Abstract

We present a new method for computing optimized channels for channelized quadratic observers (CQO) that is feasible for high-dimensional image data. The method for calculating channels is applicable in general and optimal for Gaussian distributed image data. Gradient-based algorithms for determining the channels are presented for five different information-based figures of merit (FOMs). Analytic solutions for the optimum channels for each of the five FOMs are derived for the case of equal mean data for both classes. The optimum channels for three of the FOMs under the equal mean condition are shown to be the same. This result is critical since some of the FOMs are much easier to compute. Implementing the CQO requires a set of channels and the first- and second-order statistics of channelized image data from both classes. The dimensionality reduction from M measurements to L channels is a critical advantage of CQO since estimating image statistics from channelized data requires smaller sample sizes and inverting a smaller covariance matrix is easier. In a simulation study we compare the performance of ideal and Hotelling observers to CQO. The optimal CQO channels are calculated using both eigenanalysis and a new gradient-based algorithm for maximizing Jeffrey’s divergence (J). Optimal channel selection without eigenanalysis makes the J-CQO on large-dimensional image data feasible.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Optimal channels for channelized quadratic estimators

Meredith K. Kupinski and Eric Clarkson
J. Opt. Soc. Am. A 33(6) 1214-1225 (2016)

Channelized Hotelling observers for the assessment of volumetric imaging data sets

Ljiljana Platiša, Bart Goossens, Ewout Vansteenkiste, Subok Park, Brandon D. Gallas, Aldo Badano, and Wilfried Philips
J. Opt. Soc. Am. A 28(6) 1145-1163 (2011)

Channelized-ideal observer using Laguerre-Gauss channels in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal

Subok Park, Harrison H. Barrett, Eric Clarkson, Matthew A. Kupinski, and Kyle J. Myers
J. Opt. Soc. Am. A 24(12) B136-B150 (2007)

References

  • View by:
  • |
  • |
  • |

  1. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379–2394 (1987).
    [Crossref]
  2. H. Barrett and K. Myers, Foundations of Image Science (Wiley, 2013).
  3. C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).
  4. M. Kupinski, E. Clarkson, and J. Hesterman, “Bias in Hotelling observer performance computed from finite data,” Proc. SPIE 6515, 65150S (2007).
  5. B. D. Gallas, “Variance of the channelized-hotelling observer from a finite number of trainers and testers,” Proc. SPIE 5034, 100–111 (2003).
  6. N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
    [Crossref]
  7. C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
    [Crossref]
  8. N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
    [Crossref]
  9. S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Statistics 22, 79–86 (1951).
    [Crossref]
  10. H. Jeffreys, “An invariant form for the prior probability in estimation problems,” Proc. R. Soc. London A 186, 453–461 (1946).
    [Crossref]
  11. A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by their probability distributions,” Bulletin Cal. Math. Soc. 35, 99–109 (1943).
  12. H. Barrett, C. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
    [Crossref]
  13. W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
    [Crossref]
  14. C. E. Metz, “Receiver operating characteristic analysis: a tool for the quantitative evaluation of observer performance and imaging systems,” J. Am. Coll. Radiol. 3, 413–422 (2006).
    [Crossref]
  15. E. Clarkson, “Asymptotic ideal observers and surrogate figures of merit for signal detection with list-mode data,” J. Opt. Soc. Am. A 29, 2204–2216 (2012).
    [Crossref]
  16. K. J. Myers and H. H. Barrett, “Addition of a channel mechanism to the ideal-observer model,” J. Opt. Soc. Am. A 4, 2447–2457 (1987).
  17. S. Park, M. A. Kupinski, E. Clarkson, and H. Barrett, “Efficient channels for the ideal observer,” Proc. SPIE 5372, 12–21 (2004).
  18. S. Park, H. Barrett, E. Clarkson, M. Kupinski, and K. Myers, “Channelized-ideal observer using Laguerre-Gauss channels in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal,” J. Opt. Soc. Am. A 24, B136–B150 (2007).
  19. J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
    [Crossref]
  20. S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
    [Crossref]
  21. P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).
  22. H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
    [Crossref]
  23. J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).
  24. D. J. Field, “What is the goal of sensory coding?” Neural Computation 6, 559–601 (1994).
    [Crossref]
  25. B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
    [Crossref]
  26. K. Fukunaga and W. L. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering,” IEEE Trans, Comput. C-19, 311–318 (1970).
    [Crossref]
  27. X. Huo, “A statistical analysis of Fukunaga-Koontz transform,” IEEE Signal Process. Lett. 11, 123–126 (2004).
    [Crossref]
  28. A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
    [Crossref]
  29. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, 1990).
  30. S. Zhang and T. Sim, “Discriminant subspace analysis: A Fukunaga-Koontz approach,” IEEE Trans Pattern Anal. Mach, Intell. 29, 1732–1745 (2007).
    [Crossref]
  31. E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A 27, 2313–2326 (2010).
    [Crossref]
  32. F. De la Torre and T. Kanade, “Multimodal oriented discriminant analysis,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 177–184.
  33. T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
    [Crossref]
  34. W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
    [Crossref]
  35. E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
    [Crossref]
  36. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [Crossref]
  37. S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
    [Crossref]
  38. M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task-specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, B25–B41 (2007).
    [Crossref]
  39. A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 47, 4457–4471 (2008).
    [Crossref]
  40. M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Detection and estimation with compressive measurements,” (Department of ECE, Rice University, 2006).
  41. A. DasGupta, Asymptotic Theory of Statistics and Probability (Springer, 2008).
  42. E. Clarkson and H. Barrett, “Approximations to ideal-observer performance on signal-detection tasks,” Appl. Opt. 39, 1783–1793 (2000).
    [Crossref]
  43. D. Husemoller, Fibre Bundles (Springer, 1994).
  44. A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
    [Crossref]
  45. S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
    [Crossref]

