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Applied Optics

Applied Optics

APPLICATIONS-CENTERED RESEARCH IN OPTICS

  • Editor: Joseph N. Mait
  • Vol. 52, Iss. 21 — Jul. 20, 2013
  • pp: 5235–5246

Cramer–Rao bounds for intensity interferometry measurements

Richard Holmes, Brandoch Calef, D. Gerwe, and P. Crabtree  »View Author Affiliations


Applied Optics, Vol. 52, Issue 21, pp. 5235-5246 (2013)
http://dx.doi.org/10.1364/AO.52.005235


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Abstract

The question of signal-to-noise ratio (SNR) in intensity interferometry has been revisited in recent years, as researchers have realized that various innovations can offer significant improvements in SNR. These innovations include improved signal processing. Two such innovations, the use of positivity and the use of knowledge of the general shape of the object, have been proposed. This paper investigates the potential gains offered by these two approaches using Cramer–Rao lower bounds (CRLBs). The CRLB on the variance of the positivity-constrained maximum likelihood (ML) estimate is at best 1/4 of the variance of the unconstrained estimator. This is compared to the positivity-constrained ML estimator, which delivers a best-case variance reduction of only (11/π)/2=34.1%. The gains offered by prior knowledge depend on the quality of such information, as might be expected from optimal weighting of such data with the measured data. Furthermore, biases are induced by the application of constraints, and these biases can eliminate some or all of the advantage of lower variances, as found when considering the total root-mean-square error. A form of CRLB for variance is presented that properly incorporates prior information.

© 2013 Optical Society of America

OCIS Codes
(030.4280) Coherence and statistical optics : Noise in imaging systems
(030.6600) Coherence and statistical optics : Statistical optics
(100.2980) Image processing : Image enhancement
(350.1270) Other areas of optics : Astronomy and astrophysics
(100.3175) Image processing : Interferometric imaging
(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

ToC Category:
Imaging Systems

History
Original Manuscript: January 23, 2013
Revised Manuscript: May 4, 2013
Manuscript Accepted: May 30, 2013
Published: July 18, 2013

Citation
Richard Holmes, Brandoch Calef, D. Gerwe, and P. Crabtree, "Cramer–Rao bounds for intensity interferometry measurements," Appl. Opt. 52, 5235-5246 (2013)
http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-52-21-5235


