Abstract

Both radio-frequency (rf) and envelope-detected signal analyses have lead to successful tissue discrimination in medical ultrasound. The extrapolation from tissue discrimination to a description of the tissue structure requires an analysis of the statistics of complex signals. To that end, first- and second-order statistics of complex random signals are reviewed, and an example is taken from rf signal analysis of the backscattered echoes from diffuse scatterers. In this case the scattering form factor of small scatterers can be easily separated from long-range structure and corrected for the transducer characteristics, thereby yielding an instrument-independent tissue signature. The statistics of the more economical envelope- and square-law-detected signals are derived next and found to be almost identical when normalized autocorrelation functions are used. Of the two nonlinear methods of detection, the square-law or intensity scheme gives rise to statistics that are more transparent to physical insight. Moreover, an analysis of the intensity-correlation structure indicates that the contributions to the total echo signal from the diffuse scatter and from the steady and variable components of coherent scatter can still be separated and used for tissue characterization. However, this analysis is not system independent. Finally, the statistical methods of this paper may be applied directly to envelope signals in nuclear-magnetic-resonance imaging because of the approximate equivalence of second-order statistics for magnitude and intensity.

© 1987 Optical Society of America

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    [CrossRef] [PubMed]
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  3. D. Nicholas, C. R. Hill, D. K. Nassiri, “Evaluation of backscattering coefficients for excised human tissues: principles and techniques,” Ultrasound Med. Biol. 8, 7–15 (1982).
    [CrossRef]
  4. D. Nicholas, “Evaluation of backscattering coefficients for excised human tissues: results, interpretation and associated measurements,” Ultrasound Med. Biol. 8, 17–28 (1982).
    [CrossRef]
  5. F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
    [CrossRef] [PubMed]
  6. R. C. Waag, “A review of tissue characterization from ultrasonic scattering,”IEEE Trans. Biomed. Eng. BME-31, 884–893 (1984).
    [CrossRef]
  7. E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
    [CrossRef] [PubMed]
  8. U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
    [CrossRef] [PubMed]
  9. M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).
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  16. P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  20. F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
    [PubMed]
  21. L. L. Fellingham, F. G. Sommer, “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen,”IEEE Trans. Sonics Ultrason. SU-31, 418–428 (1984).
    [CrossRef]
  22. R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
    [CrossRef]
  23. M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
    [CrossRef]
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    [CrossRef]
  25. M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
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  28. J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), Chap. 3.
  29. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9.
  30. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. XXIII, 282–332 (1944); Bell Syst. Tech. J. XXIV, 46–158 (1945).
  31. D. O. North, “The modification of noise by certain nonlinear devices,” (1943) [reprinted in Proc. IEEE51, 1016 (1963)].
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    [CrossRef]
  33. P. D. Burns, “Measurement of random and periodic image noise in raster-written images,” in Second International Congress on Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984).
  34. E. Jakeman, “Speckle statistics with small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
    [CrossRef]
  35. J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Chap. 7.
  36. D. Middleton, “Some general results in the theory of noise through non-linear devices,”Q. Appl. Math. 5, 445–498 (1948).
  37. M. Abramowitz, I. A. Stegun, Handbood of Mathematical Functions, Natl. Bur. Standards Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

1986 (5)

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

1985 (1)

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

1984 (3)

R. C. Waag, “A review of tissue characterization from ultrasonic scattering,”IEEE Trans. Biomed. Eng. BME-31, 884–893 (1984).
[CrossRef]

L. L. Fellingham, F. G. Sommer, “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen,”IEEE Trans. Sonics Ultrason. SU-31, 418–428 (1984).
[CrossRef]

E. Jakeman, “Speckle statistics with small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

1983 (2)

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

1982 (3)

M. F. Insana, J. A. Zagzebski, E. L. Madsen, “Acoustic backscattering from ultrasonically tissuelike media,” Med. Phys. 9, 848–855 (1982).
[CrossRef] [PubMed]

D. Nicholas, C. R. Hill, D. K. Nassiri, “Evaluation of backscattering coefficients for excised human tissues: principles and techniques,” Ultrasound Med. Biol. 8, 7–15 (1982).
[CrossRef]

D. Nicholas, “Evaluation of backscattering coefficients for excised human tissues: results, interpretation and associated measurements,” Ultrasound Med. Biol. 8, 17–28 (1982).
[CrossRef]

1981 (1)

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

1979 (1)

J. C. Bamber, “Theoretical modelling of the acoustic scattering structure of human liver,” Acoustic Lett. 3, 114–119 (1979).