2013 (2)

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
[Crossref]

2012 (3)

T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
[Crossref]

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

E. Clarkson, “Asymptotic ideal observers and surrogate figures of merit for signal detection with list-mode data,” J. Opt. Soc. Am. A 29, 2204–2216 (2012).
[Crossref]

2010 (2)

E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A 27, 2313–2326 (2010).
[Crossref]

J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
[Crossref]

2009 (3)

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
[Crossref]

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

2008 (2)

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[Crossref]

A. Ashok, P. K. Baheti, and M. A. Neifeld, “Compressive imaging system design using task-specific information,” Appl. Opt. 47, 4457–4471 (2008).
[Crossref]

2007 (4)

M. A. Neifeld, A. Ashok, and P. K. Baheti, “Task-specific information for imaging system analysis,” J. Opt. Soc. Am. A 24, B25–B41 (2007).
[Crossref]

S. Park, H. Barrett, E. Clarkson, M. Kupinski, and K. Myers, “Channelized-ideal observer using Laguerre-Gauss channels in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal,” J. Opt. Soc. Am. A 24, B136–B150 (2007).

S. Zhang and T. Sim, “Discriminant subspace analysis: A Fukunaga-Koontz approach,” IEEE Trans Pattern Anal. Mach, Intell. 29, 1732–1745 (2007).
[Crossref]

M. Kupinski, E. Clarkson, and J. Hesterman, “Bias in Hotelling observer performance computed from finite data,” Proc. SPIE 6515, 65150S (2007).

2006 (4)

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

C. E. Metz, “Receiver operating characteristic analysis: a tool for the quantitative evaluation of observer performance and imaging systems,” J. Am. Coll. Radiol. 3, 413–422 (2006).
[Crossref]

2004 (3)

S. Park, M. A. Kupinski, E. Clarkson, and H. Barrett, “Efficient channels for the ideal observer,” Proc. SPIE 5372, 12–21 (2004).

X. Huo, “A statistical analysis of Fukunaga-Koontz transform,” IEEE Signal Process. Lett. 11, 123–126 (2004).
[Crossref]

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

2003 (1)

B. D. Gallas, “Variance of the channelized-hotelling observer from a finite number of trainers and testers,” Proc. SPIE 5034, 100–111 (2003).

2000 (1)

1998 (3)

B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
[Crossref]

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

H. Barrett, C. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[Crossref]

1997 (2)

C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

1994 (1)

D. J. Field, “What is the goal of sensory coding?” Neural Computation 6, 559–601 (1994).
[Crossref]

1993 (1)

H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref]

1987 (2)

1970 (1)

K. Fukunaga and W. L. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering,” IEEE Trans, Comput. C-19, 311–318 (1970).
[Crossref]

1954 (1)

W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
[Crossref]

1951 (1)

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Statistics 22, 79–86 (1951).
[Crossref]

1946 (1)

H. Jeffreys, “An invariant form for the prior probability in estimation problems,” Proc. R. Soc. London A 186, 453–461 (1946).
[Crossref]

1943 (1)

A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by their probability distributions,” Bulletin Cal. Math. Soc. 35, 99–109 (1943).

Abbey, C.

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
[Crossref]

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

H. Barrett, C. Abbey, and E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[Crossref]

Abbey, C. K.

C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).

Arias, T. A.

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

Ashok, A.

Baheti, P. K.

Baraniuk, R. G.

M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Detection and estimation with compressive measurements,” (Department of ECE, Rice University, 2006).

Barrett, H.

Barrett, H. H.

C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).

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

Belongie, S.

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

Berry, M. J.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Bhattacharyya, A.

A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by their probability distributions,” Bulletin Cal. Math. Soc. 35, 99–109 (1943).

Bialek, W.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Birdsall, T.

W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
[Crossref]

Brankov, J.

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

Calderbank, R.

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Carin, L.

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[Crossref]

Carson, W. R.

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

Chen, H.

T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
[Crossref]

Chen, M.

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

Clarkson, E.

DasGupta, A.

