Abstract

Maximum-likelihood (ML) estimation in wavefront sensing requires careful attention to all noise sources and all factors that influence the sensor data. We present detailed probability density functions for the output of the image detector in a wavefront sensor, conditional not only on wavefront parameters but also on various nuisance parameters. Practical ways of dealing with nuisance parameters are described, and final expressions for likelihoods and Fisher information matrices are derived. The theory is illustrated by discussing Shack–Hartmann sensors, and computational requirements are discussed. Simulation results show that ML estimation can significantly increase the dynamic range of a Shack–Hartmann sensor with four detectors and that it can reduce the residual wavefront error when compared with traditional methods.

© 2007 Optical Society of America

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2006 (2)

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

2005 (2)

L. Chen and H. H. Barrett, "Non-Gaussian noise in X-ray and gamma-ray detectors," in Proc. SPIE 5745, 366-376 (2005).
[CrossRef]

S. T. Smith, "Covariance, subspace, and intrinsic Cramer-Rao bounds," IEEE Trans. Signal Process. 53, 1610-1630 (2005).
[CrossRef]

2003 (2)

2001 (1)

P. Stoica and T. L. Marzetta, "Parameter estimation problems with singular information matrices," IEEE Trans. Signal Process. 49, 87-89 (2001).
[CrossRef]

2000 (1)

1999 (2)

J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999).
[CrossRef]

R. Irwan and R. G. Lane, "Analysis of optimal centroid estimation applied to Shack-Hartmann sensing," Appl. Opt. 38, 6737-6743 (1999).
[CrossRef]

1998 (2)

L. Parra and H. H. Barrett, "List-mode likelihood: EM algorithm and noise estimation demonstrated on 2D-PET," IEEE Trans. Med. Imaging MI-17, 228-235 (1998).
[CrossRef]

H. H. Barrett, C. K. 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 (3)

1995 (4)

1993 (1)

D. Gagnon, "Maximum likelihood positioning in the scintillation camera using depth of interaction," IEEE Trans. Med. Imaging MI-12, 101-107 (1993).
[CrossRef]

1992 (2)

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," Proc. IEEE 70, 229-245 (1992).
[CrossRef]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations using phase diversity," J. Opt. Soc. Am. A 7, 1072-1085 (1992).
[CrossRef]

1990 (2)

H. H. Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7, 1266-1278 (1990).
[CrossRef] [PubMed]

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

1988 (3)

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, "Fast-phase diversity wave-front sensing for mirror control," in Proc. SPIE 3353, 952-963 (1988).
[CrossRef]

F. Roddier, "Curvature sensing and compensation: a new concept in adaptive optics," Appl. Opt. 27, 1223-1225 (1988).
[CrossRef] [PubMed]

1987 (2)

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

M. Rabbani, R. Shaw, and R. van Metter, "Detective quantum efficiency of imaging systems with amplifying and scattering mechanisms," J. Opt. Soc. Am. A 4, 895-901 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (1)

1983 (1)

1982 (3)

G. A. Tyler and D. L. Fried, "Image-position error associated with a quadrant detector," J. Opt. Soc. Am. 72, 804 (1982).
[CrossRef]

H. White, "Maximum likelihood estimation of misspecified models," Econometrica , 50, 1-126 (1982).
[CrossRef]

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

1977 (1)

1976 (2)

R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-210 (1976).
[CrossRef]

R. M. Gray and A. Macovski, "Maximum a posteriori estimation of position in scintillation cameras," IEEE Trans. Nucl. Sci. NS-23, 849-852 (1976).
[CrossRef]

1974 (3)

B. E. A. Saleh, "Estimations based on instants of occurrence of photon counts of low level light," Proc. IEEE 62, 530-531 (1974).
[CrossRef]

B. E. A. Saleh, "Joint probability of occurrence of photon events and estimation of optical parameters," J. Phys. A 7, 1360-1368 (1974).
[CrossRef]

B. E. A. Saleh, "Estimation of the location of an optical object with photodetectors limited by quantum noise," Appl. Opt. 13, 1824-1827 (1974).
[CrossRef] [PubMed]

1973 (1)

R. K. Swank, "Absorption and noise in X-ray phosphors," J. Appl. Phys. 44, 4199-4203 (1973).
[CrossRef]

1959 (1)

R. E. Burgess, "Homophase and heterophase fluctuations in semiconducting crystals," Discuss. Faraday Soc. 21, 51-158 (1959).

Aarsvold, J. N.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Abbey, C. K.