1977 (1)

J. C. Gore, S. Leeman, “Ultrasonic backscattering from human tissue: a realistic model,” Phys. Med. Biol. 22, 317–326 (1977).
[CrossRef] [PubMed]

1973 (1)

S. Fields, F. Dunn, “Correlation of echographic visualizability of tissue with biological composition and physiological state,”J. Acoust. Soc. Am. 54, 809–811 (1973).
[CrossRef] [PubMed]

1970 (1)

1951 (1)

J. J. Faran, “Sound scattering by solid cylinders and spheres,”J. Acoust. Soc. Am. 23, 405–418 (1951).
[CrossRef]

1948 (1)

D. Middleton, “Some general results in the theory of noise through non-linear devices,”Q. Appl. Math. 5, 445–498 (1948).

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. XXIII, 282–332 (1944); Bell Syst. Tech. J. XXIV, 46–158 (1945).

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbood of Mathematical Functions, Natl. Bur. Standards Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

Arsenault, H.

Bamber, J. C.

J. C. Bamber, “Theoretical modelling of the acoustic scattering structure of human liver,” Acoustic Lett. 3, 114–119 (1979).

Bendat, J. S.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), Chap. 3.

Bjorken, J. D.

J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Chap. 7.

Brown, D. G.

R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

Burns, P. D.

P. D. Burns, “Measurement of random and periodic image noise in raster-written images,” in Second International Congress on Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984).

Carroll, B. A.

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

Coleman, D. J.

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

Drell, S. D.

J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Chap. 7.

Dunn, F.

S. Fields, F. Dunn, “Correlation of echographic visualizability of tissue with biological composition and physiological state,”J. Acoust. Soc. Am. 54, 809–811 (1973).
[CrossRef] [PubMed]

Elbaum, M.

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

Faran, J. J.

J. J. Faran, “Sound scattering by solid cylinders and spheres,”J. Acoust. Soc. Am. 23, 405–418 (1951).
[CrossRef]

Feleppa, E. J.

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

Fellingham, L. L.

L. L. Fellingham, F. G. Sommer, “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen,”IEEE Trans. Sonics Ultrason. SU-31, 418–428 (1984).
[CrossRef]

Fellingham-Joynt, L.

L. Fellingham-Joynt, “A stochastic approach to ultrasonic tissue characterization,” Ph.D. dissertation, (Stanford University, Stanford, Calif., 1979).

Fields, S.

S. Fields, F. Dunn, “Correlation of echographic visualizability of tissue with biological composition and physiological state,”J. Acoust. Soc. Am. 54, 809–811 (1973).
[CrossRef] [PubMed]

Garbutt, P.

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

Garra, B. S.

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).

Goodman, J. W.

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975).
[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gore, J. C.

J. C. Gore, S. Leeman, “Ultrasonic backscattering from human tissue: a realistic model,” Phys. Med. Biol. 22, 317–326 (1977).
[CrossRef] [PubMed]

Greenebaum, M.

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

Hall, T. J.

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

Hill, C. R.

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

D. Nicholas, C. R. Hill, D. K. Nassiri, “Evaluation of backscattering coefficients for excised human tissues: principles and techniques,” Ultrasound Med. Biol. 8, 7–15 (1982).
[CrossRef]

Ingard, K. U.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).

Insana, M. F.

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

M. F. Insana, J. A. Zagzebski, E. L. Madsen, “Acoustic backscattering from ultrasonically tissuelike media,” Med. Phys. 9, 848–855 (1982).
[CrossRef] [PubMed]

M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).

Jakeman, E.

E. Jakeman, “Speckle statistics with small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

Joynt, L. F.

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

Kittel, C.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966), p. 66.

Kommerell, B.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Leeman, S.

J. C. Gore, S. Leeman, “Ultrasonic backscattering from human tissue: a realistic model,” Phys. Med. Biol. 22, 317–326 (1977).
[CrossRef] [PubMed]

Limberg, B.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Lizzi, F. L.

