Abstract

Asymptotic behavior of temporal autocorrelation functions of speckle intensity fluctuations induced by tissue scanning with a focused probe beam is experimentally studied for the transition from a single-scattering to a multiple-scattering mode. Such parameters as the exponential factor (or the Hurst coefficient) and the Hausdorff dimension are proposed for the characterization and the visualization of the variations of the studied tissues’ optical properties in generalized form. We studied reversible transition between various scattering modes stimulated by the application of certain chemical agents to the human sclera samples using speckle intensity correlation analysis; corresponding results are presented. The possibilities of the scattering structures imaging with local estimations of the exponential factor of speckle intensity fluctuations are shown in in vitro experiments with samples of human skin epidermis.

© 1997 Optical Society of America

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References

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  1. J. D. Briers, “Speckle fluctuations and biomedical optics: implications and applications,” Opt. Eng. 32, 277–284 (1993).
    [Crossref]
  2. J. D. Briers, “Monitoring biomedical motion and flow by means of coherent light fluctuations,” in CIS Selected Papers: Coherence-Domain Methods in Biomedical Optics, V. V. Tuchin, ed., Proc. SPIE2732, 2–15 (1996).
  3. E. Feder, Fractals (Plenum, New York, 1988).
    [Crossref]
  4. S. M. Rhytov, U. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics, Vol. 2, Random Fields (Nauka, Moscow, 1978).
  5. E. L. Church, “Fractal surface finish,” Appl. Opt. 27, 1518–1526 (1988).
    [Crossref] [PubMed]
  6. M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
    [Crossref]
  7. M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
    [Crossref] [PubMed]
  8. I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
    [Crossref] [PubMed]
  9. P. O. Rol, “Optics for transscleral laser applications,” Doctor of Natural Sciences dissertation (Institute of Biomedical Engineering, Zürich, 1992).
  10. P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).
  11. V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
    [Crossref]
  12. D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).
  13. V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
    [Crossref]
  14. V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
    [Crossref]
  15. M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
    [Crossref] [PubMed]
  16. N. Takai, T. Iwai, T. Asakura, “Correlation distance of dynamic speckles,” Appl. Opt. 22, 170–177 (1983).
    [Crossref] [PubMed]
  17. D. A. Zimnyakov, V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. 35, 4425–4433 (1996).
    [Crossref]
  18. B. Ruth, “Noncontact blood flow determination using a laser speckle method,” in Tissue Optics: Applications in Medical Diagnostics and Therapy, V. V. Tuchin, ed., Milestone Reprint Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1994), Vol. MS102, pp. 471–478.

1996 (1)

D. A. Zimnyakov, V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. 35, 4425–4433 (1996).
[Crossref]

1995 (1)

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

1994 (2)

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
[Crossref]

1993 (1)

J. D. Briers, “Speckle fluctuations and biomedical optics: implications and applications,” Opt. Eng. 32, 277–284 (1993).
[Crossref]

1990 (1)

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

1989 (1)

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

1988 (1)

1985 (1)

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

1983 (1)

1979 (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[Crossref]

Asakura, T.

Bakutkin, V. V.

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

Berry, M. V.

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[Crossref]

Briers, J. D.

J. D. Briers, “Speckle fluctuations and biomedical optics: implications and applications,” Opt. Eng. 32, 277–284 (1993).
[Crossref]

J. D. Briers, “Monitoring biomedical motion and flow by means of coherent light fluctuations,” in CIS Selected Papers: Coherence-Domain Methods in Biomedical Optics, V. V. Tuchin, ed., Proc. SPIE2732, 2–15 (1996).

Church, E. L.

Durr, U.

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Fankhauser, F.

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Feder, E.

E. Feder, Fractals (Plenum, New York, 1988).
[Crossref]

Fine, I.

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

Hammer, M.

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

Henchoz, P.-D.

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Iwai, T.

Jacques, S.

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

Kon, I. L.

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

Kravtsov, U. A.

S. M. Rhytov, U. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics, Vol. 2, Random Fields (Nauka, Moscow, 1978).

Loewinger, E.

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

Maksimova, I. L.

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

Mavlyitov, A. K.

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

Mishin, A. A.

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

Muller, G.

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

Niederer, P.

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Rhytov, S. M.

S. M. Rhytov, U. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics, Vol. 2, Random Fields (Nauka, Moscow, 1978).

Roggan, A.

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

Rol, P.

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Rol, P. O.

P. O. Rol, “Optics for transscleral laser applications,” Doctor of Natural Sciences dissertation (Institute of Biomedical Engineering, Zürich, 1992).

Ruth, B.

B. Ruth, “Noncontact blood flow determination using a laser speckle method,” in Tissue Optics: Applications in Medical Diagnostics and Therapy, V. V. Tuchin, ed., Milestone Reprint Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1994), Vol. MS102, pp. 471–478.

