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

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2012 (2)

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]

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]

2011 (1)

P. J. McNicholl and P. D. Dao, “Improved correlation determination for intensity interferometers,” Proc. SPIE 8033, 803313 (2011).
[CrossRef]

2010 (3)

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]

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]

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]

2009 (2)

É. Thiébaut, “Image reconstruction with optical interferometers,” New Astron. Rev. 53, 312–328 (2009).
[CrossRef]

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]

2008 (2)

D. Dravins and S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

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]

2006 (3)

A. Ofir and E. N. Ribak, “Off-line, multi-detector intensity interferometers I: theory,” Mon. Not. R. Astron. Soc. 368, 1646–1651 (2006).
[CrossRef]

S. LeBohec and J. Holder, “Optical intensity interferometry with atmospheric Cerenkov telescope arrays,” Astrophys. J. 649, 399–405 (2006).
[CrossRef]

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]

2005 (1)

B. Calef, “Quantifying the benefits of positivity,” Proc. SPIE 5896, 589605 (2005).
[CrossRef]

2004 (1)

2003 (2)

1999 (1)

1998 (2)

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]

P. Stoica and B. C. Ng, “On the Cramér–Rao bound under parametric constraints,” IEEE Signal Process. Lett. 5, 177–179 (1998).
[CrossRef]

1996 (2)

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]

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]

1995 (1)

1994 (2)

P. Yi-Shi Shao and W. E. Strawderman, “Improving on the James-Stein positive part estimator,” Ann. Statist. 22, 1517–1538 (1994).
[CrossRef]

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]

1993 (2)

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]

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]

1991 (2)

1990 (1)

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Trans. Inf. Theory 36, 1285–1301 (1990).
[CrossRef]

1987 (4)

R. G. Lane and R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

B. Z. Bobrovsky, E. Mayer-Wolf, and M. Zakai, “Some classes of global Cramer-Rao bounds,” Ann. Statist. 15, 1421–1438 (1987).
[CrossRef]

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]

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]

1985 (1)

1983 (1)

P. R. Fontana, “Multi-detector intensity interferometers,” J. Appl. Phys. 54, 473–480 (1983).
[CrossRef]

1982 (1)

1978 (1)

R. Miller and C. Chang, “A modified Cramér-Rao bound and its applications,” IEEE Trans. Inf. Theor. 24, 398–400 (1978).
[CrossRef]

1971 (1)

V. Herrero, “Design of optical telescope arrays for intensity interferometry,” Astron. J. 76, 198–201 (1971).
[CrossRef]

1969 (2)

R. Q. Twiss, “Applications of intensity interferometry in physics and astronomy,” Opt. Acta 16, 423–451 (1969).
[CrossRef]

R. H. T. Bates, “Contributions to the theory of intensity interferometry,” Mon. Not. R. Astron. Soc. 142, 413–428 (1969).

1958 (1)

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]

1957 (1)

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]

1952 (1)

R. Hanbury Brown, R. C. Jennison, and M. K. Das Gupta, “Apparent angular sizes of discrete radio sources,” Nature 170, 1061–1063 (1952).
[CrossRef]

Asakura, T.

Barbieri, C.

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]

Bates, R. H. T.

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]

R. G. Lane and R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

R. H. T. Bates, “Contributions to the theory of intensity interferometry,” Mon. Not. R. Astron. Soc. 142, 413–428 (1969).

Bauschke, H. H.

Belen’kii, M. S.

Ben Haim, Z.

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]

Bobrovsky, B. Z.

B. Z. Bobrovsky, E. Mayer-Wolf, and M. Zakai, “Some classes of global Cramer-Rao bounds,” Ann. Statist. 15, 1421–1438 (1987).
[CrossRef]

Bones, P. J.

Calef, B.

B. Calef, “Quantifying the benefits of positivity,” Proc. SPIE 5896, 589605 (2005).
[CrossRef]

B. Calef, “Estimator properties and performance bounds,” EVITA Technical Report (2005).

Casella, G.

