Abstract

A number of task-specific approaches to the assessment of image quality are treated. Both estimation and classification tasks are considered, but only linear estimators or classifiers are permitted. Performance on these tasks is limited by both quantum noise and object variability, and the effects of postprocessing or image-reconstruction algorithms are explicitly included. The results are expressed as signal-to-noise ratios (SNR’s). The interrelationships among these SNR’s are considered, and an SNR for a classification task is expressed as the SNR for a related estimation task times four factors. These factors show the effects of signal size and contrast, conspicuity of the signal, bias in the estimation task, and noise correlation. Ways of choosing and calculating appropriate SNR’s for system evaluation and optimization are also discussed.

© 1990 Optical Society of America

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References

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  1. L. B. Lusted, “Signal detectability and medical decisionmaking,” Science 171, 1217–1219 (1971).
    [Crossref] [PubMed]
  2. C. F. Metz, “ROC methodology in radiologic imaging,” Invest. Radiol. 21, 720–733 (1986).
    [Crossref] [PubMed]
  3. J. A. Swets, “Measuring the accuracy of diagnostic systems,” Science 240, 1285–1293 (1988).
    [Crossref] [PubMed]
  4. J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems (Academic, New York, 1982).
  5. K. M. Hanson, “popart—performance optimized algebraic reconstruction technique,” in Visual Communications and Image Processing III, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 318–325 (1989).
  6. K. M. Hanson, “Optimization for object localization of the constrained algebraic reconstruction technique,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 146–153 (1989);“Method of evaluating image-recovery algorithms based on task performance,” J. Opt. Soc. A 7, 1294–1304 (1990).
    [Crossref]
  7. S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).
  8. A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).
  9. H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
    [Crossref]
  10. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  11. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  12. H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981), Vols. I and II.
  13. J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  14. W. E. Smith, H. H. Barrett, “Linear estimation theory applied to the evaluation of a priori information and system optimization in coded-aperture imaging,” J. Opt. Soc. Am. A 5, 315–330 (1988).
    [Crossref] [PubMed]
  15. H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968).
  16. R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
    [Crossref]
  17. R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
    [Crossref]
  18. T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).
  19. A. Rose, “The sensitivity performance of the human eye on an absolute scale,” J. Opt. Soc. Am. 38, 196–208 (1948).
    [Crossref] [PubMed]
  20. W. E. Smith, H. H. Barrett, “Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
    [Crossref]
  21. A. E. Burgess, B. Colbourne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617–627 (1988).
    [Crossref] [PubMed]
  22. 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]
  23. K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
    [Crossref] [PubMed]
  24. K. J. Myers, H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
    [Crossref] [PubMed]
  25. R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
    [Crossref] [PubMed]
  26. T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).
  27. D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.
  28. T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).
  29. K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Effect of a spatially varying background on aperture choice for optimized image quality,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
    [Crossref] [PubMed]

1990 (1)

1989 (1)

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

1988 (3)

1987 (2)

1986 (2)

1985 (2)

1981 (1)

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]

1971 (1)

L. B. Lusted, “Signal detectability and medical decisionmaking,” Science 171, 1217–1219 (1971).
[Crossref] [PubMed]

1948 (1)

1931 (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[Crossref]

Albert, A.

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Barber, H. B.

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

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.

K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Effect of a spatially varying background on aperture choice for optimized image quality,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
[Crossref] [PubMed]

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

W. E. Smith, H. H. Barrett, “Linear estimation theory applied to the evaluation of a priori information and system optimization in coded-aperture imaging,” J. Opt. Soc. Am. A 5, 315–330 (1988).
[Crossref] [PubMed]

K. J. Myers, H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
[Crossref] [PubMed]

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[Crossref] [PubMed]

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

K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
[Crossref] [PubMed]

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981), Vols. I and II.

Borgstrom, M. C.

Brown, D. G.

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

R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[Crossref]

Burgess, A. E.

A. E. Burgess, B. Colbourne, “Visual signal detection. IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617–627 (1988).
[Crossref] [PubMed]

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]

R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[Crossref]

Calvert, T. W.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

Cargill, E. B.

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

Cohn, D. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Colbourne, B.

Fiete, R. D.

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[Crossref] [PubMed]

Fukunaga, K.

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

Hall, J. N.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

Hanson, K. M.

