Abstract

Current optical interferometers are affected by unknown turbulent phases on each telescope. In the field of radio interferometry, the self-calibration technique is a powerful tool to process interferometric data with missing phase information. This paper intends to revisit the application of self-calibration to optical long-baseline interferometry (OLBI). We cast rigorously the OLBI data processing problem into the self-calibration framework and demonstrate the efficiency of the method on a real astronomical OLBI data set.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Monnier, “An introduction to closure phases,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 13, pp. 203-239.
  2. A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio-Astronomy (Wiley Interscience, 1986).
  3. A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183-198 (1997).
    [CrossRef]
  4. J. Hogbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417-426 (1974).
  5. T. J. Cornwell and P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067-1086 (1981).
  6. A. Lannes, “Weak-phase imaging in optical interferometry,” J. Opt. Soc. Am. A 15, 811-824 (1998).
    [CrossRef]
  7. J. W. Goodman, Statistical Optics (Wiley-Interscience, 1985).
  8. D. L. Fried, “Statistics of a geometric representation of wavefront distortion,” J. Opt. Soc. Am. 55, 1427-1435 (1965).
    [CrossRef]
  9. A. Quirrenbach, “Phase referencing,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 9, pp. 143-160.
  10. R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276-284 (1958).
  11. T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
    [CrossRef]
  12. G. Perrin, “The calibration of interferometric visibilities obtained with single-mode optical interferometers. Computation of error bars and correlations,” Astron. Astrophys. 400, 1173-1181 (2003).
    [CrossRef]
  13. C. Haniff, “Least-squares Fourier phase estimation from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 8, 134-140 (1991).
    [CrossRef]
  14. G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).
  15. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
    [CrossRef]
  16. R. Nityananda and R. Narayan, “Maximum entropy image reconstruction--a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419-450 (1982).
    [CrossRef]
  17. E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
    [CrossRef]
  18. J. Navaza, “Accurate solutions of the maximum entropy equations. Their impact on the foundations of direct methods,” in Crystallographic Computing 5: From Chemistry to Biology, Oxford Internaturial Union of Crystallography Book Series (Oxford University Press, 1991), pp. 317-323.
  19. Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995).
  20. S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Convex approximation of the likelihood in optical interferometry,” J. Opt. Soc. Am. A 22, 2348-2356 (2005).
    [CrossRef]
  21. L. M. Mugnier, G. Le Besnerais, and S. Meimon, “Inversion in optical imaging through atmospheric turbulence,” in Bayesian Approach to Inverse Problems, J.Idier, ed., Digital Signal and Image Processing Series (ISTE/Wiley, 2008), Chap. 10, pp. 243-283.
  22. A. Lannes, “Integer ambiguity resolution in phase closure imaging,” J. Opt. Soc. Am. A 18, 1046-1055 (2001).
    [CrossRef]
  23. P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
    [CrossRef]
  24. J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
    [CrossRef]
  25. F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
    [CrossRef]
  26. G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
    [CrossRef]
  27. S. Lacour, “Imagerie des étoiles évoluées par interférométrie. Réarrangement de pupille,” Ph.D. thesis (University of Paris VI, 2007).

2006

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

2005

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Convex approximation of the likelihood in optical interferometry,” J. Opt. Soc. Am. A 22, 2348-2356 (2005).
[CrossRef]

2004

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

2003

J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
[CrossRef]

E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
[CrossRef]

G. Perrin, “The calibration of interferometric visibilities obtained with single-mode optical interferometers. Computation of error bars and correlations,” Astron. Astrophys. 400, 1173-1181 (2003).
[CrossRef]

2001

1998

1997

A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183-198 (1997).
[CrossRef]

1991

1989

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
[CrossRef]

1982

R. Nityananda and R. Narayan, “Maximum entropy image reconstruction--a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419-450 (1982).
[CrossRef]

1981

T. J. Cornwell and P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067-1086 (1981).

1974

J. Hogbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417-426 (1974).

1965

1958

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276-284 (1958).

Anterrieu, E.

A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183-198 (1997).
[CrossRef]

Bar-Shalom, Y.

Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995).

Benson, J. A.

J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
[CrossRef]

Berger, J.-P.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Carleton, N. P.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell and P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067-1086 (1981).