E. L. Lehmann and G. Casella, Theory of Point Estimation (Springer, 1998).

Chang, C.

R. Miller and C. Chang, “A modified Cramér-Rao bound and its applications,” IEEE Trans. Inf. Theor. 24, 398–400 (1978).
[CrossRef]

Chen, P.

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]

Combettes, P. L.

Crabtree, P. N.

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]

Dao, P. D.

P. J. McNicholl and P. D. Dao, “Improved correlation determination for intensity interferometers,” Proc. SPIE 8033, 803313 (2011).
[CrossRef]

Das Gupta, M. K.

R. Hanbury Brown, R. C. Jennison, and M. K. Das Gupta, “Apparent angular sizes of discrete radio sources,” Nature 170, 1061–1063 (1952).
[CrossRef]

de Wit, W.-J.

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]

Deighton, H. B.

Dravins, D.

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]

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]

D. Dravins and S. LeBohec, “Toward a diffraction-limited square-kilometer optical telescope: digital revival of intensity interferometry,” Proc. SPIE 6986, 698609 (2008).
[CrossRef]

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]

Ebstein, S. M.

Eldar, Y. C.

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]

Feautrier, P.

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]

Fessler, J. A.

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]

Fiddy, M. A.

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]

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]

Fienup, J. R.

Foellmi, C.

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]

Fontana, P. R.

P. R. Fontana, “Multi-detector intensity interferometers,” J. Appl. Phys. 54, 473–480 (1983).
[CrossRef]

Fright, W. R.

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]

Ghiglia, D. C.

Glindemann, A.

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]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985), Section 9.5.

Gorman, J. D.

J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Trans. Inf. Theory 36, 1285–1301 (1990).
[CrossRef]

Greenaway, A. H.

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]

Haji, A.

Hall, J.

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]

Hanbury Brown, R.

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]

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J. D. Gorman and A. O. Hero, “Lower bounds for parametric estimation with constraints,” IEEE Trans. Inf. Theory 36, 1285–1301 (1990).
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V. Herrero, “Design of optical telescope arrays for intensity interferometry,” Astron. J. 76, 198–201 (1971).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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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).
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H. Jensen, D. Dravins, S. LeBohec, and P. D. Nuñez, “Stellar intensity interferometry: optimizing air Cherenkov telescope array layouts,” Proc. SPIE 7734, 77341T (2010).
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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).
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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).
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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).
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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).
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Figures (6)

Fig. 1.
Fig. 1.

Normalized CRB and ML variance versus normalized parameter θ/σ, where θ=|O(k)|2. Solid line: CRB0, Eq. (15); dashed line: CRB0m, with modified PDF, Eq. (18); dotted line, variance of positivity-constrained ML estimator, Eq. (19). X’s: simulation results: 104 realizations, for σ2=1/30, for θ in [0, 1].

Fig. 2.
Fig. 2.

Normalized CRBs versus normalized parameter θ/σ, where θ=|O(k)|2, suitably normalized. Solid line: CRB0Mm with averaging and then applying positivity, Eq. (21b); dashed line: CRB0Mm from applying positivity and then averaging, Eq. (25). M=30 measurements, σ=0.2 for each measurement.

Fig. 3.
Fig. 3.

Normalized RMS error of estimated minus truth parameter value, versus SNR where θ=|O(k)|2=1, for various averaging and positivity combinations, from simulation. Solid curve: averaging only; dashed curve: averaging, then application of positivity; dashed–dotted curve: positivity, then averaging; dotted curve: Normalized standard deviation for positivity, then averaging. M=100 measurements.

Fig. 4.
Fig. 4.

Plot of CRB01, Eq. (32), with both a lower bound of zero and an upper bound of one, versus normalized measurement. Solid line: σ2=0.01; dashed line: σ2=0.09; dotted line: σ=1. Heavy solid and dashed lines are the corresponding unconstrained variances for σ2=0.01 and σ2=0.09, respectively.

Fig. 5.
Fig. 5.

