Abstract

The Hotelling trace criterion (HTC) is used to find a set of linear features that optimally separate two classes of objects. The objects used in our study were simulated livers with and without tumors, with noise, blur, and object variability. Using the receiver-operating-characteristic parameter da as our measure, we have found that the ability of the HTC to separate these objects into their correct classes, by detecting the presence or absence of a tumor, has a correlation of 0.988 with the ability of humans to separate the same two classes of objects. This suggests, therefore, that the HTC can be used as a figure of merit for optimizing system parameters, since it calculates a single, scalar figure of merit that has a high correlation with human-observer performance.

© 1987 Optical Society of America

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References

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  1. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).
  2. C. E. Metz, “Basic principles of ROC analysis,” Sem. Nucl. Med. 8, 283–298 (1978).
    [CrossRef]
  3. J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–112 (1979).
    [CrossRef] [PubMed]
  4. J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems: Methods from Signal Detection Theory (Academic, New York, 1982).
  5. C. E. Metz, “ROC methodology in radiological imaging,” Invest. Rad. 26, 720–733 (1986).
    [CrossRef]
  6. R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
    [CrossRef]
  7. J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
    [CrossRef]
  8. W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
    [CrossRef]
  9. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
    [CrossRef] [PubMed]
  10. H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
    [CrossRef]
  11. W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,”J. Opt. Soc. Am. 3, 717–725 (1986).
    [CrossRef]
  12. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  13. Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
    [CrossRef]
  14. H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
    [CrossRef]
  15. H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV; Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).
  16. R. K. O’Toole, H. Stark, “Comparative study of optical-digital vs. all-digital techniques in texural pattern recognition,” Appl. Opt. 19, 2496–2506 (1980).
    [CrossRef]
  17. C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. 60637 (personal communication, October1986).
  18. W. B. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972), p. 12.
  19. A. Ben-Isreal, T. N. E. Greville, Generalized Inverses: Theories and Application (Wiley, New York, 1974), p. 174.
  20. G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
    [CrossRef]
  21. A. E. Burgess, “On observer internal noise,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 208–213 (1986).

1986

C. E. Metz, “ROC methodology in radiological imaging,” Invest. Rad. 26, 720–733 (1986).
[CrossRef]

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,”J. Opt. Soc. Am. 3, 717–725 (1986).
[CrossRef]

1985

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

1984

Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

1982

G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
[CrossRef]

1981

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

1980

1979

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–112 (1979).
[CrossRef] [PubMed]

1978

C. E. Metz, “Basic principles of ROC analysis,” Sem. Nucl. Med. 8, 283–298 (1978).
[CrossRef]

1961

J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
[CrossRef]

1958

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

1931

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Barlow, H. B.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Barrett, H. H.

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,”J. Opt. Soc. Am. 3, 717–725 (1986).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV; Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

Ben-Isreal, A.

A. Ben-Isreal, T. N. E. Greville, Generalized Inverses: Theories and Application (Wiley, New York, 1974), p. 174.

Birdsall, T. G.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Borgstrom, M.

G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
[CrossRef]

Brown, D. G.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Burgess, A. E.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

A. E. Burgess, “On observer internal noise,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 208–213 (1986).

Egan, J. P.

J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
[CrossRef]

Fiete, R. D.

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Greenberg, G. Z.

J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
[CrossRef]

Greville, T. N. E.

A. Ben-Isreal, T. N. E. Greville, Generalized Inverses: Theories and Application (Wiley, New York, 1974), p. 174.

Gu, Z. H.

Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Hotelling, H.

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Lee, S.

Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Mazzeo, J.

G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
[CrossRef]

Meisel, W. B.

W. B. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972), p. 12.

Metz, C. E.

C. E. Metz, “ROC methodology in radiological imaging,” Invest. Rad. 26, 720–733 (1986).
[CrossRef]

C. E. Metz, “Basic principles of ROC analysis,” Sem. Nucl. Med. 8, 283–298 (1978).
[CrossRef]

C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. 60637 (personal communication, October1986).

Meyers, K. J.

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

Milster, T. D.

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

Myers, K. J.

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV; Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

O’Toole, R. K.

Pickett, R. M.

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems: Methods from Signal Detection Theory (Academic, New York, 1982).

Schulman, A. I.

J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
[CrossRef]

Seeley, G. W.

G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
[CrossRef]

Smith, W. E.

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,”J. Opt. Soc. Am. 3, 717–725 (1986).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

Stark, H.

Swets, J. A.

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–112 (1979).
[CrossRef] [PubMed]

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems: Methods from Signal Detection Theory (Academic, New York, 1982).

Tanner, W. P.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Wagner, R. F.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV; Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

Ann. Math. Stat.

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Appl. Opt.

Behav. Res. Methods Instrum.

G. W. Seeley, M. Borgstrom, J. Mazzeo, “A general interactive computer program for running signal detection experiments,” Behav. Res. Methods Instrum. 14, 555–556 (1982).
[CrossRef]

Invest. Rad.