Cotton, W.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Cotton, W. D.

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Coudé du Foresto, V.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
[CrossRef]

Foy, R.

E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
[CrossRef]

Fried, D. L.

Garcia, P. J. V.

E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
[CrossRef]

Goodman, J. W.

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

Hagenauer, P.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Haniff, C.

Hogbom, J.

J. Hogbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417-426 (1974).

Hummel, C. A.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
[CrossRef]

Jennison, R. C.

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276-284 (1958).

Kern, P. Y.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Labeye, P. R.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Lacasse, M.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Lacasse, M. G.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Lacour, S.

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

S. Lacour, “Imagerie des étoiles évoluées par interférométrie. Réarrangement de pupille,” Ph.D. thesis (University of Paris VI, 2007).

Lannes, A.

Lawson, P. R.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Le Besnerais, G.

S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Convex approximation of the likelihood in optical interferometry,” J. Opt. Soc. Am. A 22, 2348-2356 (2005).
[CrossRef]

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

L. M. Mugnier, G. Le Besnerais, and S. Meimon, “Inversion in optical imaging through atmospheric turbulence,” in Bayesian Approach to Inverse Problems, J.Idier, ed., Digital Signal and Image Processing Series (ISTE/Wiley, 2008), Chap. 10, pp. 243-283.

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

Li, X.-R.

Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995).

Malbet, F.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Maréchal, P.

A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183-198 (1997).
[CrossRef]

Meimon, S.

S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Convex approximation of the likelihood in optical interferometry,” J. Opt. Soc. Am. A 22, 2348-2356 (2005).
[CrossRef]

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

L. M. Mugnier, G. Le Besnerais, and S. Meimon, “Inversion in optical imaging through atmospheric turbulence,” in Bayesian Approach to Inverse Problems, J.Idier, ed., Digital Signal and Image Processing Series (ISTE/Wiley, 2008), Chap. 10, pp. 243-283.

Meimon, S. C.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Mennesson, B.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Millan-Gabet, R.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Millan-Galbet, R.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Monnier, J. D.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

J. D. Monnier, “An introduction to closure phases,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 13, pp. 203-239.

Moran, J. M.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio-Astronomy (Wiley Interscience, 1986).

Mozurkewich, D.

J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
[CrossRef]

Mugnier, L.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Mugnier, L. M.

S. Meimon, L. M. Mugnier, and G. Le Besnerais, “Convex approximation of the likelihood in optical interferometry,” J. Opt. Soc. Am. A 22, 2348-2356 (2005).
[CrossRef]

L. M. Mugnier, G. Le Besnerais, and S. Meimon, “Inversion in optical imaging through atmospheric turbulence,” in Bayesian Approach to Inverse Problems, J.Idier, ed., Digital Signal and Image Processing Series (ISTE/Wiley, 2008), Chap. 10, pp. 243-283.

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

Narayan, R.

R. Nityananda and R. Narayan, “Maximum entropy image reconstruction--a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419-450 (1982).
[CrossRef]

Navaza, J.

J. Navaza, “Accurate solutions of the maximum entropy equations. Their impact on the foundations of direct methods,” in Crystallographic Computing 5: From Chemistry to Biology, Oxford Internaturial Union of Crystallography Book Series (Oxford University Press, 1991), pp. 317-323.

Nityananda, R.

R. Nityananda and R. Narayan, “Maximum entropy image reconstruction--a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419-450 (1982).
[CrossRef]

Pauls, T. A.

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

Pearlman, M. R.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Pedretti, E.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Perrin, G.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

G. Perrin, “The calibration of interferometric visibilities obtained with single-mode optical interferometers. Computation of error bars and correlations,” Astron. Astrophys. 400, 1173-1181 (2003).
[CrossRef]

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

Quirrenbach, A.

A. Quirrenbach, “Phase referencing,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 9, pp. 143-160.

Ragland, S. D.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Ridgway, S.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Rousselet-Perraut, K.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Schloerb, F. P.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Schuller, P.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Schuller, P. A.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Swenson, G. W.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio-Astronomy (Wiley Interscience, 1986).

Thiébaut, E.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
[CrossRef]

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

Thompson, A. R.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio-Astronomy (Wiley Interscience, 1986).