A. DasGupta, Asymptotic Theory of Statistics and Probability (Springer, 2008).

Davenport, M. A.

M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Detection and estimation with compressive measurements,” (Department of ECE, Rice University, 2006).

De la Torre, F.

F. De la Torre and T. Kanade, “Multimodal oriented discriminant analysis,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 177–184.

Dollár, P.

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Eckstein, M. P.

C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).

Edelman, A.

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

El-Naqa, I.

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

Field, D. J.

Fox, W.

W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
[Crossref]

Fukunaga, K.

K. Fukunaga and W. L. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering,” IEEE Trans, Comput. C-19, 311–318 (1970).
[Crossref]

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, 1990).

Gallas, B. D.

B. D. Gallas, “Variance of the channelized-hotelling observer from a finite number of trainers and testers,” Proc. SPIE 5034, 100–111 (2003).

Hesterman, J.

M. Kupinski, E. Clarkson, and J. Hesterman, “Bias in Hotelling observer performance computed from finite data,” Proc. SPIE 6515, 65150S (2007).

Huo, X.

X. Huo, “A statistical analysis of Fukunaga-Koontz transform,” IEEE Signal Process. Lett. 11, 123–126 (2004).
[Crossref]

Husemoller, D.

D. Husemoller, Fibre Bundles (Springer, 1994).

Insana, M.

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
[Crossref]

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

Jeffreys, H.

H. Jeffreys, “An invariant form for the prior probability in estimation problems,” Proc. R. Soc. London A 186, 453–461 (1946).
[Crossref]

Ji, S.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[Crossref]

Kanade, T.

F. De la Torre and T. Kanade, “Multimodal oriented discriminant analysis,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 177–184.

Koontz, W. L.

K. Fukunaga and W. L. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering,” IEEE Trans, Comput. C-19, 311–318 (1970).
[Crossref]

Kullback, S.

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Statistics 22, 79–86 (1951).
[Crossref]

Kupinski, M.

Kupinski, M. A.

S. Park, M. A. Kupinski, E. Clarkson, and H. Barrett, “Efficient channels for the ideal observer,” Proc. SPIE 5372, 12–21 (2004).

Leibler, R.

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Statistics 22, 79–86 (1951).
[Crossref]

Lindfors, K.

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

Liu, J.

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

Mahalanobis, A.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

Meister, M.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Metz, C. E.

C. E. Metz, “Receiver operating characteristic analysis: a tool for the quantitative evaluation of observer performance and imaging systems,” J. Am. Coll. Radiol. 3, 413–422 (2006).
[Crossref]

Muise, R. R.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

Müller, K.-R.

B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
[Crossref]

Myers, K.

J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
[Crossref]

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
[Crossref]

S. Park, H. Barrett, E. Clarkson, M. Kupinski, and K. Myers, “Channelized-ideal observer using Laguerre-Gauss channels in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal,” J. Opt. Soc. Am. A 24, B136–B150 (2007).

H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref]

H. Barrett and K. Myers, Foundations of Image Science (Wiley, 2013).

Myers, K. J.

Neifeld, M. A.

Nguyen, N.

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
[Crossref]

Park, S.

J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
[Crossref]

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
[Crossref]

S. Park, H. Barrett, E. Clarkson, M. Kupinski, and K. Myers, “Channelized-ideal observer using Laguerre-Gauss channels in detection tasks involving non-Gaussian distributed lumpy backgrounds and a Gaussian signal,” J. Opt. Soc. Am. A 24, B136–B150 (2007).

S. Park, M. A. Kupinski, E. Clarkson, and H. Barrett, “Efficient channels for the ideal observer,” Proc. SPIE 5372, 12–21 (2004).

Perona, P.

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

Peterson, W.

W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
[Crossref]

Rodrigues, M. R.

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

Rolland, J.

H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Schölkopf, B.

B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
[Crossref]

Shen, F.

Sim, T.

S. Zhang and T. Sim, “Discriminant subspace analysis: A Fukunaga-Koontz approach,” IEEE Trans Pattern Anal. Mach, Intell. 29, 1732–1745 (2007).
[Crossref]

Smirnakis, S. M.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Smith, S. T.

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

Smola, A.

B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
[Crossref]

Stanfill, S. R.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Tu, Z.

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

Van Nevel, A.

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

Varshney, P. K.

T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
[Crossref]

Wakin, M. B.

M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Detection and estimation with compressive measurements,” (Department of ECE, Rice University, 2006).

Warland, D. K.

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Wei, L.

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

Wernick, M.

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

Wimalajeewa, T.

T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
[Crossref]

Witten, J.

J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
[Crossref]

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
[Crossref]

Xue, Y.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[Crossref]

Yang, Y.

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

Yao, J.

H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref]

Zemp, R.

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

Zhang, S.