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

Lopez, H.

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

Lorenz, A.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Lorenz, W. J.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Lowenthal, S.

Macovski, A.

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

Madsen, E. L.

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

M. F. Insana, J. A. Zagzebski, E. L. Madsen, “Acoustic backscattering from ultrasonically tissuelike media,” Med. Phys. 9, 848–855 (1982).
[CrossRef] [PubMed]

Middleton, D.

D. Middleton, “Some general results in the theory of noise through non-linear devices,”Q. Appl. Math. 5, 445–498 (1948).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9.

Morse, P. M.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).

Nassiri, D. K.

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

D. Nicholas, C. R. Hill, D. K. Nassiri, “Evaluation of backscattering coefficients for excised human tissues: principles and techniques,” Ultrasound Med. Biol. 8, 7–15 (1982).
[CrossRef]

Nicholas, D.

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

D. Nicholas, “Evaluation of backscattering coefficients for excised human tissues: results, interpretation and associated measurements,” Ultrasound Med. Biol. 8, 17–28 (1982).
[CrossRef]

D. Nicholas, C. R. Hill, D. K. Nassiri, “Evaluation of backscattering coefficients for excised human tissues: principles and techniques,” Ultrasound Med. Biol. 8, 7–15 (1982).
[CrossRef]

North, D. O.

D. O. North, “The modification of noise by certain nonlinear devices,” (1943) [reprinted in Proc. IEEE51, 1016 (1963)].

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 10.

Piersol, A. G.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), Chap. 3.

Raeth, U.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. XXIII, 282–332 (1944); Bell Syst. Tech. J. XXIV, 46–158 (1945).

Sandrik, J. M.

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

Schlaps, D.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Shawker, T. H.

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).

Smith, S. W.

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

Sommer, F. G.

L. L. Fellingham, F. G. Sommer, “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen,”IEEE Trans. Sonics Ultrason. SU-31, 418–428 (1984).
[CrossRef]

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbood of Mathematical Functions, Natl. Bur. Standards Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

Thijssen, J. M.

J. M. Thijssen, “Scattering of ultrasound: general aspects,” in Ultrasonic Tissue Characterization and Echographic Imaging 5, Proceedings of the Fifth European Communities Workshop, J. M. Thijssen, V. Mazzeo, eds. (Faculty of Medicine Printing Office, University of Nijmegen, Nijmegen, The Netherlands, 1986), pp. 7–17.

van Kaick, G.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

Waag, R. C.

R. C. Waag, “A review of tissue characterization from ultrasonic scattering,”IEEE Trans. Biomed. Eng. BME-31, 884–893 (1984).
[CrossRef]

R. C. Waag, University of Rochester, Rochester, New York 14627 (personal communication, 1986).

Wagner, R. F.

R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).

Yaremko, M. M.

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

Zagzebski, J. A.

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

M. F. Insana, J. A. Zagzebski, E. L. Madsen, “Acoustic backscattering from ultrasonically tissuelike media,” Med. Phys. 9, 848–855 (1982).
[CrossRef] [PubMed]

Zuna, I.

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

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J. C. Bamber, “Theoretical modelling of the acoustic scattering structure of human liver,” Acoustic Lett. 3, 114–119 (1979).

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IEEE Trans. Biomed. Eng. (1)

R. C. Waag, “A review of tissue characterization from ultrasonic scattering,”IEEE Trans. Biomed. Eng. BME-31, 884–893 (1984).
[CrossRef]

IEEE Trans. Sonics Ultrason. (2)

R. F. Wagner, S. W. Smith, J. M. Sandrik, H. Lopez, “Statistics of speckle in ultrasound B-scans,”IEEE Trans. Sonics Ultrason. SU-30, 156–163 (1983).
[CrossRef]

L. L. Fellingham, F. G. Sommer, “Ultrasonic characterization of tissue structure in the in vivo human liver and spleen,”IEEE Trans. Sonics Ultrason. SU-31, 418–428 (1984).
[CrossRef]

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[CrossRef]

M. F. Insana, E. L. Madsen, T. J. Hall, J. A. Zagzebski, “Tests of the accuracy of a data reduction method for determination of acoustic backscatter coefficients,”J. Acoust. Soc. Am. 79, 1230–1236 (1986).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