Thorsteinsson, H.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Traub, W.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Traub, W. A.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Tuthill, P. G.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Verhoelst, T.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Wallace, G.

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Wilkinson, P. N.

T. J. Cornwell and P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067-1086 (1981).

Woillez, J.

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Young, J. S.

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Zhao, M.

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

Astron. Astrophys.

G. Perrin, “The calibration of interferometric visibilities obtained with single-mode optical interferometers. Computation of error bars and correlations,” Astron. Astrophys. 400, 1173-1181 (2003).
[CrossRef]

G. Perrin, S. Ridgway, B. Mennesson, W. Cotton, J. Woillez, T. Verhoelst, P. Schuller, V. Coudé du Foresto, W. Traub, R. Millan-Galbet, and M. Lacasse, “Unveiling Mira stars behind the molecules. Confirmation of the molecular layer model with narrow band near-infrared interferometry,” Astron. Astrophys. 426, 279-296 (2004).
[CrossRef]

Astron. Astrophys. Suppl. Ser.

A. Lannes, E. Anterrieu, and P. Maréchal, “Clean and wipe,” Astron. Astrophys. Suppl. Ser. 123, 183-198 (1997).
[CrossRef]

J. Hogbom, “Aperture synthesis with a non-regular distribution of interferometer baselines,” Astron. Astrophys. Suppl. Ser. 15, 417-426 (1974).

Astrophys. Space Sci.

E. Thiébaut, P. J. V. Garcia, and R. Foy, “Imaging with Amber/VLTI: the case of microjets,” Astrophys. Space Sci. 286, 171-176 (2003).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024-2036 (1989).
[CrossRef]

J. Astrophys. Astron.

R. Nityananda and R. Narayan, “Maximum entropy image reconstruction--a practical non-information-theoretic approach,” J. Astrophys. Astron. 3, 419-450 (1982).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Mon. Not. R. Astron. Soc.

T. J. Cornwell and P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067-1086 (1981).

R. C. Jennison, “A phase sensitive interferometer technique for the measurement of the Fourier transforms of spatial brightness distribution of small angular extent,” Mon. Not. R. Astron. Soc. 118, 276-284 (1958).

Proc. SPIE

P. R. Lawson, W. D. Cotton, C. A. Hummel, J. D. Monnier, M. Zhao, J. S. Young, H. Thorsteinsson, S. C. Meimon, L. Mugnier, G. Le Besnerais, E. Thiébaut, and P. G. Tuthill, “An interferometry imaging beauty contest,” Proc. SPIE 5491, 886-899 (2004).
[CrossRef]

J. A. Benson, C. A. Hummel, and D. Mozurkewich, “Simultaneous 6-station observations with the NPOI,” Proc. SPIE 4838, 358-368 (2003).
[CrossRef]

F. P. Schloerb, J.-P. Berger, N. P. Carleton, P. Hagenauer, P. Y. Kern, P. R. Labeye, M. G. Lacasse, F. Malbet, R. Millan-Gabet, J. D. Monnier, M. R. Pearlman, E. Pedretti, K. Rousselet-Perraut, S. D. Ragland, P. A. Schuller, W. A. Traub, and G. Wallace, “IOTA: recent science and technology,” Proc. SPIE 6268, 62680I (2006).
[CrossRef]

Publ. Astron. Soc. Pac.

T. A. Pauls, J. S. Young, W. D. Cotton, and J. D. Monnier, “A data exchange standard for optical (visible/IR) interferometry,” Publ. Astron. Soc. Pac. 117, 1255-1262 (2005).
[CrossRef]

Other

J. D. Monnier, “An introduction to closure phases,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 13, pp. 203-239.

A. R. Thompson, J. M. Moran, and G. W. Swenson, Jr., Interferometry and Synthesis in Radio-Astronomy (Wiley Interscience, 1986).

A. Quirrenbach, “Phase referencing,” in Principles of Long Baseline Stellar Interferometry, P.R.Lawson, ed. (Jet Propulsion Laboratory, 1999), Chap. 9, pp. 143-160.

