Abstract

We address the optimization of the relative arrangement (aperture configuration) of a phased array of optical telescopes, coherently combined to form images of extended objects in a common focal plane. A novel optimality criterion, which is directly linked to the restoration error of the original object from the recorded image, is derived. This criterion is then refined into a second criterion to accommodate the possible knowledge of the noise spectrum. The optimal configuration is a function of the maximum spatial frequency of interest (or desired resolution) and takes into account the diameters of the elementary telescopes. Simulations illustrate the usefulness of this criterion for designing a synthetic-aperture optical instrument with three, four, and five telescopes.

© 1996 Optical Society of America

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References

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  1. A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propag. AP-16, 172–175 (1968).
    [CrossRef]
  2. M. J. E. Golay, “Point arrays having compact, nonredun-dant autocorrelations,” J. Opt. Soc. Am. 61, 272–273 (1971).
    [CrossRef]
  3. T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propag. AP-36, 1165–1167 (1988).
    [CrossRef]
  4. A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
    [CrossRef]
  5. R. V. Shack, J. D. Rancourt, H. Morrow, “Effects of dilution on a six-element synthetic aperture,” Appl. Opt. 10, 257–259 (1971).
    [CrossRef] [PubMed]
  6. A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, Chap. 5, pp. 149–201.
    [CrossRef]
  7. J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Proc. SPIE643, 194–207 (1986).
    [CrossRef]
  8. A. Labeyrie, G. Lemaitre, L. Kœchlin, “The optical very large array,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 323–332 (1986).
    [CrossRef]
  9. J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased arrays and thinned aperture optical telescopes,” Opt. Eng. (Bellingham) 27, 762–768 (1988).
    [CrossRef]
  10. S. M. Watson, J. P. Mills, “Two-point resolution criterion for multiaperture optical telescopes,” J. Opt. Soc. Am. A 5, 893–903 (1988).
    [CrossRef]
  11. P. Y. Bely, “HARDI: a high angular resolution deployable interferometer for space,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 92–100 (1989).
    [CrossRef]
  12. ESO/VLT Interferometry Panel, “The VLT Interferometer Implementation Plan,” VLT Report 59b (European Southern Observatory, Garching, Germany, 1989).
  13. M. Faucherre, F. Merkle, F. Vakili, “Beam combination in aperture synthesis from space: field of view limitations and (u, υ) plane coverage optimization,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 138–145 (1989).
    [CrossRef]
  14. J. P. Fitch, T. W. Lawrence, “Placement of multiple apertures for imaging telescopes,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. SPIE1237, 61–69 (1990).
    [CrossRef]
  15. L. Damé, T.-D. Guyenne, “Study of an optimized configuration for interferometric imaging of complex and extended solar structures,” in Targets for Space-Based Interferometry (European Space Agency, Noordwijk, The Netherlands, 1992), Vol. SP-354, pp. 201–208.
  16. J. M. Beckers, F. Merkle, eds., High-Resolution Imaging by Interferometry II. Part II: Multiple Aperture Interferometry, No. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1992).
  17. J. B. Breckinridge, ed., Amplitude and Intensity Spatial In-terferometry II, Proc. SPIE 2200 (1994).
  18. A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
    [CrossRef]
  19. L. M. Mugnier, G. Rousset, “Pupil configuration opti-mality criterion in synthetic aperture optics,” in Spaceborne Interferometry II, R. D. Reasenberg, ed., Proc. SPIE2477, 124–131 (1995).
    [CrossRef]
  20. W. A. Traub, “Combining beams from separated telescopes,” Appl. Opt. 25, 528–532 (1986).
    [CrossRef] [PubMed]
  21. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  22. V. Trénoguine, Analyse Fonctionnelle (Mir, Moscow, 1985).
  23. L. Schwartz, Analyse—Topologie Générale et Analyse Fonctionnelle, No. 11 of Collection Enseignement des Sciences (Hermann, Paris, 1970).
  24. A. K. Katsaggelos, “Iterative image restoration algorithms,” Opt. Eng. (Bellingham) 28, 735–748 (1989).
    [CrossRef]
  25. B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comp. C-22, 805–812 (1973).
    [CrossRef]
  26. C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), Chap. 1.
  27. A. Papoulis, Signal Analysis (McGraw-Hill, New-York, 1977).
  28. C. Perrier, “Amplitude estimation from speckle interferometry,” in Proceedings of the NATO Advanced Study Institute on Diffraction-Limited Imaging with Very Large Telescopes, D. M. Allouin, J.-M. Mariotti, eds., Vol. 274 of Series C: Mathematical and Physical Sciences(Kluwer, Dordrecht, The Netherlands, 1989), pp. 99–111.
    [CrossRef]