F. L. Lizzi, M. Greenebaum, E. J. Feleppa, M. Elbaum, “Theoretical framework for spectrum analysis in ultrasonic tissue characterization,”J. Acoust. Soc. Am. 73, 1366–1373 (1983).
[CrossRef] [PubMed]

J. Clin. Ultrasound (1)

U. Raeth, D. Schlaps, B. Limberg, I. Zuna, A. Lorenz, G. van Kaick, W. J. Lorenz, B. Kommerell, “Diagnostic accuracy of computerized B-scan texture analysis and conventional ultrasonography in diffuse parenchymal and malignant liver disease,”J. Clin. Ultrasound 13, 87–99 (1985).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (1)

Med. Phys. (1)

M. F. Insana, J. A. Zagzebski, E. L. Madsen, “Acoustic backscattering from ultrasonically tissuelike media,” Med. Phys. 9, 848–855 (1982).
[CrossRef] [PubMed]

Opt. Eng. (3)

R. F. Wagner, M. F. Insana, D. G. Brown, “Unified approach to the detection and classification of speckle texture in diagnostic ultrasound,” Opt. Eng. 25, 738–742 (1986).
[CrossRef]

M. F. Insana, R. F. Wagner, B. S. Garra, D. G. Brown, T. H. Shawker, “Analysis of ultrasound image texture via generalized Rician statistics,” Opt. Eng. 25, 743–748 (1986).
[CrossRef]

E. Jakeman, “Speckle statistics with small number of scatterers,” Opt. Eng. 23, 453–461 (1984).
[CrossRef]

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[CrossRef] [PubMed]

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Radiology (1)

F. G. Sommer, L. F. Joynt, B. A. Carroll, A. Macovski, “Ultrasonic characterization of abdominal tissues via digital analysis of backscattered waveforms,” Radiology 141, 811–817 (1981).
[PubMed]

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[CrossRef]

D. Nicholas, “Evaluation of backscattering coefficients for excised human tissues: results, interpretation and associated measurements,” Ultrasound Med. Biol. 8, 17–28 (1982).
[CrossRef]

E. J. Feleppa, F. L. Lizzi, D. J. Coleman, M. M. Yaremko, “Diagnostic spectrum analysis in ophthalmology: a physical perspective,” Ultrasound Med. Biol. 12, 623–631 (1986).
[CrossRef] [PubMed]

D. Nicholas, D. K. Nassiri, P. Garbutt, C. R. Hill, “Tissue characterization from ultrasound B-scan data,” Ultrasound Med. Biol. 12, 135–143 (1986).
[CrossRef] [PubMed]

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[CrossRef]

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

L. Fellingham-Joynt, “A stochastic approach to ultrasonic tissue characterization,” Ph.D. dissertation, (Stanford University, Stanford, Calif., 1979).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), Chap. 10.

C. Kittel, Introduction to Solid State Physics (Wiley, New York, 1966), p. 66.

P. M. Morse, K. U. Ingard, Theoretical Acoustics (McGraw-Hill, New York, 1968).

M. F. Insana, R. F. Wagner, B. S. Garra, T. H. Shawker, “A statistical approach to an expert diagnostic ultrasonic system,” in Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 24–29 (1986).

R. C. Waag, University of Rochester, Rochester, New York 14627 (personal communication, 1986).

J. M. Thijssen, “Scattering of ultrasound: general aspects,” in Ultrasonic Tissue Characterization and Echographic Imaging 5, Proceedings of the Fifth European Communities Workshop, J. M. Thijssen, V. Mazzeo, eds. (Faculty of Medicine Printing Office, University of Nijmegen, Nijmegen, The Netherlands, 1986), pp. 7–17.

J. S. Bendat, A. G. Piersol, Random Data: Analysis and Measurement Procedures (Wiley, New York, 1971), Chap. 3.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill, New York, 1960), Chap. 9.

M. Abramowitz, I. A. Stegun, Handbood of Mathematical Functions, Natl. Bur. Standards Appl. Math. Ser. 55 (U.S. Government Printing Office, Washington, D.C., 1964; Dover, New York, 1965).