Barber, H. B.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

Barrett, H. H.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

L. Chen and H. H. Barrett, "Non-Gaussian noise in X-ray and gamma-ray detectors," in Proc. SPIE 5745, 366-376 (2005).
[CrossRef]

L. Parra and H. H. Barrett, "List-mode likelihood: EM algorithm and noise estimation demonstrated on 2D-PET," IEEE Trans. Med. Imaging MI-17, 228-235 (1998).
[CrossRef]

H. H. Barrett, C. K. 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]

H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," in Proc. SPIE 3032, 110-124 (1997).
[CrossRef]

H. H. Barrett, L. Parra, and T. A. White, "List-mode likelihood," J. Opt. Soc. Am. A 14, 2914-2923 (1997).
[CrossRef]

H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, "Objective assessment of image quality: II. Fisher information, Fourier crosstalk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995).
[CrossRef]

H. H. Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7, 1266-1278 (1990).
[CrossRef] [PubMed]

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

R. G. Paxman, H. H. Barrett, W. E. Smith, and T. D. Milster, "Image reconstruction from coded data: II. Code design," J. Opt. Soc. Am. A 2, 501-509 (1985).
[CrossRef] [PubMed]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

H. H. Barrett, "Detectors for small-animal SPECT: II. Statistical limitations and estimation methods," in Small-Animal SPECT Imaging, M.Kupinski and H.Barrett eds. (Springer, 2005), Chap. 3.
[CrossRef]

L. R. Furenlid, J. Y. Hesterman, and H. H. Barrett, "Real time data acquisition and maximum-likelihood estimation for gamma cameras," in Proceedings of the 14th IEEE-NPSS Real-Time Conference (IEEE, 2005), pp. 498-501.

Berger, J. O.

J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999).
[CrossRef]

Bibby, J. M.

K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 1979).

Black, K. A.

Blanc, A.

Burgess, R. E.

R. E. Burgess, "Homophase and heterophase fluctuations in semiconducting crystals," Discuss. Faraday Soc. 21, 51-158 (1959).

Cannon, R. C.

Chen, J.

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Chen, L.

L. Chen and H. H. Barrett, "Non-Gaussian noise in X-ray and gamma-ray detectors," in Proc. SPIE 5745, 366-376 (2005).
[CrossRef]

Clarkson, E.

Clinthorne, N. H.

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

Cohn, D. L.

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

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).

Cunningham, P. R.

Dainty, J. C.

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

Denny, J. L.

Devaney, N.

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

Deville, J. H.

Dolne, J. J.

Duncan, A. L.

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, "Fast-phase diversity wave-front sensing for mirror control," in Proc. SPIE 3353, 952-963 (1988).
[CrossRef]

Elbaum, M.

Ellerbroek, B. L.

Faisal, M.

Fienup, J. R.

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations using phase diversity," J. Opt. Soc. Am. A 7, 1072-1085 (1992).
[CrossRef]

Fried, D. L.

Furenlid, L. R.

L. R. Furenlid, J. Y. Hesterman, and H. H. Barrett, "Real time data acquisition and maximum-likelihood estimation for gamma cameras," in Proceedings of the 14th IEEE-NPSS Real-Time Conference (IEEE, 2005), pp. 498-501.

Gagnon, D.

D. Gagnon, "Maximum likelihood positioning in the scintillation camera using depth of interaction," IEEE Trans. Med. Imaging MI-12, 101-107 (1993).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Gray, R. M.

R. M. Gray and A. Macovski, "Maximum a posteriori estimation of position in scintillation cameras," IEEE Trans. Nucl. Sci. NS-23, 849-852 (1976).
[CrossRef]

Greenebaum, M.

Helstrom, C. W.

Hesterman, J. Y.

L. R. Furenlid, J. Y. Hesterman, and H. H. Barrett, "Real time data acquisition and maximum-likelihood estimation for gamma cameras," in Proceedings of the 14th IEEE-NPSS Real-Time Conference (IEEE, 2005), pp. 498-501.

J. Y. Hesterman (University of Arizona, jyh@email.arizona.edu, personal communication, 2005).

Idell, P. S.

Idier, J.

Irwan, R.

Johnson, N. L.

N. L. Johnson and S. Kotz, Discrete Distributions (Wiley, 1969).

Kcral, K. F.

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

Kent, J. T.

K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 1979).

Kotz, S.

N. L. Johnson and S. Kotz, Discrete Distributions (Wiley, 1969).

Landesman, A. L.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Lane, R. G.

Lanterman, A. D.

Liseo, B.

J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999).
[CrossRef]

Löfdahl, M. G.

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, "Fast-phase diversity wave-front sensing for mirror control," in Proc. SPIE 3353, 952-963 (1988).
[CrossRef]

Macovski, A.

R. M. Gray and A. Macovski, "Maximum a posteriori estimation of position in scintillation cameras," IEEE Trans. Nucl. Sci. NS-23, 849-852 (1976).
[CrossRef]

Mar, L. S.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

Mardia, K. V.

K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 1979).

Marzetta, T. L.

P. Stoica and T. L. Marzetta, "Parameter estimation problems with singular information matrices," IEEE Trans. Signal Process. 49, 87-89 (2001).
[CrossRef]

Melsa, J. L.

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

Miller, B. W.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

Milster, T. D.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

R. G. Paxman, H. H. Barrett, W. E. Smith, and T. D. Milster, "Image reconstruction from coded data: II. Code design," J. Opt. Soc. Am. A 2, 501-509 (1985).
[CrossRef] [PubMed]

Mugnier, L. M.

Myers, K. J.

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," in Proc. SPIE 3032, 110-124 (1997).
[CrossRef]

H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, "Objective assessment of image quality: II. Fisher information, Fourier crosstalk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995).
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Nagarkar, V. V.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

Noll, R. J.