1994 (1)

J. B. Breckinridge, ed., Amplitude and Intensity Spatial In-terferometry II, Proc. SPIE 2200 (1994).

1989 (2)

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

A. K. Katsaggelos, “Iterative image restoration algorithms,” Opt. Eng. (Bellingham) 28, 735–748 (1989).
[CrossRef]

1988 (3)

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased arrays and thinned aperture optical telescopes,” Opt. Eng. (Bellingham) 27, 762–768 (1988).
[CrossRef]

S. M. Watson, J. P. Mills, “Two-point resolution criterion for multiaperture optical telescopes,” J. Opt. Soc. Am. A 5, 893–903 (1988).
[CrossRef]

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propag. AP-36, 1165–1167 (1988).
[CrossRef]

1987 (1)

A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

1986 (1)

1973 (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comp. C-22, 805–812 (1973).
[CrossRef]

1971 (2)

1968 (1)

A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propag. AP-16, 172–175 (1968).
[CrossRef]

Anterrieu, É.

A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
[CrossRef]

Bely, P. Y.

P. Y. Bely, “HARDI: a high angular resolution deployable interferometer for space,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 92–100 (1989).
[CrossRef]

Bunner, A. N.

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Proc. SPIE643, 194–207 (1986).
[CrossRef]

Casanove, M.-J.

A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Cornwell, T. J.

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propag. AP-36, 1165–1167 (1988).
[CrossRef]

Damé, L.

L. Damé, T.-D. Guyenne, “Study of an optimized configuration for interferometric imaging of complex and extended solar structures,” in Targets for Space-Based Interferometry (European Space Agency, Noordwijk, The Netherlands, 1992), Vol. SP-354, pp. 201–208.

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]

Faucherre, M.

M. Faucherre, F. Merkle, F. Vakili, “Beam combination in aperture synthesis from space: field of view limitations and (u, υ) plane coverage optimization,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 138–145 (1989).
[CrossRef]

Fitch, J. P.

J. P. Fitch, T. W. Lawrence, “Placement of multiple apertures for imaging telescopes,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. SPIE1237, 61–69 (1990).
[CrossRef]

Fitoussi, G.

A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
[CrossRef]

Golay, M. J. E.

Guyenne, T.-D.

L. Damé, T.-D. Guyenne, “Study of an optimized configuration for interferometric imaging of complex and extended solar structures,” in Targets for Space-Based Interferometry (European Space Agency, Noordwijk, The Netherlands, 1992), Vol. SP-354, pp. 201–208.

Harvey, J. E.

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased arrays and thinned aperture optical telescopes,” Opt. Eng. (Bellingham) 27, 762–768 (1988).
[CrossRef]

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Proc. SPIE643, 194–207 (1986).
[CrossRef]

Hunt, B. R.

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comp. C-22, 805–812 (1973).
[CrossRef]

Katsaggelos, A. K.

A. K. Katsaggelos, “Iterative image restoration algorithms,” Opt. Eng. (Bellingham) 28, 735–748 (1989).
[CrossRef]

Kœchlin, L.

A. Labeyrie, G. Lemaitre, L. Kœchlin, “The optical very large array,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 323–332 (1986).
[CrossRef]

Koechlin, L.

A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
[CrossRef]

Labeyrie, A.

A. Labeyrie, G. Lemaitre, L. Kœchlin, “The optical very large array,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 323–332 (1986).
[CrossRef]

Lannes, A.

A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
[CrossRef]

Lawrence, T. W.

J. P. Fitch, T. W. Lawrence, “Placement of multiple apertures for imaging telescopes,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. SPIE1237, 61–69 (1990).
[CrossRef]

Lemaitre, G.