J. D. Bjorken, S. D. Drell, Relativistic Quantum Mechanics (McGraw-Hill, New York, 1964), Chap. 7.

P. D. Burns, “Measurement of random and periodic image noise in raster-written images,” in Second International Congress on Advances in Non-Impact Printing Technologies (Society of Photographic Scientists and Engineers, Springfield, Va., 1984).

D. O. North, “The modification of noise by certain nonlinear devices,” (1943) [reprinted in Proc. IEEE51, 1016 (1963)].

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the noise cloud that results from the random walk in the complex plane (from Goodman12).

Fig. 2
Fig. 2

Schematic representation of the noise cloud that results from a constant phasor (or coherent signal) plus a random walk in the complex plane (from Goodman12).

Fig. 3
Fig. 3

(a) Rayleigh backscattered intensity as a function of frequency (dotted curve, f4); normalized squared form factors for Gaussian shapes with diameters (four standard deviations) shown (solid curve); typical 3-MHz pulse with FWHM equal to 2 MHz (dashed curve). (b) Product of f4 and squared form factors from (a); vertical arrows indicate location of specular scattering strength. (c) Product of functions from (b) and attenuation factor (see text). (d) Product of functions from (c) and pulse from (a).

Fig. 4
Fig. 4

(a) Solid curve, f4 weighted by squared Gaussian form factor; dotted curve, f4 weighted by squared hard-sphere form factor; dashed curve, expected backscattered intensity for scattering from glass spheres in gelatin, using the exact method of calculation derived by Faran24 (see text for parameters). All results are plotted as a function of the product of wave number k and particle radius a. (b) Solid curves, same as (a) with particle diameter as a parameter; dashed curves, same as (a) with particle diameter as a parameter.

Fig. 5
Fig. 5

Autocorrelation functions for intensity as a function of distance, in units of the standard deviation of the underlying Gaussian spread function. The curve parameter, from bottom to top, is r = Īs/Id = 0, 1, 2, 8, 18. The absolute units are determined by the value of Īd = 2 [cf. Eq. (2)].

Fig. 6
Fig. 6

Normalized autocovariance functions for intensity corresponding to absolute curves of Fig. 5. When the autocovariance functions for magnitude (derived in Appendix A) are plotted on the same scale, they are almost indistinguishable from these functions, except for the case r = 0 in which the greatest difference is between 0.03 and 0.04 (see Ref. 15).

Fig. 7
Fig. 7

Effect of a sinusoidal variation in coherent signal strength on the noise cloud of Fig. 2 (cf. Goodman12).

Fig. 8
Fig. 8

Top panel, autocorrelation function in intensity for simulated Rician speckle with single-frequency structured specularity of period d (other parameters defined in text). Middle panel, corresponding speckle power spectrum showing Gaussian fit to Rician noise, which integrates to the Rician variance, and peak corresponding to structured specularity, which integrates to var (Is) = ∑s2. Bottom panel, measured power spectrum for human liver in vivo. Error bars are ±1 standard error. Darkened area is Rician variance. (After Refs. 22 and 23.)

Tables (2)

Tables Icon

Table 1 Rf Tissue Signature

Tables Icon

Table 2 Intensity Tissue Structure Signaturea

Equations (85)