Parra, L.

L. Parra and H. H. Barrett, "List-mode likelihood: EM algorithm and noise estimation demonstrated on 2D-PET," IEEE Trans. Med. Imaging MI-17, 228-235 (1998).
[CrossRef]

H. H. Barrett, L. Parra, and T. A. White, "List-mode likelihood," J. Opt. Soc. Am. A 14, 2914-2923 (1997).
[CrossRef]

Patton, D. D.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Paxman, R. G.

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations using phase diversity," J. Opt. Soc. Am. A 7, 1072-1085 (1992).
[CrossRef]

R. G. Paxman, H. H. Barrett, W. E. Smith, and T. D. Milster, "Image reconstruction from coded data: II. Code design," J. Opt. Soc. Am. A 2, 501-509 (1985).
[CrossRef] [PubMed]

Pennington, T. L.

Rabbani, M.

Roddier, F.

Rogers, W. L.

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

Roggemann, M. C.

Roney, T. J.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Rousset, G.

G. Rousset, "Wavefront sensing," in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999).

Rowe, R. K.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Saleh, B. E. A.

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," Proc. IEEE 70, 229-245 (1992).
[CrossRef]

B. E. A. Saleh, "Estimation of the location of an optical object with photodetectors limited by quantum noise," Appl. Opt. 13, 1824-1827 (1974).
[CrossRef] [PubMed]

B. E. A. Saleh, "Estimations based on instants of occurrence of photon counts of low level light," Proc. IEEE 62, 530-531 (1974).
[CrossRef]

B. E. A. Saleh, "Joint probability of occurrence of photon events and estimation of optical parameters," J. Phys. A 7, 1360-1368 (1974).
[CrossRef]

Sallberg, S. A.

Scharf, L. L.

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time-Series Analysis (Addison-Wesley, 1991).

Scharmer, G. B.

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, "Fast-phase diversity wave-front sensing for mirror control," in Proc. SPIE 3353, 952-963 (1988).
[CrossRef]

Schulz, T. J.

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations using phase diversity," J. Opt. Soc. Am. A 7, 1072-1085 (1992).
[CrossRef]

Seacat, R. H.

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Shao, L.

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

Shaw, R.

Shestakova, I.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

Singh, B.

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

Smith, S. T.

S. T. Smith, "Covariance, subspace, and intrinsic Cramer-Rao bounds," IEEE Trans. Signal Process. 53, 1610-1630 (2005).
[CrossRef]

Smith, W. E.

Snyder, D. L.

Stoica, P.

P. Stoica and T. L. Marzetta, "Parameter estimation problems with singular information matrices," IEEE Trans. Signal Process. 49, 87-89 (2001).
[CrossRef]

Strimbu, L. M.

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

Swank, R. K.

R. K. Swank, "Absorption and noise in X-ray phosphors," J. Appl. Phys. 44, 4199-4203 (1973).
[CrossRef]

Tansey, R. J.

Teich, M. C.

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," Proc. IEEE 70, 229-245 (1992).
[CrossRef]

Tyler, G. A.

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics (Academic Press, 1998).

van Dam, M. A.

M. A. van Dam and R. G. Lane, "Wave-front slope estimation," J. Opt. Soc. Am. A 17, 1319-1324 (2000).
[CrossRef]

M. A. van Dam, "Wave-front sensing for adaptive optics in astronomy," Ph. D. thesis (University of Canterbury, 2002).

van Metter, R.

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Vol. I (Wiley, 1968).

Wagner, R. F.

Wallner, E. P.

Welsh, B. M.

White, H.

H. White, "Maximum likelihood estimation of misspecified models," Econometrica , 50, 1-126 (1982).
[CrossRef]

White, T. A.

Widen, K. C.

Winick, K. A.

Wolpert, R. L.

J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999).
[CrossRef]

Appl. Opt. (6)

Discuss. Faraday Soc. (1)

R. E. Burgess, "Homophase and heterophase fluctuations in semiconducting crystals," Discuss. Faraday Soc. 21, 51-158 (1959).

Econometrica (1)

H. White, "Maximum likelihood estimation of misspecified models," Econometrica , 50, 1-126 (1982).
[CrossRef]

IEEE Trans. Med. Imaging (2)

L. Parra and H. H. Barrett, "List-mode likelihood: EM algorithm and noise estimation demonstrated on 2D-PET," IEEE Trans. Med. Imaging MI-17, 228-235 (1998).
[CrossRef]

D. Gagnon, "Maximum likelihood positioning in the scintillation camera using depth of interaction," IEEE Trans. Med. Imaging MI-12, 101-107 (1993).
[CrossRef]

IEEE Trans. Nucl. Sci. (2)

N. H. Clinthorne, W. L. Rogers, L. Shao, and K. F. Kcral, "A hybrid maximum likelihood position computer for scintillation cameras," IEEE Trans. Nucl. Sci. 34, 97-101 (1987).
[CrossRef]