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References

  1. R. Hanbury Brown, The Intensity Interferometer—Its Application to Astronomy (Halstead, 1974).
  2. A. Labeyrie, S. G. Lipson, and P. Nisenson, An Introduction to Astronomical Interferometry (Cambridge University, 2006), Chap. 7.
  3. R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” Opt. Acta 16, 423–451 (1969). [CrossRef]
  4. R. Hanbury Brown, R. C. Jennison, and M. K. Das Gupta, “Apparent angular sizes of discrete radio sources,” Nature 170, 1061–1063 (1952). [CrossRef]
  5. R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity in light fluctuations. I. Basic theory: the correlation between photons in coherent beams of radiation,” Proc. R. Soc. A 242, 300–324 (1957). [CrossRef]
  6. R. Hanbury Brown and R. Q. Twiss, “Interferometry of the intensity fluctuations in light. II. An experimental test of the theory for partially coherent light,” Proc. R. Soc. A 243, 291–319 (1958). [CrossRef]
  7. V. Herrero, “Design of optical telescope arrays for intensity interferometry,” Astron. J. 76, 198–201 (1971). [CrossRef]
  8. P. R. Fontana, “Multi-detector intensity interferometers,” J. Appl. Phys. 54, 473–480 (1983). [CrossRef]
  9. S. M. Ebstein, “Recovery of spatial coherence modulus and phase from complex field correlations: fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1442–1449 (1991). [CrossRef]
  10. S. M. Ebstein, “High-light-level variance of estimators for intensity interferometry and fourth-order correlation interferometry,” J. Opt. Soc. Am. A 8, 1450–1456 (1991). [CrossRef]
  11. A. Ofir and E. N. Ribak, “Off-line, multi-detector intensity interferometers I: theory,” Mon. Not. R. Astron. Soc. 368, 1646–1651 (2006). [CrossRef]
  12. S. LeBohec and J. Holder, “Optical intensity interferometry with atmospheric Cerenkov telescope arrays,” Astrophys. J. 649, 399–405 (2006). [CrossRef]
  13. D. Dravins and S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008). [CrossRef]
  14. S. LeBohec, C. Barbieri, W.-J. de Wit, D. Dravins, P. Feautrier, C. Foellmi, A. Glindemann, J. Hall, J. Holder, R. Holmes, P. Kervella, D. Kieda, E. Le Coarer, S. Lipson, F. Malbet, S. Morel, P. Nunez, A. Ofir, E. Ribak, S. Saha, M. Schoeller, B. Zhilyaev, and H. Zinnecker, “Toward a revival of stellar intensity interferometry,” Proc. SPIE 7013, 70132E (2008). [CrossRef]
  15. H. Jensen, D. Dravins, S. LeBohec, and P. D. Nuñez, “Stellar intensity interferometry: optimizing air Cherenkov telescope array layouts,” Proc. SPIE 7734, 77341T (2010). [CrossRef]
  16. P. D. Nuñez, S. LeBohec, D. Kieda, R. Holmes, H. Jensen, and D. Dravins, “Stellar intensity interferometry: imaging capabilities of air Cherenkov telescope arrays,” Proc. SPIE 7734, 77341C (2010). [CrossRef]
  17. P. D. Nuñez, R. Holmes, D. Kieda, and S. LeBohec, “High angular resolution imaging with stellar intensity interferometry using air Cherenkov telescope arrays,” Mon. Not. R. Astron. Soc. 419, 172–183 (2012). [CrossRef]
  18. P. J. McNicholl and P. D. Dao, “Improved correlation determination for intensity interferometers,” Proc. SPIE 8033, 803313 (2011). [CrossRef]
  19. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), Section 9.5.
  20. R. H. T. Bates, “Contributions to the theory of intensity interferometry,” Mon. Not. R. Astron. Soc. 142, 413–428 (1969).
  21. A. S. Marathay, Y. Hu, and L. Shao, “Phase function of spatial coherence from second-, third-, fourth-order intensity correlations,” Opt. Eng. 33, 3265–3271 (1994). [CrossRef]
  22. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982). [CrossRef]
  23. H. B. Deighton, M. S. Scivier, and M. A. Fiddy, “Solution of the two-dimensional phase-retrieval problem,” Opt. Lett. 10, 250–251 (1985). [CrossRef]
  24. D. Israelevitz and J. S. Lim, “A new direct algorithm for image reconstruction from Fourier transform magnitude,” IEEE Trans. Acoust. Speech Signal Process 35, 511–519 (1987). [CrossRef]
  25. R. G. Lane, W. R. Fright, and R. H. T. Bates, “Direct phase retrieval,” IEEE Trans. Acoust. Speech Signal Process 35, 520–526 (1987). [CrossRef]
  26. R. G. Lane and R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987). [CrossRef]