A. Labeyrie, G. Lemaitre, L. Kœchlin, “The optical very large array,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 323–332 (1986).
[CrossRef]

Meinel, A. B.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, Chap. 5, pp. 149–201.
[CrossRef]

Meinel, M. P.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, Chap. 5, pp. 149–201.
[CrossRef]

Merkle, F.

M. Faucherre, F. Merkle, F. Vakili, “Beam combination in aperture synthesis from space: field of view limitations and (u, υ) plane coverage optimization,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 138–145 (1989).
[CrossRef]

Mills, J. P.

Moffet, A. T.

A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propag. AP-16, 172–175 (1968).
[CrossRef]

Morrow, H.

Mugnier, L. M.

L. M. Mugnier, G. Rousset, “Pupil configuration opti-mality criterion in synthetic aperture optics,” in Spaceborne Interferometry II, R. D. Reasenberg, ed., Proc. SPIE2477, 124–131 (1995).
[CrossRef]

Papoulis, A.

A. Papoulis, Signal Analysis (McGraw-Hill, New-York, 1977).

Perrier, C.

C. Perrier, “Amplitude estimation from speckle interferometry,” in Proceedings of the NATO Advanced Study Institute on Diffraction-Limited Imaging with Very Large Telescopes, D. M. Allouin, J.-M. Mariotti, eds., Vol. 274 of Series C: Mathematical and Physical Sciences(Kluwer, Dordrecht, The Netherlands, 1989), pp. 99–111.
[CrossRef]

Rancourt, J. D.

Rockwell, R. A.

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased arrays and thinned aperture optical telescopes,” Opt. Eng. (Bellingham) 27, 762–768 (1988).
[CrossRef]

Roques, S.

A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Rousset, G.

L. M. Mugnier, G. Rousset, “Pupil configuration opti-mality criterion in synthetic aperture optics,” in Spaceborne Interferometry II, R. D. Reasenberg, ed., Proc. SPIE2477, 124–131 (1995).
[CrossRef]

Rushforth, C. K.

C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), Chap. 1.

Schwartz, L.

L. Schwartz, Analyse—Topologie Générale et Analyse Fonctionnelle, No. 11 of Collection Enseignement des Sciences (Hermann, Paris, 1970).

Shack, R. V.

Traub, W. A.

Trénoguine, V.

V. Trénoguine, Analyse Fonctionnelle (Mir, Moscow, 1985).

Vakili, F.

M. Faucherre, F. Merkle, F. Vakili, “Beam combination in aperture synthesis from space: field of view limitations and (u, υ) plane coverage optimization,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 138–145 (1989).
[CrossRef]

Watson, S. M.

Wissinger, A. B.

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Proc. SPIE643, 194–207 (1986).
[CrossRef]

Woolf, N. J.

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, Chap. 5, pp. 149–201.
[CrossRef]

Amplitude and Intensity Spatial In-terferometry II, Proc. SPIE (1)

J. B. Breckinridge, ed., Amplitude and Intensity Spatial In-terferometry II, Proc. SPIE 2200 (1994).

Appl. Opt. (2)

IEEE Trans. Acoust. Speech Signal Process. (1)

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

IEEE Trans. Antennas Propag. (2)

A. T. Moffet, “Minimum-redundancy linear arrays,” IEEE Trans. Antennas Propag. AP-16, 172–175 (1968).
[CrossRef]

T. J. Cornwell, “A novel principle for optimization of the instantaneous Fourier plane coverage of correlation arrays,” IEEE Trans. Antennas Propag. AP-36, 1165–1167 (1988).
[CrossRef]

IEEE Trans. Comp. (1)

B. R. Hunt, “The application of constrained least squares estimation to image restoration by digital computer,” IEEE Trans. Comp. C-22, 805–812 (1973).
[CrossRef]

J. Mod. Opt. (1)

A. Lannes, S. Roques, M.-J. Casanove, “Stabilized reconstruction in image and signal processing; part I: partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (Bellingham) (2)