G. Le Besnerais, S. Lacour, L. M. Mugnier, E. Thiébaut, G. Perrin, and S. Meimon, “Advanced imaging methods for long-baseline optical interferometry,” IEEE J. Sel. Top. Signal Process. (to be published).

J. Navaza, “Accurate solutions of the maximum entropy equations. Their impact on the foundations of direct methods,” in Crystallographic Computing 5: From Chemistry to Biology, Oxford Internaturial Union of Crystallography Book Series (Oxford University Press, 1991), pp. 317-323.

Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Principles and Techniques (YBS Publishing, 1995).

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

L. M. Mugnier, G. Le Besnerais, and S. Meimon, “Inversion in optical imaging through atmospheric turbulence,” in Bayesian Approach to Inverse Problems, J.Idier, ed., Digital Signal and Image Processing Series (ISTE/Wiley, 2008), Chap. 10, pp. 243-283.

S. Lacour, “Imagerie des étoiles évoluées par interférométrie. Réarrangement de pupille,” Ph.D. thesis (University of Paris VI, 2007).

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

Contour plots of a polar Gaussian distribution and of its Cartesian Gaussian approximation.

Fig. 2
Fig. 2

WISARD algorithm loop.

Fig. 3
Fig. 3

Synthetic object (right) and frequency coverage (left) from the Imaging Beauty Contest 2004.

Fig. 4
Fig. 4

Reconstructions with WISARD. Upper row, under-regularized quadratic model (left), over-regularized quadratic model (center), quadratic model with correct regularization parameter (right). Lower row, white L 2 L 1 u u model with automatically set scale and delta parameters (left), white L 2 L 1 u u model with half-scale (center), white L 2 L 1 u u model with half-delta (right). Each image field is 12.1 × 12.1 mas .

Fig. 5
Fig. 5

Goodness of fit at WISARD convergence.

Fig. 6
Fig. 6

Frequency coverage (left) and reconstruction of the star χ Cygni (right).

Fig. 7
Fig. 7

Behavior of a ̂ in function of a 2 with a unit σ s .

Fig. 8
Fig. 8

Behavior of Var ( a ̂ ) in function of a 2 with a unit σ s .

Equations (94)