Plot of prior-averaged CRB, CRBpa,01(σ), Eq. (34), with both a lower bound of zero and an upper bound of one, versus measurement standard deviation (σ), when measurements θ are normalized so that the maximum measured value (at 0 spatial frequency (DC) in the u-v plane) is 1.

Fig. 6.
Fig. 6.

Analytic mean of the constrained ML estimate over ensemble of noise realizations, Eq. (38), in the case of constraints at both 0 and 1, for various values of the parameter |O(k)|2/|O(k=0)|2. Dashed line: σ/|O(k=0)|2=0.1; dotted line: σ/|O(k=0)|2=0.3; dotted–dashed line: σ/|O(k=0)|2=1.

Equations (51)

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CRB(θ)=diag{(I+θb)F1(I+θb)T},
Fij=θiln(f(x|θ)θjln(f(x|θ)x=θiθjln(f(x|θ)x,
θb=θ[T(x)f(x|θ)dxθ]=T(x)θf(x|θ)dxI,
f(x|θ)=[1/(2πdet(Σ))M/2]exp[(1/2)(xθ)Σ1(xθ)T],
θln[f(x|θ)]=1Σ1(xθ)T,
θ2ln[f(x|θ)]=1Σ11T,
F(θ)=f(x|θ)θ2ln[f(x|θ)]dx=1Σ11T,
CRB(θ)=11Σ11T,
CRB(θ)=1Σi=1M1σi2,
CRB(θ)=σ2/M.
Tpc(x)=0x0=xforx>0
CRB(θ)=[θΨT(θ)]2/F(θ),
ΨT,pc(θ)=T(x|θ)x=T(x)f(x|θ)dx=00f(x|θ)dx+0xf(x|θ)dx=0xexp[(xθ)2/(2σ2)]dx/2πσ2=[σ2/2π]1/2exp[θ2/(2σ2)]+θ/2{1+erf[θ/(2σ)]}.
θΨT,pc(θ)=(1/2){1+erf[θ/(2σ)]}
CRB0(θ)=(σ2/4){1+erf[θ/(2σ)]}2.
fm(x|θ)=[0exp[(xθ)2(2σ2)]dxδ(x)+exp[(xθ)2(2σ2)]H(x)]/2πσ2,
Fm(θ)=(1/σ2)((1/2){1+erf[θ/(2σ)]}[θ/(2πσ2)1/2]exp[θ2/(2σ2)]+exp(θ2/σ2)erfc[θ/(2σ)]/π)1/σ2F0(θ,σ).
CRB0m(θ)=(σ2/4){1+erf[θ/(2σ)]}2/F0(θ,σ).
Var(T(x|θ))x=T(x|θ)2x[T(x|θ)x]2.
Tpc(x|θ)2x=Tpc(x)2f(x|θ)dx=002f(x|θ)dx+0x2f(x|θ)dx=0x2exp[(xθ)2/(2σ2)]dx/2πσ2=0θx2exp[x2(2σ2)]dx/2πσ2+σ2/2+2θ(σ2/(2π))1/2exp[θ2/(2σ2)]+θ2/2{1+erf[θ/(2σ)]}.
CRBM0(θ)=[(σ2/(4M)]{1+erf[(θ/σ)/2/M]}2,
CRBM0m(θ)=[(σ2/(4M)]{1+erf[(θ/σ)/2/M]}2/F0(θ,σ/M).
f(x|θ)=Πi=1M{0exp[(xiθ)2/(2σ2)]dxδ(xi)+exp[(xiθ)2(2σ2)]H(xi)}/2πσ2M,
θΨM(θ)x=(1/2){1+erf[θ/(2σ)]}.