Schweitzer, D.

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

Semyonova, T. N.

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

Star, W. M.

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

Sterenborg, H. J. M.

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

Takai, N.

Tatarsky, V. I.

S. M. Rhytov, U. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics, Vol. 2, Random Fields (Nauka, Moscow, 1978).

Tuchin, V. V.

D. A. Zimnyakov, V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. 35, 4425–4433 (1996).
[Crossref]

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
[Crossref]

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

Utz, S. R.

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
[Crossref]

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).

van Gemert, M. J. C.

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

Weinreb, A.

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

Wenberger, D.

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

Yaroslavsky, I. V.

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
[Crossref]

Zimnyakov, D. A.

D. A. Zimnyakov, V. V. Tuchin, “Fractality of speckle intensity fluctuations,” Appl. Opt. 35, 4425–4433 (1996).
[Crossref]

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

Appl. Opt. (3)

IEEE Trans. Biomed. Eng. (1)

M. J. C. van Gemert, S. Jacques, H. J. M. Sterenborg, W. M. Star, “Skin optics,” IEEE Trans. Biomed. Eng. 36, 1146–1154 (1989).
[Crossref] [PubMed]

J. Phys. A (1)

M. V. Berry, “Diffractals,” J. Phys. A 12, 781–797 (1979).
[Crossref]

Laser Light Ophthalmol. (1)

P. Rol, P. Niederer, U. Durr, P.-D. Henchoz, F. Fankhauser, “Experimental investigations on the light scattering properties of the human sclera,” Laser Light Ophthalmol. 3, 201–212 (1990).

Opt. Eng. (2)

V. V. Tuchin, S. R. Utz, I. V. Yaroslavsky, “Tissue optics, light distribution and spectroscopy,” Opt. Eng. 10, 3178–3188 (1994).
[Crossref]

J. D. Briers, “Speckle fluctuations and biomedical optics: implications and applications,” Opt. Eng. 32, 277–284 (1993).
[Crossref]

Opt. Spectrosc. (1)

D. A. Zimnyakov, V. V. Tuchin, S. R. Utz, “Investigation of the statistical properties of partially developed speckle fields in applications to the diagnostics of the human skin structure,” Opt. Spectrosc. 76, 838–844 (1994).

Phys. Med. Biol. (2)

M. Hammer, A. Roggan, D. Schweitzer, G. Muller, “Optical properties of ocular fundus tissues—an in vitro study using the double-integrating-sphere technique and inverse Monte Carlo simulation,” Phys. Med. Biol. 40, 963–978 (1995).
[Crossref] [PubMed]

I. Fine, E. Loewinger, A. Weinreb, D. Wenberger, “Optical properties of the sclera,” Phys. Med. Biol. 30, 565–571 (1985).
[Crossref] [PubMed]

Other (7)

P. O. Rol, “Optics for transscleral laser applications,” Doctor of Natural Sciences dissertation (Institute of Biomedical Engineering, Zürich, 1992).

V. V. Tuchin, I. L. Maksimova, D. A. Zimnyakov, I. L. Kon, A. K. Mavlyitov, A. A. Mishin, “Light propagation in tissues with controlled optical properties,” in Photon Propagation in Tissues II, D. A. Benaron, B. Chance, G. J. Mueller, eds., Proc. SPIE2925, 118–142 (1996).
[Crossref]

V. V. Bakutkin, I. L. Maksimova, T. N. Semyonova, V. V. Tuchin, I. L. Kon, “Controlling optical properties of sclera,” in Ophthalmic Technologies V, J. M. Parel, Q. Ren, K. M. Joos, eds., Proc. SPIE2393, 137–141 (1995).
[Crossref]

J. D. Briers, “Monitoring biomedical motion and flow by means of coherent light fluctuations,” in CIS Selected Papers: Coherence-Domain Methods in Biomedical Optics, V. V. Tuchin, ed., Proc. SPIE2732, 2–15 (1996).

E. Feder, Fractals (Plenum, New York, 1988).
[Crossref]

S. M. Rhytov, U. A. Kravtsov, V. I. Tatarsky, Introduction to Statistical Radiophysics, Vol. 2, Random Fields (Nauka, Moscow, 1978).

B. Ruth, “Noncontact blood flow determination using a laser speckle method,” in Tissue Optics: Applications in Medical Diagnostics and Therapy, V. V. Tuchin, ed., Milestone Reprint Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Wash., 1994), Vol. MS102, pp. 471–478.

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Figures (9)

Fig. 1
Fig. 1

Optical scheme of the spatial speckle correlometer: 1, monomode He–Ne laser; 2, telescopic system; 3, focusing micro-objective; 4, tissue under study; 5, two-dimensional scanning device; 6, manually rotating polarizer; 7, photodetector (photomultiplier with pinhole diaphragm).

