further comparison of these processing results is made. The calculated values in Figs. 7 and 8 are normalized by dividing each series by their maximum value, and the normalized results can be compared in the same plot as shown in Fig. 9. The black squares, red circles, green up-triangles, and blue down-triangles represent the normalized values corresponding to the results of fractal, pulse counting, autocorrelation, and FFT analyses, respectively. The lines that connect a series of normalized values represent the four analysis methods used in speckle signal processing. The slope of the measuring line shows the measurement sensitivity. It is seen that in the range of velocities from 100mm/s to 600mm/s, the normalized parameter of the fractal analysis is more sensitive to the variation of the velocity than the others. This indicates that fractal analysis has a higher measure sensitivity compared with pulse counting, the autocorrelation function, and FFT. Why did fractal analysis have better performance? A reasonable explanation is that the conventional measures that rely on extracting characteristic scales (mean pulse frequency, correlation time, and mean spectral frequency) are not adequate for characterizing such speckle signals. Fractal analysis is based on structure, which can simultaneously consider the frequency and amplitude of the signal. Fractal BC is a more suitable parameter for describing existing differences in speckle signals.

5. Conclusion

This paper proposed the fractal method to analyze self-mixing speckle signals in velocity sensing. The algorithm of box counting was introduced theoretically. The validity of this method is demonstrated experimentally by processing speckle signals obtained from numerical simulation and experiments. The processing result shows a better linearity and higher measurement sensitivity compared with those of pulse counting, autocorrelation, and FFT. The results indicate that the fractal method can be used as a valid analysis tool for self-mixing speckle signals in velocity sensing.

Acknowledgement

This work was supported by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education grant 20050319007.

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]

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]

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,” in Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference (IEEE 1999), pp. 1756–1760.

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.

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]

6.

D. Han, M. Wang, and J. Zhou, “Self-mixing speckle interference in DFB lasers,” Opt. Express 14, 3312–3317, (2006). [CrossRef] [PubMed]

7.

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

8.

J. Uozumi, M. Ibrahim, and T. Asakura, “Fractal speckles,” Opt. Commun. 156, 350–358 (1998). [CrossRef]

9.

Z. Li, H. Li, and Y. Qiu, “Fractal analysis of laser speckle for measuring roughness,” Proc. SPIE 6027, 470–476 (2006).

10.

H. Funamizu and J. Uozumi, “Generation of fractal speckles by means of a spatial light modulator,” Opt Express 15, 7415–7422 (2007). [CrossRef] [PubMed]

11.

B.B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

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

  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]
  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]
  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," in Proceedings of the 16th IEEE Instrumentation and Measurement Technology Conference (IEEE 1999), pp. 1756-1760.
  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. 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]
  6. D. Han, M. Wang, and J. Zhou, "Self-mixing speckle interference in DFB lasers," Opt. Express 14, 3312-3317, (2006). [CrossRef] [PubMed]
  7. D. A. Zimnyakov and V. V. Tuchin, "Fractality of speckle intensity fluctuations," Appl. Opt. 35, 4325-4333 (1996). [CrossRef] [PubMed]
  8. J. Uozumi, M. Ibrahim, and T. Asakura, "Fractal speckles," Opt. Commun. 156, 350-358 (1998). [CrossRef]
  9. Z. Li, H. Li, and Y. Qiu, "Fractal analysis of laser speckle for measuring roughness," Proc. SPIE 6027, 470-476 (2006).
  10. H. Funamizu and J. Uozumi, "Generation of fractal speckles by means of a spatial light modulator," Opt Express 15, 7415-7422 (2007). [CrossRef] [PubMed]
  11. B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, 1982).

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