S. Zhang and T. Sim, “Discriminant subspace analysis: A Fukunaga-Koontz approach,” IEEE Trans Pattern Anal. Mach, Intell. 29, 1732–1745 (2007).
[Crossref]

Ann. Math. Statistics (1)

S. Kullback and R. Leibler, “On information and sufficiency,” Ann. Math. Statistics 22, 79–86 (1951).
[Crossref]

Appl. Opt. (2)

BMVC (1)

P. Dollár, Z. Tu, P. Perona, and S. Belongie, “Integral channel features,” BMVC 2, 5 (2009).

Bulletin Cal. Math. Soc. (1)

A. Bhattacharyya, “On a measure of divergence between two statistical populations defined by their probability distributions,” Bulletin Cal. Math. Soc. 35, 99–109 (1943).

IEEE Signal Process. Lett. (1)

X. Huo, “A statistical analysis of Fukunaga-Koontz transform,” IEEE Signal Process. Lett. 11, 123–126 (2004).
[Crossref]

IEEE Trans Pattern Anal. Mach, Intell. (1)

S. Zhang and T. Sim, “Discriminant subspace analysis: A Fukunaga-Koontz approach,” IEEE Trans Pattern Anal. Mach, Intell. 29, 1732–1745 (2007).
[Crossref]

IEEE Trans, Comput. (1)

K. Fukunaga and W. L. Koontz, “Application of the Karhunen-Loeve expansion to feature selection and ordering,” IEEE Trans, Comput. C-19, 311–318 (1970).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

A. Mahalanobis, R. R. Muise, S. R. Stanfill, and A. Van Nevel, “Design and application of quadratic correlation filters for target detection,” IEEE Trans. Aerosp. Electron. Syst. 40, 837–850 (2004).
[Crossref]

IEEE Trans. Inf. Theory (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

IEEE Trans. Med. Imaging (6)

J. Brankov, Y. Yang, L. Wei, I. El-Naqa, and M. Wernick, “Learning a channelized observer for image quality assessment,” IEEE Trans. Med. Imaging 28, 991–999 (2009).

J. Witten, S. Park, and K. Myers, “Partial least squares: a method to estimate efficient channels for the ideal observers,” IEEE Trans. Med. Imaging 29, 1050–1058 (2010).
[Crossref]

S. Park, J. Witten, and K. Myers, “Singular vectors of a linear imaging system as efficient channels for the Bayesian ideal observer,” IEEE Trans. Med. Imaging 28, 657–668 (2009).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality I: Task information,” IEEE Trans. Med. Imaging 32, 683–690 (2013).
[Crossref]

C. Abbey, R. Zemp, J. Liu, K. Lindfors, and M. Insana, “Observer efficiency in discrimination tasks simulating malignant and benign breast lesions imaged with ultrasound,” IEEE Trans. Med. Imaging 25, 198–209 (2006).
[Crossref]

N. Nguyen, C. Abbey, and M. Insana, “Objective assessment of sonographic quality II: Acquisition information spectrum,” IEEE Trans. Med. Imaging 32, 691–698 (2013).
[Crossref]

IEEE Trans. Signal Process. (2)

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[Crossref]

T. Wimalajeewa, H. Chen, and P. K. Varshney, “Performance limits of compressive sensing-based signal classification,” IEEE Trans. Signal Process. 60, 2758–2770 (2012).
[Crossref]

J. Am. Coll. Radiol. (1)

C. E. Metz, “Receiver operating characteristic analysis: a tool for the quantitative evaluation of observer performance and imaging systems,” J. Am. Coll. Radiol. 3, 413–422 (2006).
[Crossref]

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

Nature (1)

S. M. Smirnakis, M. J. Berry, D. K. Warland, W. Bialek, and M. Meister, “Adaptation of retinal processing to image contrast and spatial scale,” Nature 386, 69–73 (1997).
[Crossref]

Neural Computation (2)

D. J. Field, “What is the goal of sensory coding?” Neural Computation 6, 559–601 (1994).
[Crossref]

B. Schölkopf, A. Smola, and K.-R. Müller, “Nonlinear component analysis as a kernel eigenvalue problem,” Neural Computation 10, 1299–1319 (1998).
[Crossref]

Proc. Natl. Acad. Sci. USA (1)

H. Barrett, J. Yao, J. Rolland, and K. Myers, “Model observers for assessment of image quality,” Proc. Natl. Acad. Sci. USA 90, 9758–9765 (1993).
[Crossref]

Proc. R. Soc. London A (1)

H. Jeffreys, “An invariant form for the prior probability in estimation problems,” Proc. R. Soc. London A 186, 453–461 (1946).
[Crossref]

Proc. SPIE (4)

C. K. Abbey, H. H. Barrett, and M. P. Eckstein, “Practical issues and methodology in assessment of image quality using model observers,” Proc. SPIE 3032, 182–194 (1997).

M. Kupinski, E. Clarkson, and J. Hesterman, “Bias in Hotelling observer performance computed from finite data,” Proc. SPIE 6515, 65150S (2007).