R. M. Gray and A. Macovski, "Maximum a posteriori estimation of position in scintillation cameras," IEEE Trans. Nucl. Sci. NS-23, 849-852 (1976).
[CrossRef]

IEEE Trans. Signal Process. (2)

P. Stoica and T. L. Marzetta, "Parameter estimation problems with singular information matrices," IEEE Trans. Signal Process. 49, 87-89 (2001).
[CrossRef]

S. T. Smith, "Covariance, subspace, and intrinsic Cramer-Rao bounds," IEEE Trans. Signal Process. 53, 1610-1630 (2005).
[CrossRef]

J. Appl. Phys. (1)

R. K. Swank, "Absorption and noise in X-ray phosphors," J. Appl. Phys. 44, 4199-4203 (1973).
[CrossRef]

J. Nucl. Med. (1)

T. D. Milster, J. N. Aarsvold, H. H. Barrett, A. L. Landesman, L. S. Mar, D. D. Patton, T. J. Roney, R. K. Rowe, and R. H. Seacat III, "A full-field modular gamma camera," J. Nucl. Med. 31, 632-639 (1990).
[PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (14)

A. Blanc, L. M. Mugnier, and J. Idier, "Marginal estimation of aberrations and image restoration by use of phase diversity," J. Opt. Soc. Am. A 20, 1035-1045 (2003).
[CrossRef]

D. L. Snyder, C. W. Helstrom, A. D. Lanterman, and M. Faisal, "Compensation for readout noise in CCD images," J. Opt. Soc. Am. A 12, 272-283 (1995).
[CrossRef]

H. H. Barrett, J. L. Denny, R. F. Wagner, and K. J. Myers, "Objective assessment of image quality: II. Fisher information, Fourier crosstalk, and figures of merit for task performance," J. Opt. Soc. Am. A 12, 834-852 (1995).
[CrossRef]

R. C. Cannon, "Global wave-front reconstruction using Shack-Hartmann sensors," J. Opt. Soc. Am. A 12, 2031-2039 (1995).
[CrossRef]

H. H. Barrett, K. J. Myers, N. Devaney, and J. C. Dainty, "Objective assessment of image quality IV. Application to adaptive optics," J. Opt. Soc. Am. A 3080-3105 (2006).
[CrossRef]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, "Joint estimation of object and aberrations using phase diversity," J. Opt. Soc. Am. A 7, 1072-1085 (1992).
[CrossRef]

M. A. van Dam and R. G. Lane, "Wave-front slope estimation," J. Opt. Soc. Am. A 17, 1319-1324 (2000).
[CrossRef]

H. H. Barrett, C. K. 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]

S. A. Sallberg, B. M. Welsh, and M. C. Roggemann, "Maximum a posteriori estimation of wave-front slopes using a Shack-Hartmann wave-front sensor," J. Opt. Soc. Am. A 14, 1347-1354 (1997).
[CrossRef]

R. G. Paxman, H. H. Barrett, W. E. Smith, and T. D. Milster, "Image reconstruction from coded data: II. Code design," J. Opt. Soc. Am. A 2, 501-509 (1985).
[CrossRef] [PubMed]

H. H. Barrett, L. Parra, and T. A. White, "List-mode likelihood," J. Opt. Soc. Am. A 14, 2914-2923 (1997).
[CrossRef]

K. A. Winick, "Cramér-Rao lower bounds on the performance of charge-coupled-device optical position estimator," J. Opt. Soc. Am. A 3, 1809-1815 (1986).
[CrossRef]

M. Rabbani, R. Shaw, and R. van Metter, "Detective quantum efficiency of imaging systems with amplifying and scattering mechanisms," J. Opt. Soc. Am. A 4, 895-901 (1987).
[CrossRef] [PubMed]

H. H. Barrett, "Objective assessment of image quality: effects of quantum noise and object variability," J. Opt. Soc. Am. A 7, 1266-1278 (1990).
[CrossRef] [PubMed]

J. Phys. A (1)

B. E. A. Saleh, "Joint probability of occurrence of photon events and estimation of optical parameters," J. Phys. A 7, 1360-1368 (1974).
[CrossRef]

Opt. Eng. (1)

R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).

Proc. IEEE (2)

B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," Proc. IEEE 70, 229-245 (1992).
[CrossRef]

B. E. A. Saleh, "Estimations based on instants of occurrence of photon counts of low level light," Proc. IEEE 62, 530-531 (1974).
[CrossRef]

Proc. SPIE (5)

J. N. Aarsvold, H. H. Barrett, J. Chen, A. L. Landesman, T. D. Milster, D. D. Patton, T. J. Roney, R. K. Rowe, R. H. Seacat III, and L. M. Strimbu, "Modular scintillation cameras: a progress report," in Proc. SPIE 914, 319-325 (1988).