FMm(θ)=M/σ2F0(θ,σ).
CRB0Mm(θ)=[σ2/(4M)]{1+erf[θ/(2σ)]}2/F0(θ,σ).
f1(x|β)dθf(x|θ)g(θ)=dθexp[(xθ)2/(2σ2)]exp[(θβ)2/(2σβ2)]/[(2πσ2)(2πσβ2)]1/2=exp[(xβ)2/(2σ2)]/(2πσ2)1/2,
f2(x|θ)=[1/(2πdet(Σ))M/2]exp[(1/2)(xθ)Σ1(xθ)T],
Σ=[σ200σβ2]
CRBp(θ)=1/(1/σ2+1/σβ2).
Var(T)[θΨT(θ)θ]2/{θln[f(x|θ)]}2xθ,
θΨT,01(θ)=(1/2){erf[(1θ)/(2σ)]+erf[θ/(2σ)]},
CRB01(θ)=(σ2/4){erf[(1θ)/(2σ)]+erf[θ/(2σ)]}2.
θΨT,01(θ)θ=(1/2)01dθ{erf[(1θ)/(2σ)]+erf[θ/(2σ)]}=(1/2){01dθerf[(1θ)/(2σ)]+01dθerf[θ/(2σ)]}=(1/2){01duerf[(1θ)/(2σ)]+01dθ  erf[θ/(2σ)]}=01duerf[u/(2σ)]=erf[1/(2σ)]+(exp{[1/(2σ)2]}1)(2σ2/π)1/2,
CRBpa,01(σ)=σ2{erf[1/(2σ)]+(exp{[1/(2σ)2]}1)(2σ2/π)1/2}2.
CRBpa(σ,0,θmax)={σ/θmax}2{θmaxerf[θmax/(2σ)]+(exp{[θmax/(2σ)]2}1)(2σ2/π)1/2}2.
CRBpa(σ,0,θmax){σ/θmax}2{θmax[θmax/(2σ)]2/π1/2+{[θmax/(2σ)]2}(2σ2/π)1/2}2=(1/2π)θmax2+O(θmax4),
SNR(2π)1/2θ/θmax.
ΨT,0,θmax(θ)=T(x)f(x|θ)dx=(σ2/(2π))1/2{exp[θ2/(2σ2)]exp[(θmaxθ)2/(2σ2)]}+(θ/2){erf[θ/(2σ)]+erf[(θmaxθ)/(2σ)]}+(θmax/2)erfc[(θmaxθ)/(2σ)].
f0,θmax(x|θ)={0exp((xθ)22σ2)dxδ(x)+θmaxexp((xθ)2(2σ2))dxδ(xθmax)+exp[(xθ)2/(2σ2)][H(x)H(xθmax)]}(2πσ2)1/2.
F0,θmax(θ,σ)1/σ2[F0(θ,σ)+F0(θmaxθ,σ)1],
ΨT(θ)=T(x)f(x|θ)dx.
θΨT(θ)=T(x)θf(x|θ)dx.
ΨT(θ)θf(x|θ)dx=ΨT(θ)θf(x|θ)dx=ΨT(θ)θf(x|θ)dx=ΨT(θ)θ(1)=0.
dθg(θ)[ΨT(θ)θf(x|θ)dx]=d(θ)g(θ)[0]=0=d(θ)g(θ)ΨT(θ)[θf(x|θ)dx]=ΨT(θ)θθf(x|θ)dx.
θΨT(θ)=[T(x)]ΨT(θ)θ]θf(x|θ)dx.
θΨT(θ)=[T(x)ΨT(θ)θ]{θln[f(x|θ)]}f(x|θ)dx.
θΨT(θ)θ=dθdx[T(x)ΨT(θ)θ]{θln[f(x|θ)]}f(x|θ)g(θ).
θΨT(θ)θ=dθdx{[T(x)ΨT(θ)θ][f(x|θ)g(θ)]1/2}{θln[f(x|θ)][f(x|θ)g(θ)]1/2}.
θΨT(θ)θ=dθdx{[T(x)ΨT(θ)θ][f(x|θ)g(θ)]1/2}{θln[f(x|θ)][f(x|θ)g(θ)]}1/2{dθdx[T(x)ΨT(θ)θ]2[f(x|θ)g(θ)]}1/2{dθdx{θln[f(x|θ)]}2[f(x|θ)g(θ)]}1/2.
Var(T)[θΨT(θ)θ]2/{θln[f(x,θ)]}2xθ

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