C. E. Metz, “ROC methodology in radiological imaging,” Invest. Rad. 26, 720–733 (1986).
[CrossRef]

Invest. Radiol.

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–112 (1979).
[CrossRef] [PubMed]

J. Acoust. Soc. Am.

J. P. Egan, G. Z. Greenberg, A. I. Schulman, “Operating characteristics, signal detectability, and the method of free response,”J. Acoust. Soc. Am. 33, 993–1007 (1961).
[CrossRef]

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

J. Opt. Soc. Am.

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,”J. Opt. Soc. Am. 3, 717–725 (1986).
[CrossRef]

Opt. Eng.

Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Phys. Med. Biol.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Science

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Sem. Nucl. Med.

C. E. Metz, “Basic principles of ROC analysis,” Sem. Nucl. Med. 8, 283–298 (1978).
[CrossRef]

Other

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems: Methods from Signal Detection Theory (Academic, New York, 1982).

H. H. Barrett, W. E. Smith, K. J. Meyers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing and Display, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.535, 65–69 (1985).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV; Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

C. E. Metz, Department of Radiology, University of Chicago, Chicago, Ill. 60637 (personal communication, October1986).

W. B. Meisel, Computer-Oriented Approaches to Pattern Recognition (Academic, New York, 1972), p. 12.

A. Ben-Isreal, T. N. E. Greville, Generalized Inverses: Theories and Application (Wiley, New York, 1974), p. 174.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

A. E. Burgess, “On observer internal noise,” in Proceedings of the Application of Optical Instrumentation in Medicine XIV: Medical Imaging, Processing, and Display, R. H. Schneider, S. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 208–213 (1986).

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Figures (7)

Fig. 1
Fig. 1

The set of 64 simulated liver images without any blur or noise degradation. Each image in a square, 64 × 64 pixel array (4 cm × 4 cm). The first 32 images have a tumor of random size, shape, location, and brightness placed in them, whereas the remaining 32 do not.

Fig. 2
Fig. 2

The 64 objects in Fig. 1 after blur and noise degradation (object set number 2). Each object is a square, 64 × 64 pixel array.

Fig. 3
Fig. 3

The 64 × 64 pixel linear feature operator Ah calculated for object set number 2. If we compare this image of the feature operator with each of the 64 × 64 pixel objects in Fig. 2, we note that the bright region in the feature operator is the region within each object where a tumor is most likely to occur. The feature operator is, therefore, simply a template that can be put over the image in a matched-filtering operation.

Fig. 4
Fig. 4

The ROC curve generated from the HTC method and the observer study on object set number 2. A ROC curve is a plot of the probability of a true positive, e.g., correctly classifying a liver as diseased, versus the probability of a false positive, e.g., incorrectly classifying a normal liver as diseased, as a function of the decision criterion. A poor classifier will have a ROC curve close to the diagonal or chance line (da = 0). A perfect classifier will have a ROC curve that follows the left-vertical and upper-horizontal edges of the ROC square (da = ∞). For the HTC ROC curve, da (Hotelling) 2.37. For the human-observer ROC curve, da (human) = 1.31.

Fig. 5
Fig. 5

Plot of J versus da2 (human) with a linear fit to the data points.

Fig. 6
Fig. 6

Plot of da (Hotelling) versus da (human) with a linear fit to the data points. The correlation is 0.988.

Fig. 7
Fig. 7

Simultaneous diagonalization of S1 and S2 for a two-dimensional case. (a) Is transformed into (b) by performing a whitening transformation, and (b) is transformed into (c) by an orthonormal transformation.12

Tables (1)

Tables Icon

Table 1 Summary of Results for All Nine Sets of Images with the Corresponding Amount of Blur and Noise Added to Each Set

Equations (68)