M. G. Löfdahl, A. L. Duncan, and G. B. Scharmer, "Fast-phase diversity wave-front sensing for mirror control," in Proc. SPIE 3353, 952-963 (1988).
[CrossRef]

B. W. Miller, H. B. Barber, H. H. Barrett, I. Shestakova, B. Singh, and V. V. Nagarkar, "Single-photon spatial resolution enhancement of columnar CsI(Tl) using centroid estimation and event discrimination," Proc. SPIE 6142, 61421T (2006).
[CrossRef]

H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," in Proc. SPIE 3032, 110-124 (1997).
[CrossRef]

L. Chen and H. H. Barrett, "Non-Gaussian noise in X-ray and gamma-ray detectors," in Proc. SPIE 5745, 366-376 (2005).
[CrossRef]

Stat. Sci. (1)

J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999).
[CrossRef]

Other (13)

H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).

M. A. van Dam, "Wave-front sensing for adaptive optics in astronomy," Ph. D. thesis (University of Canterbury, 2002).

L. R. Furenlid, J. Y. Hesterman, and H. H. Barrett, "Real time data acquisition and maximum-likelihood estimation for gamma cameras," in Proceedings of the 14th IEEE-NPSS Real-Time Conference (IEEE, 2005), pp. 498-501.

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

H. L. Van Trees, Detection, Estimation, and Modulation Theory, Vol. I (Wiley, 1968).

L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time-Series Analysis (Addison-Wesley, 1991).

R. K. Tyson, Principles of Adaptive Optics (Academic Press, 1998).

G. Rousset, "Wavefront sensing," in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999).

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

H. H. Barrett, "Detectors for small-animal SPECT: II. Statistical limitations and estimation methods," in Small-Animal SPECT Imaging, M.Kupinski and H.Barrett eds. (Springer, 2005), Chap. 3.
[CrossRef]

J. Y. Hesterman (University of Arizona, jyh@email.arizona.edu, personal communication, 2005).

K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 1979).

N. L. Johnson and S. Kotz, Discrete Distributions (Wiley, 1969).

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Figures (6)

Fig. 1
Fig. 1

Block diagram of a generic wavefront sensor and reconstructor.

Fig. 2
Fig. 2

Plots of pr [ g m k ¯ m ] for mixed Poisson and Gaussian noise: (a) and (b) show pr [ g m k ¯ m ] versus g m for fixed k ¯ m ; (c) and (d) show pr [ g m k ¯ m ] versus k ¯ m for fixed g m . Plots (a) and (c) are for small electronic noise ( σ = 0.2 in electron units), and plots (b) and (d) are for larger electronic noise ( σ = 2.0 ) .

Fig. 3
Fig. 3

Display of the response functions f m ( τ ) used in simulation of a Shack–Hartmann sensor with a single lenslet and a 2 × 2 array of photodetectors. Each plot represents the mean response of one photodetector as a function of the x and y components of the wavefront tilt.

Fig. 4
Fig. 4

Left, centroid estimates of an 8 × 8 array of tilts from Poisson data in a quad-cell Shack–Hartmann sensor; right, ML estimates from the same data.

Fig. 5
Fig. 5

Comparison of traditional LS estimation of wavefront coefficients from centroid data versus direct ML estimation from photodetector outputs. Parameters used in the simulation include: λ = 680 nm ; pupil diameter = 24 μ m × 128 = 3072 μ m ; lenslet size = 192 μ m ; CCD pixel size = 24 μ m ; and focal length = 9.9 mm . The wavefront was sampled at 1726 points across the pupil diameter, and 322 rows and columns of zeros were used to pad the wavefront function to a 2048 × 2048 array before computing the FFT. The markers represent the mean, and the error bars represent the standard deviation of the residual wavefront rms of the 50 estimations for each light level.

Fig. 6
Fig. 6

Same as Fig. 5 except that global tip and tilt were not removed from the simulated wavefront and were also included in the coefficients to estimate.

Tables (1)

Tables Icon

Table 1 Vectors Relevant to Wavefront Sensing

Equations (114)