K. M. Hanson, “popart—performance optimized algebraic reconstruction technique,” in Visual Communications and Image Processing III, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 318–325 (1989).

K. M. Hanson, “Optimization for object localization of the constrained algebraic reconstruction technique,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 146–153 (1989);“Method of evaluating image-recovery algorithms based on task performance,” J. Opt. Soc. A 7, 1294–1304 (1990).
[Crossref]

Hickernell, T. S.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

Holman, B. L.

S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).

Hotelling, H.

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[Crossref]

Hunt, B. R.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

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]

Ker, M.

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

Kijewski, M. F.

S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).

Kwo, D. P.

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

Lusted, L. B.

L. B. Lusted, “Signal detectability and medical decisionmaking,” Science 171, 1217–1219 (1971).
[Crossref] [PubMed]

Melsa, J. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Metz, C. F.

C. F. Metz, “ROC methodology in radiologic imaging,” Invest. Radiol. 21, 720–733 (1986).
[Crossref] [PubMed]

Moore, S. C.

S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).

Mueller, S. P.

S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).

Myers, K. J.

Patton, D. D.

Pickett, R. M.

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems (Academic, New York, 1982).

Rolland, J. P.

Rose, A.

Seeley, G. W.

Smith, W. E.

Swets, J. A.

J. A. Swets, “Measuring the accuracy of diagnostic systems,” Science 240, 1285–1293 (1988).
[Crossref] [PubMed]

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems (Academic, New York, 1982).

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981), Vols. I and II.

Tapiovaara, M. J.

R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[Crossref]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968).

Wagner, R. F.

K. J. Myers, J. P. Rolland, H. H. Barrett, R. F. Wagner, “Effect of a spatially varying background on aperture choice for optimized image quality,” J. Opt. Soc. Am. A 7, 1279–1293 (1990).
[Crossref] [PubMed]

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]

Wagner, R. K.

R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[Crossref]

White, T. A.

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

Woolfenden, J. M.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

Young, T. Y.

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

Ann. Math. Stat. (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360–378 (1931).
[Crossref]

Invest. Radiol. (1)

C. F. Metz, “ROC methodology in radiologic imaging,” Invest. Radiol. 21, 720–733 (1986).
[Crossref] [PubMed]

J. Nucl. Med. (1)

T. A. White, H. H. Barrett, E. B. Cargill, R. D. Fiete, M. Ker, “The use of the Hotelling trace to optimize collimator performance,” J. Nucl. Med. 30, 892(A) (1989).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (7)

Phys. Med. Biol. (1)

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

Science (3)

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]

J. A. Swets, “Measuring the accuracy of diagnostic systems,” Science 240, 1285–1293 (1988).
[Crossref] [PubMed]

L. B. Lusted, “Signal detectability and medical decisionmaking,” Science 171, 1217–1219 (1971).
[Crossref] [PubMed]

Other (14)

D. P. Kwo, H. B. Barber, H. H. Barrett, T. S. Hickernell, J. M. Woolfenden, “Comparison of NaI(T1), HgI2and CdTe surgical probes—II: Effect of scatter compensation on probe performance,” submitted to Med. Phys.

T. S. Hickernell, H. H. Barrett, H. B. Barber, J. N. Hall, J. M. Woolfenden, “Probability modelling of a radiation-detector-probe system for statistical analysis,” Phys. Med. Biol. (to be published).

J. A. Swets, R. M. Pickett, Evaluation of Diagnostic Systems (Academic, New York, 1982).

K. M. Hanson, “popart—performance optimized algebraic reconstruction technique,” in Visual Communications and Image Processing III, T. R. Hsing, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1001, 318–325 (1989).

K. M. Hanson, “Optimization for object localization of the constrained algebraic reconstruction technique,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 146–153 (1989);“Method of evaluating image-recovery algorithms based on task performance,” J. Opt. Soc. A 7, 1294–1304 (1990).
[Crossref]

S. P. Mueller, M. F. Kijewski, S. C. Moore, B. L. Holman, “Maximum-likelihood estimation—a model for optimal quantitation in nuclear medicine,” J. Nucl. Med. (to be published).