Fig. 2
Fig. 2

Bitmap images of speckle patterns in the detection plane for two stages of the sclera enlightenment; time elapsed after Trazograph application is, a, ~2.5 min and, b, ~10 min, the distance between the sample and the detection plane is equal to 200 mm.

Fig. 3
Fig. 3

Evolution of the form of autocorrelation peak with the sclera turbidity decay for arbitrary selected sclera sample; sample thickness is equal to 0.7 mm; signal recording time after Trazograph application: ■, 120 s; +, 220 s; □, 420 s; ×, 520 s; scanning velocity is equal to 4 mm/s.

Fig. 4
Fig. 4

Typical dependencies of, a, exponential factor νI and, b, normalized value of the average intensity /I0 measured in the detection point on time after the Trazograph application; I0 is the intensity of the inscattered probe beam in the detection point.

Fig. 5
Fig. 5

Typical examples of the speckle intensity time series with different polarization states caused by the sample scanning.

Fig. 6
Fig. 6

Evolution of form of the cross-correlation coefficient of speckle intensity fluctuations with various polarization states caused by the decrease of human sclera turbidity: a, 200 s after Trazograph application; b, 300 s after Trazograph application; c, 400 s after Trazograph application.

Fig. 7
Fig. 7

Human epidermis structure images with exponential factor of the speckle intensity fluctuations as visualization parameter: a, normal epidermis; b, middle stage of the psoriasis plaque progress.

Fig. 8
Fig. 8

Evolution of histograms of νI with the progress of psoriasis: a, normal epidermis; b, early stage of the disease; c, middle stage; d, later stage (desquamation).

Fig. 9
Fig. 9

Theoretical [see Eq. (12)] and experimental (○) values of intensity autocorrelation decay for later stages of the sclera enlightenment process. Error bars for 90% confidence level correspond to variations of gI(τ) for sample series that contains 10 specimens; corresponding time interval after Trazograph application, 600–800 s; theoretical dependencies (1–5) correspond to different values of waist radius: 1, 2.0 µm; 2, 2.5 µm; 3, 3.0 µm; 4, 3.5 µm; 5, 4.0 µm; scanning velocity, 4 mm/s.

Tables (1)

Tables Icon

Table 1 Mean Values and Standard Deviations of the Exponential Factor for Generated νI Images at Various Stages of Disease

Equations (14)

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

D I ( τ ) C T 2 ν I | τ | ν I ,
D I ( K j ) = 1 N K j i = 1 N K j ( I i I i + K j ) 2 , j = 1 ,   ,   L ,
g I ( K j ) = i = 1 N K j I ˜ i I ˜ i + K j i = 1 N K j I ˜ i 2 ,
I ˜ i = I i 1 N i = 1 N I i .
ν I = i = 1 L   ln   K i i = 1 L   ln   D I ( K i ) L i = 1 L   ln   K i   ln   D I ( K i ) i = 1 L   ln   K i 2 L i = 1 L ( ln   K i ) 2 .
D I ( K j ,   m ) = 1 2 P + 1 K j i = m P i = m + P K j ( I i I i + K j ) 2 ,
r ( τ ) = I ˜ ( t ) I ˜ ( t + τ ) ¯ / [ I ˜ ( t ) I ˜ ( t + τ ) ] ¯ max ,
G v ( τ ) = 8 π 2   exp [ j ( ω + q 0 v ) τ ] × P + ( R ) P * ( R ) exp ( j k ξ R τ ) Φ ε ( q ) d 2 R ,
G v ( τ ) = exp [ i ( ω + q 0 v ) τ ] 8 π 3 Φ ε ( q ¯ 0 ) P + P * d 3 R .
P ( R ) const w 0 w ( z )   exp x 2 + y 2 w 2 ( z ) × exp i k ( x 2 + y 2 ) 2 R ( z ) ,
P + const w 0 w ( z )   exp ( x + | v | τ ) 2 + y 2 w 2 ( z ) × exp i k [ ( x + | v | τ ) 2 + y 2 ] 2 R ( z ) ; P * const w 0 w ( z )   exp ( x | v | τ ) 2 + y 2 w 2 ( z ) × exp i k [ ( x | v | τ ) 2 + y 2 ] 2 R ( z ) ,
G v ( τ ) const × exp [ j ( ω + q 0 v ) ] exp 2 | v | 2 τ 2 w 0 2 .
g I ( τ ) const exp 4 | v | 2 τ 2 w 0 2 .
G v ( τ ) = exp [ j ( ω + q 0 v ) τ ] | P ( R ) | 2 × exp ( j k ξ R τ ) Φ ε ( q ) d 3 R ,

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