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

v ¯ = H θ or v = H θ + n ,
L ( θ g ) = pr ( g θ ) .
θ ̂ ¯ = d P θ ̂ pr ( θ ̂ θ ) θ ̂ .
θ ̂ ¯ = d M g pr ( g θ ) θ ̂ ( g ) θ ̂ ( g ) g θ .
b ( θ ) θ ̂ ¯ θ d M g [ θ ̂ ( g ) θ ] pr ( g θ ) = d P θ [ θ ̂ θ ] pr ( θ ̂ θ ) .
Var ( θ ̂ p ) [ θ ̂ p θ ̂ p ] [ θ ̂ p θ ̂ p ] * g θ = d M g θ ̂ p ( g ) θ ̂ p ( g ) 2 pr ( g θ ) = d P θ θ ̂ p θ ̂ p 2 pr ( θ ̂ θ ) ,
[ K θ ̂ ] p p = [ θ ̂ p θ ̂ p ] [ θ ̂ p θ ̂ p ] * g θ
K θ ̂ = ( θ ̂ θ ̂ ¯ ) ( θ ̂ θ ̂ ¯ ) g θ ,
MSE = θ ̂ θ 2 g θ = d M g θ ̂ ( g ) θ 2 pr ( g θ ) = tr [ K θ ̂ ] + tr [ bb ] ,
EMSE = θ ̂ ( g ) θ 2 g θ θ .
θ = ( α β ) ,
F j k = [ θ j ln pr ( g θ ) ] [ θ k ln pr ( g θ ) ] g θ = d M g pr ( g θ ) [ 1 pr ( g θ ) θ j pr ( g θ ) ] [ 1 pr ( g θ ) θ k pr ( g θ ) ] .
[ K θ ̂ ] n n = Var { θ ̂ n } [ F 1 ] n n .
K θ ̂ F 1 .
K θ ̂ ( θ b + I ) F 1 ( θ b + I ) t ,
θ ̂ ML argmax θ pr ( g θ ) ,
θ ̂ ML argmax θ ln [ pr ( g θ ) ] .
a ̂ M L = a ( θ ̂ M L ) .
Pr ( g θ ) = m = 1 M exp [ g ¯ m ( θ ) ] [ g ¯ m ( θ ) ] g m g m ! ,
ln Pr ( g θ ) = m = 1 M { g ¯ m ( θ ) + g m ln [ g ¯ m ( θ ) ] ln ( g m ! ) } .
θ j ln Pr ( g θ ) = m = 1 M { 1 + g m g ¯ m ( θ ) } g ¯ m ( θ ) θ j .
[ g m g ¯ m ( θ ) ] [ g m g ¯ m ( θ ) ] = g ¯ m ( θ ) δ m m ,
F j k = m = 1 M 1 g ¯ m ( θ ) g ¯ m ( θ ) θ j g ¯ m ( θ ) θ k .
pr ( G θ ) = pr ( { r k } , K θ ) = pr ( { r k } K , θ ) Pr ( K θ ) ,
pr ( G θ ) = Pr ( K θ ) k = 1 K pr ( r k θ ) ,
pr ( r k θ ) = b ( r k ; θ ) d e t d 2 r b ( r ; θ ) ,
pr ( G θ ) = exp [ K ¯ ( θ ) ] [ K ¯ ( θ ) ] K K ! k = 1 K b ( r k ; θ ) d e t d 2 r b ( r ; θ ) = exp [ K ¯ ( θ ) ] K ! k = 1 K b ( r ; θ ) ,
ln pr ( G θ ) = K ¯ ( θ ) ln K ! + k = 1 K ln b ( r k ; θ ) .
pr ( g θ ) = m = 1 M 1 2 π σ 2 exp [ [ g m g ¯ m ( θ ) ] 2 2 σ 2 ] ,
ln pr ( g θ ) = 1 2 M ln ( 2 π σ 2 ) 1 2 σ 2 m = 1 M [ g m g ¯ m ( θ ) ] 2 .
pr ( g θ ) = m = 1 M 1 2 π σ m 2 exp [ [ g m g ¯ m ( θ ) ] 2 2 σ m 2 ] .
F j k = m = 1 M 1 σ m 2 g ¯ m ( θ ) θ j g ¯ m ( θ ) θ k .
pr ( g m θ ) = k m = 1 pr ( g m k m ) P r ( k m θ ) ,
pr ( g θ ) = m = 1 M pr ( g m θ ) = m = 1 M 1 2 π σ 2 k m = 0 exp [ ( g m R k m ) 2 2 σ 2 ] exp [ k ¯ m ( θ ) ] [ k ¯ m ( θ ) ] k m k m ! .
F j k m = 1 M R 2 σ 2 + R 2 k ¯ m ( θ ) k ¯ m ( θ ) θ j k ¯ m ( θ ) θ k ,
pr ( g ) = [ ( 2 π ) M det ( K ) ] 1 2 exp [ 1 2 ( g g ¯ ) t K 1 ( g g ¯ ) ] ,
pr ( g θ ) = [ ( 2 π ) M det [ K ( θ ) ] ] 1 2 exp { 1 2 [ g g ¯ ( θ ) ] t [ K ( θ ) ] 1 [ g g ¯ ( θ ) ] } .
W ( r ) = n = 1 γ n u n ( r ) ,
W ( r ) = n = 1 N α n ψ n ( r ) + Δ W ( r ) .
χ k ( r ) = { S ( r r j ) ( x x j ) j = ( k + 1 ) 2 if k odd S ( r r j ) ( y y j ) j = k 2 if k even .
W ( r ) = k = 1 2 J τ k χ k ( r ) + δ W ( r ) .
Δ W ( r ) = k = 1 β k i n t Ξ k ( r ) .
pr ( g α ) = d K + L β pr ( g α , β ) pr ( β α ) ,
