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

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References

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  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).

2007 (1)

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

2006 (3)

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

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

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]

2000 (1)

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]

1999 (1)

1998 (1)

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

1996 (2)

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]

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

Alanis, E. E.

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]

Asakura, T.

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

Funamizu, H.

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

Han, D.

Hao, H.

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]

Ibrahim, M.

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

Ikeda, H.

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]

Ito, S.

Li, H.

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

Li, Z.

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

Lu, M.

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]

Qiu, Y.

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

Rabal, H. J.

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]

Romero, G. G.

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]

Sawaki, T.

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]

Shibata, T.

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]

Shinohara, S.

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]

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]

Sumi, M.

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]

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]

Tuchin, V. V.

Uozumi, J.

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

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

Wang, M.

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]

Yoshida, H.

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]

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]

Zdemir, S. K.

Zhou, J.

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

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]

Zimnyakov, D. A.

Appl. Opt. (2)

IEEE Trans. Instrum. Meas. (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]

Opt Express (1)

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

Opt. Commun. (2)

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]

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

Opt. Eng. (1)

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]

Opt. Express (1)

Proc. SPIE (1)

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

Other (2)

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.

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

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Figures (9)

Fig. 1.
Fig. 1.

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

Fig. 2.
Fig. 2.

Plot of log Nδ(F)versus log δ .

Fig. 3.
Fig. 3.

Plot of slope versus scale δ.

Fig. 4.
Fig. 4.

Group of self-mixing speckle signals produced by numerical simulation.

Fig. 5.
Fig. 5.

Linear relationship between BC and velocity, the result to the fractal analysis of the simulated speckle signal.

Fig. 6.
Fig. 6.

Group of self-mixing speckle signals produced by experiments.

Fig. 7.
Fig. 7.

Linear relationship between BC and velocity, the result of the fractal analysis of simulated speckle signal.

Fig. 8.
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.
Fig. 9.

Comparing results using normalized parameters for each analysis.

Equations (10)

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Dim = lim δ 0 log N δ ( F ) log δ ,
Dim = Δ log N δ ( F ) Δ log δ δ ( δ 1 , δ 2 ) ,
[ ( m 1 ) δ + 1 , m δ ] , m = 1 , 2 , . . . , M δ .
[ ( m 1 ) δ + 1 , m δ ] , m = 1 , 2 , . . . , floor ( M δ ) .
[ ( floor ( M δ ) · δ + 1 ) , M ] the last interval
N δ ( m ) = max [ f ( ( m 1 ) δ + 1 ) , f ( m δ ) ] δ
min [ f ( ( m 1 ) δ + 1 ) , f ( m δ ) ] δ + 1
( max [ f ( ( m 1 ) δ + 1 ) , f ( m δ ) ] δ 0 ) .
N δ ( F ) = m = 1 M δ N δ ( m ) .
N = i = 1 n f ( i ) f ( i + 1 ) δ , i = 1 , 2 , 3 . . . ( n 1 ) .

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