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p ( a r , a i ) = ( 2 π σ 2 ) - 1 exp [ - ( a r 2 + a i 2 ) / 2 σ 2 ] .
a a * = a r 2 + a i 2 = 2 σ 2 = I d
p ( a r , a i ) = ( 2 π σ 2 ) - 1 exp { - [ ( a r - I s ) 2 + a i 2 ] / 2 σ 2 } .
a ( x 1 ) = a 1 r + i a 1 i .
a ( x 2 ) = a 2 r + i a 2 i .
R a ( x 1 , x 2 ) = a ( x 1 ) a * ( x 2 ) .
A ( x ) = h ( x ) * a ( x ) ,
R A ( Δ x ) = h ( - Δ x ) * R a ( Δ x ) * h * ( Δ x ) ,
a ( x 1 ) = i = 1 N a i e i ϕ i
a ( x 2 ) = j = 1 N a j e i ϕ j ,
a ( x 1 ) a * ( x 2 ) = i = 1 N j = 1 N a i a j exp [ i ( ϕ i - ϕ j ) ] = N a i 2 δ ( Δ x ) = I d δ ( Δ x ) ,
R A ( Δ x ) = I d [ h ( - Δ x ) * h * ( Δ x ) ] .
W A ( k ) = H ( k ) 2 W a ( k ) .
H ( k ) FT h ( Δ x )
W a ( k ) = I d , W A ( k ) = I d H ( k ) 2 .
W A ( k ) FT R A ( Δ x ) W a ( k ) R a ( Δ x ) , H ( k ) 2 h ( - Δ x ) * h * ( Δ x ) .
h ( x , k 0 ) = h t ( x , y , k 0 ) h r ( z , k 0 ) , k 0 = k 0 .
h t ( x , y , k 0 ) = h t ( x , k 0 ) .
h r ( z , k 0 ) = A exp ( - z 2 / 2 σ 0 2 ) exp ( i k 0 z ) ,
H r ( k ) FT h r ( z , k 0 ) , H r ( k ) = exp [ - σ 0 2 ( k - k 0 ) 2 / 2 ] .
κ 1 / ρ c 2 ρ c = 1 / κ c Δ ( ρ c ) / ρ c = - Δ ( κ c ) / κ c } ,             Δ ρ 0 { - Δ κ / κ = 2 Δ c / c - γ κ .
Φ b ( 2 k ) = 2 π 2 k 2 Γ κ ( 2 k ) , Γ κ ( 2 k ) = 1 ( 2 π ) 3 d x γ κ ( x ) exp ( - 2 i k · x ) ,
Φ b ( 2 k ) = k 2 0 a 0 d r γ κ ( r ) sin ( 2 k r ) r 2 / ( 2 k r ) .
Φ b ( 2 k ) Φ 0 ( 2 k ) = 1 3 k 2 a 0 3 γ ¯ κ ,             2 k a 0 1 ,
d σ 0 = Φ 0 2 = 1 9 k 4 a 0 6 γ ¯ κ 2 = a 2 ,
- 2 k 2 Γ κ ( 2 k ) = 1 ( 2 π ) 3 d x γ κ ( x ) ( x · x ) exp ( - 2 i k · x ) k 0 x 2 .
R γ ( x 1 x 2 ) = γ κ ( x 1 ) γ κ ( x 2 ) .
R γ ( x 1 x 2 ) = A exp ( - Δ z 2 / 2 σ 0 2 ) exp ( i k 0 Δ z ) * γ κ ( x 1 ) γ κ ( x 2 ) * exp ( - Δ z 2 / 2 σ 0 2 ) exp ( - i k 0 Δ z ) .
W b ( k ) = d σ 0 Γ 0 2 ( 2 k ) H r ( k ) 2 = a i 2 H r ( k ) 2
Γ 0 2 ( 2 k ) = ( 9 / 16 π 2 a 0 6 γ ¯ κ 2 ) × d x 1 d x 2 γ κ ( x 1 ) γ κ ( x 2 ) exp ( - 2 i k · Δ x ) ,
C a ( Δ x ) = R a ( Δ x ) - a 2 ,
ρ ( Δ x ) = C a ( Δ x ) / C a ( 0 ) .
I 1 = a 1 r 2 + a 1 i 2 ,             I 2 = a 2 r 2 + a 2 i 2 ,             I = 2 σ 2 .