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

u k l ( t ) r l ( t ) r k ( t ) ,
y k l data ( t ) a k l data ( t ) e i ϕ k l data ( t ) .
y k l data ( t ) = η k l ( t ) FT [ x ( ξ ) ] ( ν k l ( t ) ) FT [ x ( ξ ) ] ( 0 ) .
N b = ( N t 2 ) = N t ( N t 1 ) 2
{ a ( x , t ) H ( t ) x , ϕ ( x , t ) arg { H ( t ) x } . }
ϕ k l data ( t ) = ϕ k l ( x , t ) + φ l ( t ) φ k ( t ) + noise [ 2 π ]
ϕ data ( t ) = ϕ ( x , t ) + B φ ( t ) + noise [ 2 π ] ,
f ( ϕ data ( t ) ) = f ( ϕ ( x , t ) + B φ ( t ) ) = f ( ϕ ( x , t ) ) .
{ ϕ k l data ( t ) = ϕ k l ( x , t ) + φ l ( t ) φ k ( t ) + noise [ 2 π ] , ϕ l m data ( t ) = ϕ l m ( x , t ) + φ m ( t ) φ l ( t ) + noise [ 2 π ] , ϕ m k data ( t ) = ϕ m k ( x , t ) + φ k ( t ) φ m ( t ) + noise [ 2 π ] , }
β k l m data ( t ) ϕ k l data ( t ) + ϕ l m data ( t ) + ϕ m k data ( t ) + noise [ 2 π ] = ϕ k l ( x , t ) + ϕ l m ( x , t ) + ϕ m k ( x , t ) + noise [ 2 π ] β k l m ( x , t ) + noise [ 2 π ] .
f ( ϕ ) = g ( C ϕ ) ,
β data ( t ) = C ϕ ( x , t ) + noise [ 2 π ] .
{ s data ( t ) = a 2 ( x , t ) + s noise ( t ) , s noise ( t ) N ( 0 , R s ( t ) ) , β data ( t ) = C ϕ ( x , t ) + β noise ( t ) [ 2 π ] , β noise ( t ) N ( 0 , R β ( t ) ) . }
J data ( x ) = t J data ( x , t ) = t χ s ( t ) 2 ( x ) + χ β ( t ) 2 ( x ) ,
J ( x ) = J data ( x ) + J prior ( x )
( p , q ) , x ( p , q ) 0 ,
p , q x ( p , q ) = 1 .
L 2 L 1 w ( x ) = δ 2 p , q x ( p , q ) s δ ln ( 1 + x ( p , q ) s δ )
s = 1 N pix ; δ = 1 .
s data ( t ) = a 2 ( x , t ) + s noise ( t ) , s noise ( t ) N ( 0 , R s ( t ) ) ,
β data ( t ) = C ϕ ( x , t ) + β noise ( t ) [ 2 π ] ,
β noise ( t ) N ( 0 , R β ( t ) )
y data ( t ) = F α ( t ) H ( t ) x + y noise ( t ) ,
a data ( t ) = a ( x , t ) + a noise ( t ) , a noise ( t ) N ( 0 , R a ( t ) ) .
C β data ( t ) = C C ϕ ( x , t ) + C β noise ( t ) + 2 π C κ ,
ϕ data ( t ) C β data ( t ) ,
ϕ ker ( t ) ( C C I d ) ϕ ( x , t ) + 2 π C κ
ϕ data ( t ) = ϕ ( x , t ) + ϕ ker ( t ) + C β noise ( t ) .
C ϕ ker ( t ) = ( C C = I d C C ) ϕ ( x , t ) + 2 π C C = I d κ = 2 κ π = 0 [ 2 π ] .
ϕ data ( t ) = ϕ ( x , t ) + B ¯ α ( t ) + C β noise ( t ) [ 2 π ] .
R ϕ ( t ) 0 C R β ( t ) C T .
C R ϕ ( t ) C T = R β ( t ) .
{ R ϕ ( t ) } i j = { 3 { R ϕ ( t ) 0 } i j if i = j 0 if i j } .
i , j { R ϕ ( t ) } i j = i , j { R ϕ ( t ) 0 } i j .
ϕ data ( t ) = ϕ ( x , t ) + B ¯ α ( t ) + ϕ noise ( t ) [ 2 π ] ,
ϕ noise ( t ) N ( 0 , R ϕ ( t ) ) .
{ a data ( t ) = a ( x , t ) + a noise ( t ) , ϕ data ( t ) = ϕ ( x , t ) + B ¯ α ( t ) + ϕ noise ( t ) [ 2 π ] , with a noise ( t ) N ( 0 , R a ( t ) ) , ϕ noise ( t ) N ( 0 , R ϕ ( t ) ) . }
y data ( t ) a data ( t ) e i ϕ data ( t ) .
y data ( t ) = H ( t ) x e i B ¯ α ( t ) + y noise ( t ) .
y m ( x , α ( t ) ) H ( t ) x e i B ¯ α ( t ) .