B. D. Gallas, “Variance of the channelized-hotelling observer from a finite number of trainers and testers,” Proc. SPIE 5034, 100–111 (2003).

S. Park, M. A. Kupinski, E. Clarkson, and H. Barrett, “Efficient channels for the ideal observer,” Proc. SPIE 5372, 12–21 (2004).

SIAM J. Imag. Sci. (1)

W. R. Carson, M. Chen, M. R. Rodrigues, R. Calderbank, and L. Carin, “Communications-inspired projection design with application to compressive sensing,” SIAM J. Imag. Sci. 5, 1185–1212 (2012).
[Crossref]

SIAM J. Matrix Anal. Appl. (1)

A. Edelman, T. A. Arias, and S. T. Smith, “The geometry of algorithms with orthogonality constraints,” SIAM J. Matrix Anal. Appl. 20, 303–353 (1998).
[Crossref]

Trans. IRE Prof. Group Inf. Theory (1)

W. Peterson, T. Birdsall, and W. Fox, “The theory of signal detectability,” Trans. IRE Prof. Group Inf. Theory 4, 171–212 (1954).
[Crossref]

Other (6)

F. De la Torre and T. Kanade, “Multimodal oriented discriminant analysis,” in Proceedings of the 22nd International Conference on Machine Learning (ACM, 2005), pp. 177–184.

M. A. Davenport, M. B. Wakin, and R. G. Baraniuk, “Detection and estimation with compressive measurements,” (Department of ECE, Rice University, 2006).

A. DasGupta, Asymptotic Theory of Statistics and Probability (Springer, 2008).

D. Husemoller, Fibre Bundles (Springer, 1994).

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, 1990).

H. Barrett and K. Myers, Foundations of Image Science (Wiley, 2013).

Supplementary Material (2)

» Media 1: MP4 (3090 KB)     
» Media 2: MP4 (3076 KB)     

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1.
Fig. 1. Top image is a sample from the first class and the bottom image is a sample from the second class. The images look more similar in (a) since Δ σ is smaller compared to (b). Movies for (a) and (b) are generated by repeatedly sampling from the ensemble (Media 1 and Media 2). (a)  Δ σ = 0.2 pixels. (b)  Δ σ = 0:5 pixels.
Fig. 2.
Fig. 2. AUC comparison for (a) equal and (b) unequal mean images for the two classes. The AUC is calculated at various values of Δ σ , the correlation length difference between the covariance matrices of the two classes. The number of channels is constant at L = 25 and the compression ratio M / L = 100 . (a) Signal absent in the mean, (b) signal present in the mean.
Fig. 3.
Fig. 3. Eigenspectrum of K 2 1 K 1 for six values of Δ σ . The highest 25 values of κ + 1 / κ are marked in red (in the top left part of each curve) to denote the Eigen-CQO selections. The number of channels is constant at L = 25 . Δ σ = 0.1 pixels (a), 0.15 pixels (b), 0.2 pixels (c), 0.25 pixels (d), 0.30 pixels (e), 0.35 pixels (f).
Fig. 4.
Fig. 4. Observer comparison for unequal mean images and varying the magnitude of the mean difference. The number of channels and second-order statistics are held constant; L = 25 and Δ σ = 0.15 pixels.
Fig. 5.
Fig. 5. Observer comparison for (a) equal and (b) unequal mean images under both classes and varying the number of channels. The second-order statistics are constant; Δ σ = 0.15 pixels. (a) Signal absent in the mean, (b) signal present in the mean.
Fig. 6.
Fig. 6. J versus iteration number in Eq. (24) for (a) equal and (b) unequal means. Gradient term in Eq. (24) versus iteration number for (c) equal and (d) unequal means. The second-order statistics are constant; Δ σ = 0.25 pixels. Discrepancies for different runs (using independent random samples for the initial solution) of the iterative gradient-based search algorithm indicate the presence of local maxima. (a), (c) Signal absent in the mean. (b), (d) Signal present in the mean.
Fig. 7.
Fig. 7. Eigenspectrum of R 12 [see Eq. (19)] for (a) equal and (b) unequal means and Δ σ = 0.25 pixels and L = 25 . Discrepancies for different runs (using independent random samples as the initial solution) of the iterative gradient-based search algorithm indicate the presence of local maxima. When a signal in the mean is introduced an eigenvalue close to one appears in the eigenspectrum. The corresponding channel is the matrix-vector product of T and the corresponding eigenvector. This channel is shown in (c) to demonstrate that the signal in the mean affects the J-CQO channels. (a) Signal absent in the mean and (b) signal present in the mean. (c)  T .
Fig. 8.
Fig. 8. Top row displays the channels (i.e., rows of the matrix T ) for the J-CQO iterative algorithm for three values of the correlation length difference between the two classes with equal means. L = 25 and each row of the channel matrix is displayed as a single image in a 5 × 5 tiling. The bottom row shows the eigenvectors of R 12 [see Eq. (19)] back projected by T . (a) J-CQO T ( Δ σ = 0.2 pixels), (b) J-CQO T ( Δ σ = 0.5 pixels), (c) J-CQO T ( Δ σ = 1 pixel), (d) backprojection of R 12 eigenvectors by T ( Δ σ = 0.2 pixels), (e) backprojection of R 12 eigenvectors by T ( Δ σ = 0.5 pixels), (f) backprojection of R 12 eigenvector by T ( Δ σ = 1 pixel).