pr ( g α , β e x t ) = d K β i n t pr ( g α , β e x t , β i n t ) pr ( β i n t α ) .
pr ( β i n t α ) pr ( β i n t α = 0 ) .
pr ( β i n t α = 0 ) = N exp [ 1 2 ( β i n t ) t C 1 ( β i n t ) ] ,
pr ( g α , β e x t ) N d K β i n t pr ( g α , β e x t , β i n t ) exp [ 1 2 ( β i n t ) t C 1 ( β i n t ) ] .
pr ( g α , β e x t ) N d K β i n t exp [ m = 1 M [ g m g ¯ m ( α , β e x t , β i n t ) ] 2 2 σ 2 ] exp [ 1 2 ( β i n t ) t C 1 ( β i n t ) ] ,
pr ( g α , β e x t ) N exp { 1 2 [ g g ¯ ( α , β e x t , 0 ) ] t K t o t 1 [ g g ¯ ( α , β e x t , 0 ) ] t } ,
K t o t σ 2 I + ACA t ,
θ = ( α β e x t ) .
ln Pr ( g α , β e x t ) = m = 1 M { g ¯ m ( α , β e x t ) + g m ln [ g ¯ m ( α , β e x t ) ] } .
g ¯ m ( α , I 0 ) = I 0 f m ( α ) ,
ln Pr ( g α , I 0 ) = I 0 m = 1 M f m ( α ) + m = 1 M g m ln [ f m ( α ) ] + N t o t ln ( I 0 ) ,
α n ln Pr ( g α , I 0 ) = m = 1 M [ g m g ¯ m ( α , I 0 ) f m ( α ) ] f m ( α ) α n ,
I 0 ln Pr ( g α , I 0 ) = 1 I 0 m = 1 M [ g m g ¯ m ( α , I 0 ) ] .
[ g m g ¯ m ( α , I 0 ) ] [ g m g ¯ m ( α , I 0 ) ] g α , I 0 = g ¯ m ( α , I 0 ) δ m m ,
F n n = I 0 m = 1 M 1 f m ( α ) f m ( α ) α n f m ( α ) α n ( n , n N ) ,
F n , N + 1 = F N + 1 , n = m = 1 M f m ( α ) α n ( n N ) ,
F N + 1 , N + 1 = N ¯ t o t I 0 2 = 1 I 0 m = 1 M f m ( α ) ,
F = [ A N × N B N × 1 B 1 × N t C 1 × 1 ] ,
g ¯ m ( α , I 0 , b ) = I 0 f m ( α ) + b ,
ln Pr ( g α , I 0 , b ) = I 0 m = 1 M f m ( α ) M b + m = 1 M g m ln [ I 0 f m ( α ) + b ] .
α n ln Pr ( g α , I 0 , b ) = I 0 m = 1 M [ g m g ¯ m ( α , I 0 , b ) g ¯ m ( α , I 0 , b ) ] f m ( α ) α n ,
I 0 ln Pr ( g α , I 0 , b ) = m = 1 M [ g m g ¯ m ( α , I 0 , b ) ] f m ( α ) I 0 f m ( α ) + b ,
b ln Pr ( g α , I 0 , b ) = m = 1 M [ g m g ¯ m ( α , I 0 , b ) g ¯ m ( α , I 0 , b ) ] .
ln pr ( g θ ) = 1 2 m = 1 M [ g m g ¯ m ( θ ) ] 2 σ m 2 ( θ ) + constant .
ln pr ( g θ ) = 1 2 m = 1 M m = 1 M [ g m g ¯ m ( θ ) ] [ K 1 ] m m [ g m g ¯ m ( θ ) ] = 1 2 [ g g ¯ ( θ ) ] t K 1 [ g g ¯ ( θ ) ] ,
f m ( τ ) = d 2 r d m ( r ) s ( r z 0 τ ) .
m = 1 M 1 f m ( τ ) f t o t = constant ,
ln Pr ( g τ , I 0 ) = f t o t I 0 + m = 1 M 1 g m ln [ f m ( τ ) ] + N t o t ln ( I 0 ) ,
m = 1 M 1 f m ( τ ) τ n = τ n m = 1 M 1 f m ( τ ) = 0 .
I ̂ 0 = m = 1 M 1 g m m = 1 M 1 f m ( τ ) = N t o t m = 1 M 1 f m ( τ ) .
m = 1 M 1 g m f m ( τ ) f m ( τ ) τ n = 0 when τ = τ ̂ .
I 0 m = 1 M 1 f m ( τ ) M b + m = 1 M 1 g m ln [ I 0 f m ( τ ) + b ] = maximum at τ = τ ̂ , I 0 = I ̂ 0 , b = b ̂ .
ln pr ( g Θ ) = m = 1 M ln pr [ g m g ¯ m ( Θ ) ] ,
m = 1 M f m ( α ) = f t o t = constant .
m = 1 M g m ln [ f m ( α ) ] = maximum .
pr ( g θ ) = 1 2 π σ 2 k = 0 exp [ ( g R k ) 2 2 σ 2 ] [ k ¯ ( θ ) ] k k ! exp [ k ¯ ( θ ) ] .