χ y ( t ) 2 ( x , α ( t ) ) [ Re { y data ( t ) y m ( x , α ( t ) ) y ¯ noise ( t ) } Im { y data ( t ) y m ( x , α ( t ) ) y ¯ noise ( t ) } ] T × R y noise ( t ) 1 [ Re { y data ( t ) y m ( x , α ( t ) ) y ¯ noise ( t ) } Im { y data ( t ) y m ( x , α ( t ) ) y ¯ noise ( t ) } ] .
J data ( x , α ) = t J data ( x , α ( t ) , t ) = t χ y ( t ) 2 ( x , α ( t ) ) .
J ( x , α ) = J data ( x , α ) + J prior ( x ) .
J data ( x , α ) = t J data ( x , α ( t ) , t ) .
B 2 [ 1 1 ] ,
B N t [ 1 N t 1 I d N t 1 O B N t 1 ] ,
C N t [ B N t 1 I d [ ( N t 1 ) ( N t 2 ) ] 2 ] ,
im B ker C .
dim im B rank B = N t 1 .
dim ker C N t N t 1 .
ker C = im B .
f ( ϕ + B φ ) = f ( ϕ ) , ( ϕ , φ )
C C = I d
C C C C = 0 C ( C C ϕ ϕ ) = 0 ,
ϕ A 7 φ , ( C C ϕ ϕ ) = B φ ,
ϕ φ , ϕ = C C ϕ B φ , ϕ .
f ( ϕ ) = f ( C C ϕ B φ ) = f ( C C ϕ ) = g ( C ϕ ) .
im B = im B ¯ ,
ker C = im B ¯ .
C ϕ ker 0 [ 2 π ] .
α 1 , ϕ ker C ( 0 [ 2 π ] ) + B ¯ α 1 [ 2 π ] .
μ [ 0 κ ]
C μ = [ I d ] [ 0 κ ] = κ .
α , ϕ ker B ¯ α [ 2 π ] .
s data = a 2 + s noise ,
a ̂ = { s data if s data > 0 0 else } ,
a ̂ = a + a noise ,
( a + σ a ) 2 = a 2 + σ s 2 a σ a σ s ,
a data = a + a noise ,
a data = { s data if s data > 0 0 else }
a ¯ = { σ s 6 if s data σ s 6 , s data if s data σ s 6 } ,
σ a = { σ s 2 if s data σ s σ s 2 s data if s data σ s } .
y α ( t ) ( x , t ) H ( t ) x e i B ¯ α ( t ) ,
{ a data ( t ) = y α ( t ) ( x , t ) + a noise ( t ) , a noise ( t ) N ( 0 , R a ( t ) ) , ϕ data ( t ) 2 π arg y α ( t ) ( x , t ) + ϕ noise ( t ) , ϕ noise ( t ) N ( 0 , R ϕ ( t ) ) . }
ϕ data = ϕ ¯ + ϕ noise ,
a data = a ¯ + a noise ,
{ y ¯ a ¯ exp i ϕ ¯ , y noise y data y ¯ , y rad n Re { y noise e i ϕ ¯ } , y tan n Im { y noise e i ϕ ¯ } , y ̿ noise [ y rad n y tan n ] , }
{ y rad n = [ a ¯ + a noise ] cos ϕ noise a ¯ , y tan n = [ a ¯ + a noise ] sin ϕ noise . }
{ y ̿ noise = E { [ y rad n y tan n ] } = [ y ¯ rad n y ¯ tan n ] , R y ̿ noise = E { [ y ¯ rad n y rad n y ¯ tan n y tan n ] [ y ¯ rad n y rad n y ¯ tan n y tan n ] T } , }
R y ̿ noise [ R rad , rad R rad , tan R rad , tan T R tan , tan ] .
E { sin ϕ i noise } = 0 ,
E { cos ϕ i noise } = exp ( R ϕ i i 2 ) ,
E { sin ϕ i noise sin ϕ j noise } = sinh R ϕ i j exp ( R ϕ i i + R ϕ j j 2 ) ,
E { cos ϕ i noise cos ϕ j noise } = cosh R ϕ i j exp ( R ϕ i i + R ϕ j j 2 ) ,
E { cos ϕ i noise sin ϕ j noise } = 0 .
E { y rad i n } = a ¯ i [ e R ϕ i i 2 1 ] ,
E { y tan i n } = 0 ,
[ R rad , rad ] i j = [ a ¯ i a ¯ j ( cosh R ϕ i j 1 ) + R a i j cosh R ϕ i j ] e [ ( R ϕ i i + R ϕ j j ) 2 ] ,
[ R rad , tan ] i j = 0 ,
[ R tan , tan ] i j = ( a ¯ i a ¯ j + R a i j ) sinh R ϕ i j e [ ( R ϕ i i + R ϕ j j ) 2 ] .
{ R a = Diag { σ a , i 2 } , R ϕ = Diag { σ ϕ , i 2 } . }
{ R rad , rad = Diag { σ rad , i 2 } , R tan , tan = Diag { σ tan , i 2 } , R rad , tan = 0 , }
σ rad , i 2 = a ¯ i 2 2 ( 1 e σ ϕ , i 2 ) 2 + σ a , i 2 2 ( 1 + e 2 σ ϕ , i 2 ) ,
σ tan , i 2 = a ¯ i 2 2 ( 1 e 2 σ ϕ , i 2 ) + σ a , i 2 2 ( 1 e 2 σ ϕ , i 2 ) .

Metrics