  27. P. Chen, M. A. Fiddy, A. H. Greenaway, and Y. Wang, “Zero estimation for blind deconvolution from noisy sampled data,” Proc. SPIE 2029, 14–22 (1993). [CrossRef]
  28. P. J. Bones, C. R. Parker, B. L. Satherley, and R. W. Watson, “Deconvolution and phase retrieval with use of zero sheets,” J. Opt. Soc. Am. A 12, 1842–1857 (1995). [CrossRef]
  29. D. C. Ghiglia, L. A. Romero, and G. A. Mastin, “Systematic approach to two-dimensional blind deconvolution by zero-sheet separation,” J. Opt. Soc. Am. A 10, 1024–1036 (1993). [CrossRef]
  30. T. Isernia, G. Leone, R. Pierri, and F. Soldovieri, “Role of support information and zero locations in phase retrieval by a quadratic approach,” J. Opt. Soc. Am. A 16, 1845–1856 (1999). [CrossRef]
  31. E. M. Vartiainen, K. Peiponen, and T. Asakura, “Phase retrieval in optical spectroscopy: resolving optical constants from power spectra,” Appl. Spectrosc. 50, 1283–1289 (1996). [CrossRef]
  32. H. H. Bauschke, P. L. Combettes, and D. Russell Luke, “Hybrid projection–reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003). [CrossRef]
  33. R. B. Holmes and M. S. Belen’kii, “Investigation of the Cauchy–Riemann equations for one-dimensional image recovery in intensity interferometry,” J. Opt. Soc. Am. A 21, 697–706 (2004). [CrossRef]
  34. É. Thiébaut, “Image reconstruction with optical interferometers,” New Astron. Rev. 53, 312–328 (2009). [CrossRef]
  35. R. B. Holmes, S. Lebohec, and P. D. Nunez, “Two-dimensional image recovery in intensity interferometry using the Cauchy-Riemann relations,” Proc. SPIE 7818, 78180O (2010). [CrossRef]
  36. J. Murray-Krezan and P. N. Crabtree, “Effects of aberrations on image reconstruction of data from hybrid intensity interferometers,” Proc. SPIE 8407, 840703 (2012). [CrossRef]
  37. J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Trans. Inf. Theory 36, 1285–1301 (1990). [CrossRef]
  38. A. O. Hero, J. A. Fessler, and M. Usman, “Exploring estimator bias-variance tradeoffs using the uniform Cramér–Rao bound,” IEEE Trans. Signal Process. 44, 2026–2041 (1996). [CrossRef]
  39. C. L. Matson and A. Haji, “Biased Cramér–Rao lower bound calculations for inequality-constrained estimators,” J. Opt. Soc. Am. A 23, 2702–2712 (2006). [CrossRef]
  40. B. Calef, “Estimator properties and performance bounds,” EVITA Technical Report (2005).
  41. B. Calef, “Quantifying the benefits of positivity,” Proc. SPIE 5896, 589605 (2005). [CrossRef]
  42. E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).
  43. J. Zmuidzinas, “Cramer–Rao sensitivity limits for astronomical instruments: implications for interferometer design,” J. Opt. Soc. Am. A 20, 218–233 (2003). [CrossRef]
  44. M. Moeneclaey, “On the true and the modified Cramer–Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters,” IEEE Trans. Commun. 46, 1536–1544 (1998). [CrossRef]
  45. S. Prasad, “On Bayesian and other related Cramer–Rao lower bounds on statistical estimation errors,” Technical report (University of New Mexico, 2004).
  46. P. Stoica and B. C. Ng, “On the Cramér–Rao bound under parametric constraints,” IEEE Signal Process. Lett. 5, 177–179 (1998). [CrossRef]
  47. Z. Ben Haim and Y. C. Eldar, “On the constrained Cramér–Rao bound with a singular Fisher information matrix,” IEEE Signal Process. Lett. 16, 453–456 (2009). [CrossRef]
  48. Y. Wang, G. Leus, and A. Van der Veen, “Cramer-Rao bound for range estimation,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing (IEEE, 2009), pp. 3301–3304.
  49. R. Miller and C. Chang, “A modified Cramér-Rao bound and its applications,” IEEE Trans. Inf. Theor. 24, 398–400 (1978). [CrossRef]
  50. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, 1968), Vol. I, pp. 54–87.
  51. B. Z. Bobrovsky, E. Mayer-Wolf, and M. Zakai, “Some classes of global Cramer-Rao bounds,” Ann. Statist. 15, 1421–1438 (1987). [CrossRef]
  52. H. L. Van Trees and K. L. Bell, eds., Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking (Wiley, 2007).
  53. P. Yi-Shi Shao and W. E. Strawderman, “Improving on the James-Stein positive part estimator,” Ann. Statist. 22, 1517–1538 (1994). [CrossRef]

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