θ n ln pr ( g θ ) = 1 pr ( g θ ) θ n pr ( g θ ) = 1 pr ( g θ ) 1 2 π σ 2 k = 0 exp [ ( g R k ) 2 2 σ 2 ] 1 k ! θ n { [ k ¯ ( θ ) ] k exp [ k ¯ ( θ ) ] } = 1 pr ( g θ ) 1 2 π σ 2 k = 0 exp [ ( g R k ) 2 2 σ 2 ] [ k ¯ ( θ ) k ] k ! exp [ k ¯ ( θ ) ] ( 1 k ¯ ( θ ) 1 ) k ¯ ( θ ) θ n .
θ n ln pr ( g θ ) = [ k = 0 exp [ ( 1 2 σ 2 ) ( g R R k ) 2 ] exp [ k ¯ ( θ ) ] [ k ¯ ( θ ) ] k k ! k = 0 exp [ ( 1 2 σ 2 ) ( g R k ) 2 ] exp [ k ¯ ( θ ) ] [ k ¯ ( θ ) ] k k ! 1 ] k ¯ ( θ ) θ n .
θ n ln pr g θ ( g ) = [ pr g θ ( g R ) pr g θ ( g ) 1 ] k ¯ ( θ ) θ n
θ n pr g θ ( g ) = [ pr g θ ( g R ) pr g θ ( g ) ] k ¯ ( θ ) θ n .
F n n = [ θ n ln pr g θ ( g ) ] [ θ n ln pr g θ ( g ) ] g θ = [ pr g θ ( g R ) pr g θ ( g ) 1 ] 2 g θ k ¯ ( θ ) θ n k ¯ ( θ ) θ n .
[ pr g θ ( g R ) pr g θ ( g ) 1 ] 2 g θ
= d g pr g θ ( g ) [ pr g θ ( g R ) pr g θ ( g ) 1 ] 2
= d g [ pr g θ ( g R ) ] 2 pr g θ ( g ) 1 ,
F j k = [ d g [ pr g θ ( g R ) ] 2 pr g θ ( g ) 1 ] k ¯ ( θ ) θ j k ¯ ( θ ) θ k .
F j k R 2 σ 2 + R 2 k ¯ ( θ ) k ¯ ( θ ) θ j k ¯ ( θ ) θ k .
pr ( θ ) = pr ( α , β ) = N θ exp [ 1 2 ( θ θ ¯ ) t K θ 1 ( θ θ ¯ ) ] ,
K θ = [ K α α K α β K α β t K β β ] ,
pr ( β α ) = N β α exp [ 1 2 ( β β ̃ ) t K β α 1 ( β β ̃ ) ] ,
β ̃ = β ¯ + K β α K α α 1 ( α α ¯ ) ,
K β α = K β β K β α K α α 1 K α β .
g ¯ m ( α , β e x t , β i n t ) g ¯ m ( α , β e x t , 0 ) + k = 1 K A m k β k i n t where A m k = g ¯ m ( α , β e x t , β i n t ) β k i n t β i n t = 0 .
g ¯ = g ¯ 0 + A β i n t .
pr ( g α , β e x t ) N d K β i n t exp [ 1 2 σ 2 g g ¯ 0 A β i n t 2 ] exp [ 1 2 ( β i n t ) t C 1 ( β i n t ) ] ,
pr ( x ) = [ ( 2 π ) M det ( K ) ] 1 2 exp [ 1 2 ( x x ¯ ) t K 1 ( x x ¯ ) ] = d M ξ exp [ 2 π i ξ t ( x x ¯ ) ] exp ( 2 π 2 ξ t K ξ ) .
pr ( g α , β e x t ) = d K β i n t d M ξ d K η exp ( 2 π 2 σ 2 ξ 2 ) exp [ 2 π i ξ t ( g ¯ 0 + A β i n t ) ] exp ( 2 π i ξ t g ) exp ( 2 π 2 η t C η ) exp ( 2 π i η t β i n t ) .
pr ( g α , β e x t ) = d M ξ exp ( 2 π 2 σ 2 ξ 2 ) exp ( 2 π 2 ξ 2 ACA t ξ ) exp [ 2 π i ξ t ( g g ¯ 0 ) ] = [ ( 2 π ) M det ( K t o t ) ] 1 2 exp { 1 2 [ g g ¯ 0 ] t K t o t 1 [ g g ¯ 0 ] t } ,
( χ k , W ) = n = 1 N α n ( χ k , ψ n ) + ( χ k , Δ W ) = 1 χ 2 τ k + ( χ k , δ W ) ,
τ k = n = 1 N M k n α n + ( χ k , Δ W ) ,
τ = M α .
α = [ M t M ] 1 M t τ B τ .
r c = d 2 r r I ( r ) d 2 r I ( r ) ,
r ̂ c ( g ) = 1 g t o t m = 1 M r m g m ,
g t o t = m = 1 M g m .
r ̂ c ( g ) = 1 g t o t Rg ,
τ ̂ ( g ) r ̂ c z 0 ,
Ψ r ̂ c θ ( ξ ) exp [ 2 π i ξ t r ̂ c ] r ̂ c θ ,
Ψ r ̂ c θ ( ξ ) = exp [ 2 π i 1 g t o t ξ t Rg ] g θ , g t o t g t o t θ .
Ψ g θ , g t o t ( ρ ) exp [ 2 π i ρ t g ] g θ , g t o t ,
Ψ r ̂ c θ ( ξ ) Ψ g ̂ θ , g t o t ( 1 g t o t R t ξ ) g t o t θ t .
Ψ g θ , g t o t ( ρ ) = [ m = 1 M p m ( θ ) exp ( 2 π i ρ m ) ] g t o t ,

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