I 1 I 2 = ( a 1 r 2 + a 1 i 2 ) ( a 2 r 2 + a 2 i 2 ) .
X 1 X 2 X 3 X 4 = X 1 X 2 X 3 X 4 + X 1 X 3 X 2 X 4 + X 1 X 4 X 2 X 3
a 1 r a 2 i = a 1 i a 2 r = 0 ,
I 1 I 2 = 2 a 1 r 2 a 2 r 2 + 4 a 1 r a 2 r 2 + 2 a 1 i 2 a 2 r 2
ρ = a 1 a 2 * / I = ( a 1 r a 2 r + a 1 i a 2 i ) / 2 σ 2 = ( a 1 r a 2 r ) / σ 2 = a 1 i a 2 i / σ 2 ,
I 1 I 2 = ( 2 σ 2 ) 2 ( 1 + ρ 2 ) , = I d 2 ( 1 + ρ 2 ) .
a 1 = a 1 r + + i a 1 i , a 2 = a 2 r + + i a 2 i ,
I 1 I 2 = ( a 1 2 + 2 a 1 r + 2 ) ( a 2 2 + 2 a 2 r + 2 ) ,
a 1 2 = a 1 r 2 + a 1 i 2 , a 2 2 = a 2 r 2 + a 2 i 2 .
a 1 2 a 2 r = a 1 r a 2 2 = 0 ,
I 1 I 2 = I 2 d ( 1 + ρ 2 ) + 2 I d I ¯ s + I ¯ s 2 + 2 I d I ¯ s ρ .
I d = d k I d H ( k ) 2 .
ρ = ( π σ 0 2 ) - 1 / 2 exp ( - z 2 / 2 σ 0 2 ) * exp ( - z 2 / 2 σ 0 2 ) .
I 2 = I 2 d + 2 I d I ¯ s + I ¯ s 2 = ( I d + I ¯ s ) 2 , I 2 = 2 I 2 d + 4 I d I ¯ s + I ¯ s 2 ,
I 2 - I 2 = I 2 d + 2 I d I ¯ s = σ R 2 ,
a 1 = a 1 r + 1 + i ( a 1 i + J 1 ) , a 2 = a 2 r + 2 + i ( a 2 i + J 2 ) ,
I 1 I 2 = I 2 d ( 1 + ρ 2 ) + I d ( I s 1 + I s 2 ) + ( 1 2 + J 1 2 ) ( 2 2 + J 2 2 ) + 2 I d ρ ( 1 2 + J 1 J 2 ) .
I ( x ) I ( x + Δ x ) = I 2 d ( 1 + ρ 2 ) + 2 I d I ¯ s + I s 1 I s 2 + 2 I d ρ ( 1 2 + J 1 J 2 ) .
I s 1 I s 2 = X - 1 I s 1 I s 2 = ( X ) - 1 x 1 I s ( x 1 ) I s ( x 1 + Δ x ) .
I s = I ¯ s + Δ I s , = ¯ + Δ , J = J ¯ + Δ J .
I s I s = I ¯ s 2 + Δ I s Δ I s .
I s I s FT I ¯ s 2 δ ( f ) + Δ ˜ I s ( f ) 2 / X , P FT ρ , etc . ,
W ( f ) = δ ( f ) ( I 2 d + 2 I d I ¯ s ) + δ ( f ) I ¯ s 2 + I 2 d P * P + Δ ˜ I s ( f ) 2 / X + 2 I d P [ ¯ 2 + J ¯ 2 ] + 2 I d P * [ Δ R ˜ ( f ) 2 + Δ J ˜ ( f ) 2 ] / X .
Var ( I ) = I 2 - I 2 > I 2             ( non - Gaussian statistics )
Var ( I ) = I 2 d + 2 I d I ¯ s = σ R 2 ( I d + I ¯ s ) 2 = I 2             ( Gaussian statistics ) ,
t = 2 I 2 d + 4 I d I ¯ s + I s 2 , p = I 2 d + 2 I d I ¯ s + I s 2 , b = ( I d + I ¯ s ) 2 ,
t - p = I 2 d + 2 I d I ¯ s = σ R 2 , p - b = I s 2 - I ¯ s 2 = Σ s 2 , b 1 / 2 = I d + I ¯ s .
˜ ( f ) = ( A 1 / 2 ) [ δ ( f - f 0 ) + δ ( f + f 0 ) ] + A 0 δ ( f ) ,
