OSA's Digital Library

Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Editor: Stephen A. Burns
  • Vol. 24, Iss. 2 — Feb. 1, 2007
  • pp: 391–414

Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions

Harrison H. Barrett, Christopher Dainty, and David Lara  »View Author Affiliations

JOSA A, Vol. 24, Issue 2, pp. 391-414 (2007)

View Full Text Article

Enhanced HTML    Acrobat PDF (317 KB)

Browse Journals / Lookup Meetings

Browse by Journal and Year


Lookup Conference Papers

Close Browse Journals / Lookup Meetings

Article Tools



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

OCIS Codes
(010.0010) Atmospheric and oceanic optics : Atmospheric and oceanic optics
(010.1080) Atmospheric and oceanic optics : Active or adaptive optics
(010.7350) Atmospheric and oceanic optics : Wave-front sensing

ToC Category:
Atmospheric and Oceanic Optics

Original Manuscript: November 22, 2005
Revised Manuscript: May 2, 2006
Manuscript Accepted: July 28, 2006

Harrison H. Barrett, Christopher Dainty, and David Lara, "Maximum-likelihood methods in wavefront sensing: stochastic models and likelihood functions," J. Opt. Soc. Am. A 24, 391-414 (2007)

Sort:  Author  |  Year  |  Journal  |  Reset  


  1. R. K. Tyson, Principles of Adaptive Optics (Academic Press, 1998).
  2. G. Rousset, "Wavefront sensing," in Adaptive Optics in Astronomy, F.Roddier, ed. (Cambridge U. Press, 1999).
  3. F. Roddier, "Curvature sensing and compensation: a new concept in adaptive optics," Appl. Opt. 27, 1223-1225 (1988). [CrossRef] [PubMed]
  4. G. A. Tyler and D. L. Fried, "Image-position error associated with a quadrant detector," J. Opt. Soc. Am. 72, 804 (1982). [CrossRef]
  5. E. P. Wallner, "Optimal wave-front correction using slope measurements," J. Opt. Soc. Am. 73, 1771-1776 (1983) [CrossRef]
  6. H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. B. E. A. Saleh, "Estimations based on instants of occurrence of photon counts of low level light," Proc. IEEE 62, 530-531 (1974). [CrossRef]
  12. B. E. A. Saleh, "Joint probability of occurrence of photon events and estimation of optical parameters," J. Phys. A 7, 1360-1368 (1974). [CrossRef]
  13. M. Elbaum and M. Greenebaum, "Annular apertures for angular tracking," Appl. Opt. 16, 2438-2440 (1977). [CrossRef] [PubMed]
  14. 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]
  15. R. Irwan and R. G. Lane, "Analysis of optimal centroid estimation applied to Shack-Hartmann sensing," Appl. Opt. 38, 6737-6743 (1999). [CrossRef]
  16. M. A. van Dam, "Wave-front sensing for adaptive optics in astronomy," Ph. D. thesis (University of Canterbury, 2002).
  17. M. A. van Dam and R. G. Lane, "Wave-front slope estimation," J. Opt. Soc. Am. A 17, 1319-1324 (2000). [CrossRef]
  18. B. M. Welsh, B. L. Ellerbroek, M. C. Roggemann, and T. L. Pennington, "Fundamental performance comparison of a Hartmann and a shearing interferometer wavefront sensor," Appl. Opt. 34, 4186-4195 (1995). [CrossRef] [PubMed]
  19. 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]
  20. 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]
  21. R. C. Cannon, "Global wave-front reconstruction using Shack-Hartmann sensors," J. Opt. Soc. Am. A 12, 2031-2039 (1995). [CrossRef]
  22. 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]
  23. R. A. Gonsalves, "Phase retrieval and diversity in adaptive optics," Opt. Eng. 21, 829-832 (1982).
  24. 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]
  25. J. J. Dolne, R. J. Tansey, K. A. Black, J. H. Deville, P. R. Cunningham, K. C. Widen, and P. S. Idell, "Practical issues in wave-front sensing by use of phase diversity," Appl. Opt. 42, 5284-5289 (2003). [CrossRef] [PubMed]
  26. 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]
  27. 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).
  28. 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]
  29. D. Gagnon, "Maximum likelihood positioning in the scintillation camera using depth of interaction," IEEE Trans. Med. Imaging MI-12, 101-107 (1993). [CrossRef]
  30. 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]
  31. 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.
  32. J. L. Melsa and D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, 1978).
  33. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Vol. I (Wiley, 1968).
  34. L. L. Scharf, Statistical Signal Processing: Detection, Estimation, and Time-Series Analysis (Addison-Wesley, 1991).
  35. 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]
  36. P. Stoica and T. L. Marzetta, "Parameter estimation problems with singular information matrices," IEEE Trans. Signal Process. 49, 87-89 (2001). [CrossRef]
  37. J. O. Berger, B. Liseo, and R. L. Wolpert, "Integrated likelihood methods for eliminating nuisance parameters," Stat. Sci. 14, 1-28 (1999). [CrossRef]
  38. H. Cramér, Mathematical Methods of Statistics (Princeton U. Press, 1946).
  39. H. H. Barrett, L. Parra, and T. A. White, "List-mode likelihood," J. Opt. Soc. Am. A 14, 2914-2923 (1997). [CrossRef]
  40. 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]
  41. 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]
  42. B. E. A. Saleh and M. C. Teich, "Multiplied Poisson noise in pulse, particle, and photon detection," Proc. IEEE 70, 229-245 (1992). [CrossRef]
  43. 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]
  44. R. E. Burgess, "Homophase and heterophase fluctuations in semiconducting crystals," Discuss. Faraday Soc. 21, 51-158 (1959).
  45. R. K. Swank, "Absorption and noise in X-ray phosphors," J. Appl. Phys. 44, 4199-4203 (1973). [CrossRef]
  46. 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]
  47. H. H. Barrett, R. F. Wagner, and K. J. Myers, "Correlated point processes in radiological imaging," in Proc. SPIE 3032, 110-124 (1997). [CrossRef]
  48. L. Chen and H. H. Barrett, "Non-Gaussian noise in X-ray and gamma-ray detectors," in Proc. SPIE 5745, 366-376 (2005). [CrossRef]
  49. 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]
  50. H. White, "Maximum likelihood estimation of misspecified models," Econometrica , 50, 1-126 (1982). [CrossRef]
  51. R. J. Noll, "Zernike polynomials and atmospheric turbulence," J. Opt. Soc. Am. 66, 207-210 (1976). [CrossRef]
  52. 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]
  53. J. Y. Hesterman (University of Arizona, jyh@email.arizona.edu, personal communication, 2005).
  54. S. T. Smith, "Covariance, subspace, and intrinsic Cramer-Rao bounds," IEEE Trans. Signal Process. 53, 1610-1630 (2005). [CrossRef]
  55. K. V. Mardia, J. T. Kent, and J. M. Bibby, Multivariate Analysis (Academic, 1979).
  56. N. L. Johnson and S. Kotz, Discrete Distributions (Wiley, 1969).

Cited By

Alert me when this paper is cited

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article  |  Next Article »

OSA is a member of CrossRef.

CrossCheck Deposited