Equations (111)

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

g = H f + n .
v = Tg ,
f ( r ) = n = 0 N f n ψ n ( r ) .
g = Hf + n .
λ I ( g ) = ln [ p r 1 ( g ) ] ln [ p r 2 ( g ) ] ,
p r i ( v ) = [ ( 2 π ) L det ( TK i T ) ] 1 2 × exp [ 1 2 ( v T g ¯ i ) ( TK i T ) 1 ( v T g ¯ i ) ] ,
λ ( T , g ) = ln [ p r 1 ( Tg ) ] ln [ p r 2 ( Tg ) ] ,
Q i ( v ) = ( v T g ¯ i ) ( TK i T ) 1 ( v T g ¯ i ) .
λ CQO ( T , g ) = 1 2 [ Q 2 ( Tg ) Q 1 ( Tg ) ln det ( TK 1 T ) + ln det ( TK 2 T ) ] ,
f ( v ) i = R L f ( v ) p r i ( v ) d L v .
F 1 ( T ) = λ ( v ) 1 = D K L ( p r 1 p r 2 ) ,
F 2 ( T ) = λ ( v ) 2 = D K L ( p r 2 p r 1 ) ;
F 3 ( T ) = λ ( v ) 1 λ ( v ) 2 .
F 4 ( T ) = 4 ln exp [ 1 2 λ ( v ) ] 1 , = 4 ln exp [ 1 2 λ ( v ) ] 2 , = 4 D B ( p r 1 , p r 2 ) .
F 5 ( T ) = 1 1 4 π | exp [ ( 1 2 + i α ) λ ( v ) ] 2 | 2 d α α 2 + 1 4 .
F ( MT ) = F ( T )
λ H ( g ) = ( K 1 s ) g ,
C i ( T ) = TK i T .
R 12 ( T ) = C 2 1 ( T ) C 1 ( T ) ,
H 2 ( T ) = s T C 2 1 ( T ) Ts .
2 F 1 ( T ) = L + tr [ R 12 ( T ) ] + H 2 ( T ) ln det [ R 12 ( T ) ] .
d d t F 1 ( T + t E ) | t = 0 = tr [ E F 1 ( T ) ] ,
Δ 1 ( T ) = C 2 1 ( T ) T ( K 1 + ss ) [ I T C 2 1 ( T ) TK 2 ] C 2 1 ( T ) TK 2 + C 1 1 ( T ) TK 1 .
T ( n + 1 ) = T ( n ) + ϵ Δ 1 ( T ( n ) ) ,
λ CQO ( T ( n ˜ ) , g ) .
T ( n + 1 ) = T ( n ) [ I + ϵ ( T ( n ) ) Δ 1 ( T ( n ) ) ] .
λ CQO ( T eig , g ) .
[ R 12 ( T ) + I ] ( TK 2 T ) 1 TK 2 = [ R 12 ( T ) + I ] ( TK 1 T ) 1 TK 1 .
C 2 1 2 ( T ) R 12 ( T ) C 2 1 2 ( T ) = C 2 1 2 ( T ) C 1 ( T ) C 2 1 2 ( T ) ,
K 2 1 K 1 T = T R 12 ( T ) .
K 2 1 K 1 T M = T M [ ( M ) 1 R 12 ( T ) M ] ,
K 2 1 K 1 T M = T M Λ ( T ) ,
F ( T ) = F ( MT ) = l ( κ l ln κ l ) ,
2 F 2 ( T ) = L + tr [ R 21 ( T ) ] + H 1 ( T ) ln det [ R 21 ( T ) ] .
Δ 2 ( T ) = C 1 1 ( T ) T ( K 2 + ss ) [ I T C 1 1 ( T ) TK 1 ] C 1 1 ( T ) TK 1 + C 2 1 ( T ) TK 2 .
2 F 3 ( T ) = 2 L + tr [ R 12 ( T ) ] + H 2 ( T ) + tr [ R 21 ( T ) ] + H 1 ( T ) .
Δ 3 ( T ) = C 1 1 ( T ) T ( K 2 + ss ) [ I T C 1 1 ( T ) TK 1 ] + C 2 1 ( T ) T ( K 1 + ss ) [ I T C 2 1 ( T ) TK 2 ] .
[ R 12 ( T ) R 21 ( T ) ] ( TK 1 T ) 1 TK 1 = [ R 12 ( T ) R 21 ( T ) ] ( TK 2 T ) 1 TK 2 .
K 2 1 K 1 T = T R 12 ( T ) .
F 3 ( T ) = 1 2 l ( κ l + 1 κ l ) L .
t l K 2 t l = δ l l , t l K 1 t l = κ l δ l l .