A. Albert, Regression and the Moore-Penrose Pseudoinverse (Academic, New York, 1972).

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

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

H. H. Barrett, W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981), Vols. I and II.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

T. Y. Young, T. W. Calvert, Classification, Estimation and Pattern Recognition (Elsevier, New York, 1974).

H. L. Van Trees, Detection, Estimation and Modulation Theory (Wiley, New York, 1968).

R. K. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, S. J. Dwyer, R. G. Jost, H. Schneider, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[Crossref]

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Tables (1)

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Table 1 Summary of All Comparisons

Equations (132)

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

g = H f + n ,
n | f d n p ( n | f ) ,
f d f p ( f ) ,
j d f p ( f | j ) ,
f = j = 1 J P j j ,
g n | f = H f ,
K n | f n n t n | f = diag ( H f ) ,
K n = K n | f f = diag ( H f ¯ ) ,
K g [ g H f ¯ ] [ g H f ¯ ] t n , f = [ H ( f f ¯ ) + n ] [ H ( f f ¯ ) + n ] t n | f f = H K f H t + K n .
f ¯ j f j ,
K j ( f f ¯ j ) ( f f ¯ j ) t j .
S 2 j = 1 J P j K j ,
S 1 j = 1 J P j ( f ¯ f ¯ j ) ( f ¯ f ¯ j ) t ,
f ¯ = j = 1 J P j f ¯ j .
S 1 = P 1 P 2 ( f ¯ 2 f ¯ 1 ) ( f ¯ 2 f ¯ 1 ) t .
K f ( f f ¯ ) ( f f ¯ ) t f ,
K f = S 1 + S 2 .
f ̂ = O { g } = O { H f + n } = O H f + O n ,
f ̂ = f + [ O H I ] f + O n = f + b + m ,
m n | f = 0 .
K m m m t n , f = O K n O t .
b [ O H I ] f = f ̂ f n | f .
B b b t f .
f ̂ ¯ f ̂ n , f ,
K ̂ ( f ̂ f ̂ ¯ ) ( f ̂ f ̂ ¯ ) t n , f = O H K f H t O t + K m .
EMSE [ f ̂ ] | f ̂ f | 2 n , f = tr ( f ̂ f ) ( f ̂ f ) t n , f = tr B + tr K m ,
f ̂ ¯ j f ̂ n , j ,
K ̂ j ( f ̂ f ̂ ¯ j ) ( f ̂ f ̂ ¯ j ) t n | j .
Ŝ 2 = j = 1 J P j K ̂ j ,
Ŝ 1 = j = 1 J P j ( f ̂ f ̂ ¯ j ) ( f ̂ f ̂ ¯ j ) t ,
K ̂ = Ŝ 1 + Ŝ 2 .
Ŝ 1 = O H S 1 H t O t ,
Ŝ 2 = O H S 2 H t O t + O K n O t .
θ = w t f ,
H + H w = w ,
θ ̂ = y t g
θ ̂ n | f = θ for all f .
A + A w = w ,
A = O H .
θ ̂ ROI = w t f ̂ .
b ROI θ ̂ ROI n | f θ = w t b ,
[ w t b ] 2 f = tr [ B W ] ,
W w w t .
var ( θ ̂ ROI ) ( θ ̂ ROI θ ̂ ¯ ROI ) 2 n , f = tr [ K ̂ W ] .