˜ ( f ) * ˜ ( f ) = d f ˜ ( f ) ˜ ( f - f ) = δ ( f ) ( A 1 2 / 2 + A 0 2 ) + A 1 A 0 [ δ ( f - f 0 ) + δ ( f + f 0 ) ] + ( A 1 2 / 4 ) [ δ ( f - 2 f 0 ) + δ ( f + 2 f 0 ) ] ,
X - 1 d f ˜ ( f ) * ˜ ( f ) 2 = A 0 4 + A 0 2 A 1 2 + A 1 4 / 4 + 2 A 0 2 A 1 2 + A 1 4 / 8 ,
I ¯ s = 2 = A 0 2 + A 1 2 / 2 , I s 2 = 4 = A 0 4 + 3 A 0 2 A 1 2 + 3 A 1 4 / 8 ,
Σ s 2 = I s 2 - I ¯ s 2 = 2 A 0 2 A 1 2 + A 1 4 / 8 ,
X - 1 d f ˜ ( f ) * ˜ ( f ) 2 = I ¯ s 2 + Σ s 2
X - 1 d f Δ ˜ I s ( f ) 2 = Σ s 2 .
d f ˜ ( f ) 2 = A 0 2 + A 1 2 / 2 = I ¯ s .
d x ( a * b ) = ( d x a ) ( d x b ) ,
2 I d d f P * ˜ ( f ) 2 2 I d I s
ρ = h ( - Δ x ) * h * ( Δ x ) / [ h * h * ] Δ x = 0 , ρ P = H ( f ) 2
a = i a i e i ϕ l ,
p 2 ( a r , a i ) = ( 2 π σ 2 ) - 1 exp [ - ( a r 2 + a i 2 ) / 2 σ 2 ]
p 1 ( V ) = ( V / σ 2 ) exp ( - V 2 / 2 σ 2 )             V 0 = 0             otherwise .
p 1 ( I ) = ( 2 σ 2 ) - 1 exp ( - I / 2 σ 2 )             I 0 = 0             otherwise
p 4 ( a ) = exp [ - ½ ( a t K 0 - 1 a ) ] ( 2 π ) 2 K 0 1 / 2 ,
K 0 = σ 2 [ 1 0 ρ 0 0 1 0 ρ ρ 0 1 0 0 ρ 0 1 ] ;             det K 0 = K 0 = σ 8 ( 1 - ρ 2 ) 2 .
ρ a 1 a 2 * ( I 1 I 2 ) 1 / 2 = a 1 r a 2 r σ 1 σ 2 = a 1 i a 2 i σ 1 σ 2 .
p 2 ( V 1 , V 2 ) = V 1 V 2 ψ 2 ( 1 - ρ 2 ) exp [ - ( V 1 2 + V 2 2 ) / 2 ψ ( 1 - ρ 2 ) ] × I 0 ( ρ V 1 V 2 / ψ [ 1 - ρ 2 ] ) ,
R ν ( Δ x ) = V 1 ν V 2 ν = 0 d V 1 d V 2 V 1 ν + 1 V 2 ν + 1 p 2 ( V 1 V 2 ) ,
V 1 ν V 2 ν = ( 2 ψ ) ν Γ 2 ( ν 2 + 1 ) { ( 1 - ρ 2 ) ν F 2 1 ( ν 2 + 1 , ν 2 + 1 ; 1 ; ρ 2 ) F 2 1 ( - ν 2 ; - ν 2 ; 1 ; ρ 2 )
V 1 2 V 1 2 = I 1 I 2 = ( 2 ψ ) 2 ( 1 + ρ 2 ) ,
p 2 ( a r , a i ) = ( 2 π σ 2 ) - 1 exp { - [ ( a r - ) 2 + a i 2 ] / 2 σ 2 } .
p 2 ( V 1 V 2 ) = V 1 V 2 ψ 2 ( 1 - ρ 2 ) exp [ - ( V 1 2 + V 2 2 ) / 2 ψ ( 1 - ρ 2 ) ] × exp [ - 2 / 2 ψ ( 1 + ρ ) ] × m = 0 m I m [ ρ V 1 V 2 ψ ( 1 - ρ 2 ) ] × I m [ V 1 ψ ( 1 + ρ ) ] I m [ V 2 ψ ( 1 + ρ ) ] ,
R ν ( Δ x ) = V 1 ν V 2 ν = ( 2 ψ ) ν ( 1 - ρ 2 ) ν + 1 × exp [ - R 2 ρ / ψ ( 1 + ρ ) ] × m = 0 m ρ m ( m ! ) 2 ( 1 - ρ 1 + ρ ) m p m × n = 0 ρ 2 n Γ 2 ( m + n + 1 + / 2 ν ) n ! ( n + m ) ! × F 1 1 [ - n - / 2 ν ; m + 1 ; - p ( 1 - ρ ) 1 + ρ ] 2 ,

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