I ( T ) = R L [ p r 1 ( v ) p r 2 ( v ) ] 1 2 d L v .
4 H ( T ) = 2 c ( T ) C 1 ( T ) c ( T ) g ¯ 1 T C 1 1 ( T ) T g ¯ 1 g ¯ 2 T C 2 1 ( T ) T g ¯ 2 .
[ v c ( T ) ] C 1 ( T ) [ v c ( T ) ] H ( T ) = 1 2 ( v T g ¯ 1 ) C 1 1 ( T ) ( v T g ¯ 1 ) + 1 2 ( v T g ¯ 2 ) C 2 1 ( T ) ( v T g ¯ 2 ) .
F 4 ( T ) = 2 ln det C ( T ) + ln det C 1 ( T ) + ln det C 2 ( T ) + H ( T ) .
F 5 ( T ) = 1 2 + 1 2 erf [ 1 2 d A ( T ) ] ,
F 5 ( T ) = 1 1 4 π | exp [ ( 1 2 + i α ) λ ( v ) ] 2 | 2 d α α 2 + 1 4 .
Λ 1 2 + i α ( v ) 2 = N ( α , T ) I ( α , T ) .
N ( α , T ) = [ 1 ( 2 π ) L det C 1 ( T ) ] 1 2 + i α × [ 1 ( 2 π ) L det C 2 ( T ) ] 1 2 i α .
| N ( α , T ) | 2 = 1 ( 2 π ) L [ 1 det C 1 ( T ) ] [ 1 det C 1 ( T ) ]
R L exp { 1 4 v [ C 1 1 ( T ) + C 2 1 ( T ) ] v i α 2 v [ C 1 1 ( T ) C 2 1 ( T ) ] v } d v .
| N ( α , MT ) | 2 = 1 ( 2 π ) L l = 1 L μ l .
| N ( α , MT ) | 2 = 1 ( 2 π ) L det R 12 ( T ) .
I ( α , MT ) = R L exp [ 1 4 l = 1 L ( 1 μ l + 1 ) u l 2 i α 2 l = 1 L ( 1 μ l 1 ) u l 2 ] d L u .
I ( α , MT ) = π L 2 l = 1 L [ 1 4 ( 1 μ l + 1 ) + i α 2 ( 1 μ l 1 ) ] 1 2 .
I ( α , MT ) = ( 2 π ) L 2 det [ ( 1 2 + i α ) R 21 ( T ) + ( 1 2 i α ) I ] ,
| Λ 1 2 + i α ( v ) 2 | 2 = [ P ( α , T ) ] 1 2
[ ( 1 2 + i α ) R 12 ( T ) + ( 1 2 i α ) I ] × [ ( 1 2 + i α ) R 21 ( T ) + ( 1 2 i α ) I ] .
P ( α , T ) = ( 1 4 + α 2 ) [ R 12 ( T ) + R 21 ( T ) ] + 2 ( 1 4 α 2 ) I .
F 5 ( T ) = 1 1 4 π [ P ( α , T ) ] 1 2 d α α 2 + 1 4 ,
F 5 ( T ) = 1 1 4 π l = 1 L [ ( 1 4 + α 2 ) ( μ l + 1 μ l ) + 2 ( 1 4 α 2 ) ] 1 2 d α α 2 + 1 4 .
K 2 1 K 1 T = T R 12 ( T ) .
l = 1 L [ ( 1 4 + α 2 ) ( κ l + 1 κ l ) + 2 ( 1 4 α 2 ) ] 1 2 d α α 2 + 1 4 .
l = 1 L [ 1 4 ( κ l + 1 ) 2 κ l + α 2 ( κ l 1 ) 2 κ l ] 1 2 d α α 2 + 1 4 .
y l = 4 κ l ( κ l + 1 ) 2 ,
l = 1 L y l 1 + 4 α 2 ( 1 y l ) d α α 2 + 1 4 .
f ( y ) = y 1 + 4 α 2 ( 1 y )
1 y l = 1 4 ( κ l + 1 κ l + 2 ) .
d 2 d t 2 F 3 ( T + t E ) | t = 0 = tr [ E D ( T ) E ]
C j ( 0 ) = TK j T ,
C j ( 0 ) = EK j T + TK j E ,
C j ( 0 ) = 2 EK j E ,
R i j ( 0 ) = C j 1 ( 0 ) [ C i ( 0 ) C j ( 0 ) R i j ( 0 ) ] ,
R i j ( 0 ) = C j 1 ( 0 ) [ C i ( 0 ) C j ( 0 ) R i j ( 0 ) 2 C j ( 0 ) R i j ( 0 ) ] ,
F 3 ( 0 ) = tr [ R 12 ( 0 ) ] + tr [ R 21 ( 0 ) ] .
C 1 ( 0 ) = B Λ + Λ B
C 2 ( 0 ) = B + B
C 1 ( 0 ) = 2 B Λ B + 2 B 0 Λ 0 B 0