EMSE [ θ ̂ ROI ] = ( θ ̂ ROI θ ) 2 n , f = tr ( B W ) + tr ( K m W ) .
[ SNR est ] 2 θ 2 n , f EMSE [ θ ̂ ]
[ SNR ROI ] 2 = tr [ F W ] tr [ B W ] + tr [ K m W ] ,
F f f t f .
[ SNR pixel ] 2 = [ f k ] 2 [ K m | f ] k k ,
W i j = δ i k δ j k .
θ ̂ GM = w t [ A t K m 1 A ] + A t K m 1 f ̂ ,
C + = [ C t C ] + C t .
θ ̂ GM = w t [ K m 1 / 2 A ] + K m 1 / 2 f ̂ v t f ̂ .
EMSE [ θ ̂ GM ] = tr [ V K m ] = tr [ ( A t K m 1 A ) + W ] ,
[ SNR GM ] 2 = tr [ F W ] tr ( A t K m 1 A ) + W ] .
θ ̂ WE = w t ( A t K m 1 A + K f 1 ) 1 A t K m 1 ( f ̂ f ̂ ¯ ) + w t f ¯ .
EMSE [ θ ̂ WE ] = tr [ ( A t K m 1 A + K f 1 ) 1 W ] ,
[ SNR WE ] 2 = tr [ F W ] tr [ ( A t K m 1 A + K m 1 ) 1 W ] .
λ = u t f ̂ .
d 2 = [ λ j = 2 λ j = 1 ] 2 P 1 var ( λ | j = 1 ) + P 2 var ( λ | j = 2 ) ,
[ SNR λ ] 2 = P 1 P 2 d 2 .
[ λ j = 2 λ j = 1 ] 2 = [ u t ( f ̂ ¯ 2 f ̂ ¯ 1 ) ] 2 = u t ( f ̂ ¯ 2 f ̂ ¯ 1 ) ( f ̂ ¯ 2 f ̂ ¯ 1 ) t u = tr [ U ( f ̂ ¯ 2 f ̂ ¯ 1 ) ( f ̂ ¯ 2 f ̂ ¯ 1 ) t ] = [ P 1 P 2 ] 1 tr [ U Ŝ 1 ] ,
U u u t .
var ( λ | j ) = u t K ̂ j u .
[ SNR λ ] 2 = tr [ U Ŝ 1 ] tr [ U Ŝ 2 ]
λ MF = s t f ̂ .
[ SNR MF ] 2 = tr [ S Ŝ 1 ] tr [ S Ŝ 2 ] ,
S s s t .
[ SNR MF ] 2 = [ tr ( Ŝ 1 ) ] 2 tr [ Ŝ 1 Ŝ 2 ] .
[ SNR MF ] 2 = [ tr ( Ŝ 1 ) ] 2 tr [ Ŝ 1 K m ] .
λ GM = s t K m 1 f ̂ .
λ GM = [ K m 1 / 2 s ] t K m 1 / 2 f ̂ .
[ SNR PW ] 2 = tr [ K m 1 Ŝ 1 K m 1 S ] 2 tr [ K m 1 Ŝ 2 K m 1 S ] .
[ SNR PW ] 2 = { tr [ K m 1 Ŝ 1 ] } 2 tr [ K m 1 Ŝ 2 K m 1 Ŝ 1 ] .
[ SNR PW ] 2 = tr ( K m 1 Ŝ 1 ) .
λ Hot = [ f ̂ ¯ 2 f ̂ ¯ 1 ] t Ŝ 2 1 f ̂ ,
[ SNR Hot ] 2 = tr [ Ŝ 2 1 Ŝ 1 ] .
[ SNR MF ] 2 [ SNR ROI ] 2 = { tr [ Ŝ 1 ] } 2 tr [ F W ] tr [ B W ] + tr [ K m W ] tr [ Ŝ 1 Ŝ 2 ] .
[ SNR MF ] 2 [ SNR ROI ] 2 = Q Rose Q bias Q conspic Q correl ,
Q Rose = tr [ Ŝ 1 ] tr [ W ] tr [ F W ] ,
Q bias = tr [ B W ] + tr [ K m W ] tr [ K m W ] ,
Q conspic = tr [ K m W ] tr [ Ŝ 2 W ] ,
Q correl = tr [ Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 Ŝ 2 ] tr [ W ] .
tr [ Ŝ 1 ] C 2 f 0 2 A disk ,
Q Rose = C 2 A disk ,
[ SNR PW ] 2 [ SNR GM ] 2 = { tr [ K m 1 Ŝ 1 ] } 2 tr [ F W ] tr [ ( A t K m 1 A ) + W ] tr [ K m 1 Ŝ 2 K m 1 Ŝ 1 ] .
[ SNR PW ] 2 [ SNR GM ] 2 = Q Rose Q conspic Q correl ,
Q conspic = tr [ ( A t K m 1 A ) + W ] tr [ Ŝ 2 W ] ,
Q correl = { tr [ K m 1 Ŝ 1 ] } 2 [ Ŝ 2 W ] tr [ K m 1 Ŝ 2 K m 1 Ŝ 1 ] tr [ Ŝ 1 ] tr [ W ] .