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

S 1 i = 1 K P i ( f i - f 0 ) ( f i - f 0 ) T ,
f 0 = i = 1 K P i f i .
S 2 i = 1 K P i C i ,
C i = ( f i - f i ) ( f i - f i ) T .
J tr ( S 2 - 1 S 1 ) ,
g i = Hf i + n i ,
S 1 g = HS 1 H T
S 2 g = HS 2 H T + C n ,
C n = i = 1 K P i n i n i T .
J g tr [ S 2 g - 1 S 1 g ] = tr [ ( HS 2 H T + C n ) - 1 HS 1 H T ] .
h = A h f .
S 2 - 1 S 1 A T = A T Λ ,
J = tr [ ( A S 2 A T ) - 1 ( A S 1 A T ) ] = i = 1 N λ i = i = 1 R λ i ,
A h T = S 2 - 1 ( f 2 - f 1 ) ,
J = P 1 ( h 1 - h 0 ) 2 + P 2 ( h 2 - h 0 ) 2 P 1 σ h 1 2 + P 2 σ h 2 2 ,
J = P 1 P 2 ( h 1 - h 2 ) 2 P 1 σ h 1 2 + P 2 σ h 2 2 .
J = ( h 1 - h 2 ) 2 2 ( σ h 1 2 + σ h 2 2 ) = 1 4 d a 2 ,
d ( h 2 - h 1 ) σ .
1 2 exp ( - J 2 ) .
C i C ˜ i = 1 M j = 1 M ( f i j - f i ) ( f i j - f i ) T ,
f i = 1 M j = 1 M f i j
S 2 S ˜ 2 = i = 1 K P i C ˜ i .
S 2 S ˜ 2 = S ˜ 2 ( nf ) + C n ,
S ˜ 2 ( nf ) = i = 1 K P i C ˜ i ( nf )
S 2 S ˜ 2 ( nf ) + σ n 2 I ,
S 2 = lim ( σ 0 ) ( S 2 T S 2 + σ I ) - 1 S 2 T .
W 1 [ P 1 ( f 1 - f 0 ) , , P i ( f i - f 0 ) , , P K ( f K - f 0 ) ]
W 2 = ( 1 / M ) [ P 1 ( f 11 - f 1 , , P 1 ( f 1 M - f 1 ) , , P K ( f K 1 - f K ) , , P K ( f K M - f K ) ]
J = tr { [ A h ( HW 2 W 2 T H T + σ n 2 I ) A h T ] - 1 × [ A h HW 1 W 1 T A h T H T ] }
J = tr { [ ( A h HW 2 ) ( A h HW 2 ) T + σ n 2 A h A h T ] - 1 × [ ( A h HW 1 ) ( A h HW 1 ) T ] } = tr ( D - 1 B ) ,
D ( A h HW 2 ) ( A h HW 2 ) T + σ n 2 A h A h T
B ( A h HW 1 ) ( A h HW 1 ) T .
B pert = B + ( 2 Δ A · A h n + Δ A 2 ) m = 1 2 M H W 1 m n 2 ,
d a L s + n - L n ( σ L s + n 2 + σ L n 2 2 ) 1 / 2 ,
S 2 - 1 S 1 A T = A T Λ ,
Y = θ - 1 / 2 Φ T X ,
S 2 Φ = Φ θ ,
Φ T Φ = I .
θ - 1 / 2 Φ T S 2 Φ θ - 1 / 2 = I
θ - 1 / 2 Φ T S 1 Φ θ - 1 / 2 = K .
Z = Ψ T Y
K Ψ = Ψ Λ
Ψ T Ψ = I .
Ψ T I Ψ = Ψ T Ψ = I
Ψ T I Ψ = Λ .
θ - 1 / 2 Φ T S 1 Φ θ - 1 / 2 Ψ = Ψ Λ
S 1 Φ θ - 1 / 2 Ψ = ( θ - 1 / 2 Φ T ) - 1 Ψ Λ .
S 1 ( Φ θ - 1 / 2 Ψ ) = S 2 ( Φ θ - 1 / 2 Ψ ) Λ
S 2 - 1 S 1 ( Ψ T θ - 1 / 2 Φ T ) T = ( Ψ T θ - 1 / 2 Φ T ) T Λ .
A = Ψ θ - 1 / 2 Φ T .
S 2 - 1 S 1 A h T = A h T J .
S 1 = P 1 ( f 1 - f 0 ) ( f 1 - f 0 ) T - P 2 ( f 2 - f 0 ) ( f 2 - f 0 ) T .
S 1 = P 2 ( 1 - P 2 ) ( f 2 - f 1 ) ( f 2 - f 1 ) T
S 1 = P ( 1 - P ) Δ f ¯ Δ f ¯ T ,
S 2 = ( 1 - P ) C 1 + P C 2 .
S 2 - 1 Δ f ¯ J = S 2 - 1 S 1 S 2 - 1 Δ f ¯ = S 2 - 1 P ( P - 1 ) Δ f ¯ Δ f ¯ T S 2 - 1 Δ f ¯ .
J = P ( P - 1 ) Δ f ¯ T S 2 - 1 Δ f ¯ ,
J = P ( P - 1 ) Δ f ¯ T [ ( 1 - P ) C 1 + P C 2 ] - 1 Δ f ¯ .
P 1 ( 1 - s ) P 2 s exp [ - μ ( s ) ] ,
μ ( s ) = ½ s ( s - 1 ) Δ f ¯ T [ ( 1 - s ) C 1 + s C 2 ] - 1 Δ f ¯ + ½ ln ( 1 - s ) C 1 + s C 2 C 1 1 - s C 2 s ,
μ ( P ) = ½ J - ½ ln ( β ) ,
β C 1 ( 1 - P ) C 2 P ( 1 - P ) C 1 + P C 2 .
P P ( 1 - P ) ( 1 - P ) exp ( - J 2 ) β .
β = ( n λ 1 n ) 1 - P ( n λ 2 n ) P n [ ( 1 - P ) λ 1 n + P λ 2 n ]
β = n λ 1 n 1 - p λ 2 n P ( 1 - P ) λ l n + P λ 2 n ,
x r y 1 - r r x + ( 1 - r ) y ;
P P ( P - 1 ) ( P - 1 ) exp ( - J 2 ) .
1 2 exp ( - J 2 ) .

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