C 2 ( 0 ) = 2 BB + 2 B 0 B 0 .
tr [ E D ( T ) E ] = 2 tr [ ( Λ 1 Λ ) ( B 0 B 0 Λ 1 B 0 Λ 0 B 0 ) ] .
tr [ E D ( T ) E ] = 2 tr [ ( B 0 T 0 ) ( Λ 1 Λ ) ( B 0 Λ 1 B 0 Λ 0 ) T 0 K 2 ] .
D ( T ) B 0 T 0 = ( Λ 1 Λ ) ( B 0 Λ 1 B 0 Λ 0 ) T 0 K 2 .
F 3 ( T ) = tr [ ( T K 2 T ) 1 ( T K 1 T ) ] + tr [ ( T K 1 T ) 1 ( T K 2 T ) ] = F 3 ( T ) ,
D ( T ) B 0 p q T 0 = ( 1 κ p κ p ) ( 1 κ 0 q κ p ) B 0 p q T 0 .
[ K i ] n , m = ( b ¯ SNR ) 2 × exp ( ( ( n x m x ) mod 50 ) 2 + ( ( n y m y ) mod 50 ) 2 2 σ i 2 )
g ¯ 2 = g ¯ 1 + A s s
g i k = g ¯ i + K i 1 / 2 w
Q 1 ( v ) 1 = tr C 1 1 ( T ) ( v T g ¯ 1 ) ( v T g ¯ 1 ) 1 = L .
Q 2 ( v ) 1 = tr { C 2 1 ( T ) [ C 1 ( T ) + Tss T ] } .
2 λ ( v ) 1 = L + tr [ C 2 1 ( T ) C 1 ( T ) ] + s T C 2 1 ( T ) Ts ln det [ C 2 1 ( T ) C 1 ( T ) ] .
2 F 1 ( T ) = L + tr [ R 12 ( T ) ] + H 2 ( T ) ln det [ R 12 ( T ) ] .
D 1 ( T ) = d d t tr [ R 12 ( T + t E ) ] | t = 0 .
2 tr { E [ R 12 ( T ) C 2 1 ( T ) TK 2 + C 2 1 ( T ) TK 1 ] } .
D 2 ( T ) = d d t ln det [ R 12 ( T + t E ) ] | t = 0 .
d d t ln det [ R 12 ( T + t E ) ] = tr [ R 12 1 ( T + t E ) d dt R 12 ( T + t E ) ] .
2 tr { E [ C 2 1 ( T ) TK 2 + C 1 1 ( T ) TK 1 ] } .
D 3 ( T ) = d d t H 2 ( T + t E ) | t = 0 ,
2 tr { E [ C 2 1 ( T ) Tss T C 2 1 ( T ) TK 2 + C 2 1 ( T ) Tss ] } .
C 2 1 ( T ) T ( K 1 + ss ) [ I T C 2 1 ( T ) TK 2 ] C 2 1 ( T ) TK 2 + C 1 1 ( T ) TK 1 .
d d t R 12 ( T + t E ) | t = 0 = C 2 1 ( T ) [ ( EK 2 T + TK 2 E ) R 12 ( T ) + ( EK 1 T + TK 1 E ) ] ,
d d t R 21 ( T + t E ) | t = 0 = C 1 1 ( T ) [ ( EK 1 T + TK 1 E ) R 21 ( T ) + ( EK 2 T + TK 2 E ) ] .
Q 1 ( α , T ) = P 1 ( α , T ) C 1 1 ( T ) = [ C 1 ( T ) P ( α , T ) ] 1
Q 2 ( α , T ) = P 1 ( α , T ) C 2 1 ( T ) = [ C 2 ( T ) P ( α , T ) ] 1 .
S 1 ( α , T ) = R 21 ( T ) Q 1 ( α , T ) = [ C 1 ( T ) P ( α , T ) R 12 ( T ) ] 1
S 2 ( α , T ) = R 12 ( T ) Q 2 ( α , T ) = [ C 2 ( T ) P ( α , T ) R 21 ( T ) ] 1 .
2 ( 1 4 + α 2 ) { [ Q 2 ( α , T ) S 1 ( α , T ) ] TK 1 + [ Q 1 ( α , T ) S 2 ( α , T ) ] TK 2 } .
U 1 ( T ) = [ P ( α , T ) ] 1 2 [ Q 2 ( α , T ) S 1 ( α , T ) ] d α ,
U 2 ( T ) = [ P ( α , T ) ] 1 2 [ Q 1 ( α , T ) S 2 ( α , T ) ] d α .
U 1 ( T ) TK 1 = U 2 ( T ) TK 2 .
K 2 1 K 1 T = T U 2 ( T ) U 1 1 ( T ) .
U 2 ( T ) U 1 1 ( T ) = R 12 ( T ) .

Metrics