Q bias = tr [ ( A t K m 1 A + K f 1 ) 1 W ] tr [ ( A t K m 1 A ) + W ] ,
Q conspic = tr [ ( A t K m 1 A ) + W ] tr [ Ŝ 2 W ] ,
Q correl = tr [ Ŝ 2 1 Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 ] tr ( W ) .
Q bias = tr [ B W ] + tr [ K m W ] tr [ K m W ] ,
Q conspic = tr [ K m W ] tr [ Ŝ 2 W ] ,
Q correl = tr [ Ŝ 2 1 Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 ] tr [ W ] .
SNR 2 T 2 α T + β T 2 .
1 [ SNR est ] 2
[ tr ( Ŝ 1 ) ] 2 tr [ ( Ŝ 1 Ŝ 2 ) ]
tr [ BW ] + tr [ K m W ] tr [ FW ]
tr [ Ŝ 1 ] tr [ W ] tr [ FW ]
tr [ BW ] + tr [ K m W ] tr [ K m W ]
tr [ K m W ] tr [ Ŝ 2 W ]
tr [ Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 Ŝ 2 ] tr [ W ]
{ tr [ K m 1 Ŝ 1 ] } 2 tr [ K m 1 Ŝ 2 K m 1 Ŝ 1 ]
tr [ ( A t K m 1 A ) + W ] tr [ FW ]
tr [ Ŝ 1 ] tr [ W ] tr [ FW ]
tr [ ( A t K m 1 A ) + W ] tr [ Ŝ 2 W ]
{ tr [ K m 1 Ŝ 1 ] } 2 tr [ Ŝ 2 W ] tr [ K m 1 Ŝ 2 K m 1 Ŝ 1 ] tr [ Ŝ 1 ] tr [ W ]
tr [ Ŝ 2 1 Ŝ 1 ]
tr [ BW ] + tr [ K m W ] tr [ FW ]
tr [ Ŝ 1 ] tr [ W ] tr [ FW ]
tr [ BW ] + tr [ K m W ] tr [ K m W ]
tr [ K m W ] tr [ Ŝ 2 W ]
tr [ Ŝ 2 1 Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 ] tr [ W ]
tr [ Ŝ 2 1 Ŝ 1 ]
tr [ ( A t K m 1 A + K f 1 ) 1 W ] tr [ F W ]
tr [ Ŝ 1 ] tr [ W ] tr [ FW ]
tr [ ( A t K m 1 A + K f 1 ) 1 W ] tr [ ( A t K m 1 A ) + W ]
tr [ ( A t K m 1 A ) + W ] tr [ Ŝ 2 W ]
tr [ Ŝ 2 1 Ŝ 1 ] tr [ Ŝ 2 W ] tr [ Ŝ 1 ] tr [ W ]
θ ̂ ROI = w t f ̂ = w t [ f + b + m ] .
var ( θ ̂ ROI ) ( θ ̂ ROI θ ̂ ¯ ROI ) 2 n , f = [ w t ( f ̂ f ̂ ¯ ) ] 2 n , f = w t ( f ̂ f ̂ ¯ ) ( f ̂ f ̂ ¯ ) t n , f w = w t K ̂ w = tr [ K ̂ w ] ,
θ ̂ ROI θ = w t b + w t m ,
EMSE [ θ ̂ ROI ] = ( θ ̂ ROI θ ) 2 n , f = w t b b t n , f w + w t m m t n , f w = w t B w + w t K m w = tr ( B W ) + tr ( K m W ) ,
θ ̂ GM θ = v t f ̂ w t f ,
v t = w t [ K m 1 / 2 A ] + K m 1 / 2 .
θ ̂ GM n | f = w t [ K m 1 / 2 A ] + K m 1 / 2 A f = w t A + A [ K m 1 / 2 A ] + K m 1 / 2 A f ,
θ ̂ GM n | f = w t A + K m 1 / 2 K m 1 / 2 A [ K m 1 / 2 A ] + K m 1 / 2 A f .
D D + D = D .
θ ̂ GM n | f = w t A + A f = w t f = θ ,
EMSE [ θ ̂ GM ] = [ θ ̂ GM ] n , f θ 2 n , f = v t f ̂ f ̂ t n , f v w t f f t f w = v t [ A F A t ] v + v t K m v w t F w ,
EMSE [ θ ̂ GM ] = v t K m v = tr [ V K m ] = tr [ ( A t K m 1 A ) + W ] ,

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