J. E. Harvey, R. A. Rockwell, “Performance characteristics of phased arrays and thinned aperture optical telescopes,” Opt. Eng. (Bellingham) 27, 762–768 (1988).
[CrossRef]

A. K. Katsaggelos, “Iterative image restoration algorithms,” Opt. Eng. (Bellingham) 28, 735–748 (1989).
[CrossRef]

Other (16)

L. M. Mugnier, G. Rousset, “Pupil configuration opti-mality criterion in synthetic aperture optics,” in Spaceborne Interferometry II, R. D. Reasenberg, ed., Proc. SPIE2477, 124–131 (1995).
[CrossRef]

C. K. Rushforth, “Signal restoration, functional analysis, and Fredholm integral equations of the first kind,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, 1987), Chap. 1.

A. Papoulis, Signal Analysis (McGraw-Hill, New-York, 1977).

C. Perrier, “Amplitude estimation from speckle interferometry,” in Proceedings of the NATO Advanced Study Institute on Diffraction-Limited Imaging with Very Large Telescopes, D. M. Allouin, J.-M. Mariotti, eds., Vol. 274 of Series C: Mathematical and Physical Sciences(Kluwer, Dordrecht, The Netherlands, 1989), pp. 99–111.
[CrossRef]

P. Y. Bely, “HARDI: a high angular resolution deployable interferometer for space,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 92–100 (1989).
[CrossRef]

ESO/VLT Interferometry Panel, “The VLT Interferometer Implementation Plan,” VLT Report 59b (European Southern Observatory, Garching, Germany, 1989).

M. Faucherre, F. Merkle, F. Vakili, “Beam combination in aperture synthesis from space: field of view limitations and (u, υ) plane coverage optimization,” in New Technologies for Astronomy, J.-P. Swings, ed., Proc. SPIE1130, 138–145 (1989).
[CrossRef]

J. P. Fitch, T. W. Lawrence, “Placement of multiple apertures for imaging telescopes,” in Amplitude and Intensity Spatial Interferometry, J. B. Breckinridge, ed., Proc. SPIE1237, 61–69 (1990).
[CrossRef]

L. Damé, T.-D. Guyenne, “Study of an optimized configuration for interferometric imaging of complex and extended solar structures,” in Targets for Space-Based Interferometry (European Space Agency, Noordwijk, The Netherlands, 1992), Vol. SP-354, pp. 201–208.

J. M. Beckers, F. Merkle, eds., High-Resolution Imaging by Interferometry II. Part II: Multiple Aperture Interferometry, No. 39 of ESO Conference and Workshop Proceedings (European Southern Observatory, Garching, Germany, 1992).

V. Trénoguine, Analyse Fonctionnelle (Mir, Moscow, 1985).

L. Schwartz, Analyse—Topologie Générale et Analyse Fonctionnelle, No. 11 of Collection Enseignement des Sciences (Hermann, Paris, 1970).

A. Lannes, É. Anterrieu, L. Koechlin, G. Fitoussi, “On the concept of field-to-resolution ratio in aperture synthesis,” in Space Optics 1994: Earth Observation and Astronomy, G. Cerutti-Maori, P. Roussel, eds., Proc. SPIE2209, 402–412 (1994).
[CrossRef]

A. B. Meinel, M. P. Meinel, N. J. Woolf, “Multiple aperture telescope diffraction images,” in Applied Optics and Optical Engineering, R. R. Shannon, J. C. Wyant, eds. (Academic, New York, 1983), Vol. 9, Chap. 5, pp. 149–201.
[CrossRef]

J. E. Harvey, A. B. Wissinger, A. N. Bunner, “A parametric study of various synthetic aperture telescope configurations for coherent imaging applications,” in Infrared, Adaptive, and Synthetic Aperture Optical Systems, R. B. Johnson, W. L. Wolfe, J. S. Fender, eds., Proc. SPIE643, 194–207 (1986).
[CrossRef]

A. Labeyrie, G. Lemaitre, L. Kœchlin, “The optical very large array,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. SPIE628, 323–332 (1986).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Optimal configurations with criterion c for three, four, and five telescopes (see Table 1).

Fig. 2
Fig. 2

Optimal configurations with criterion c′ for three, four, and five telescopes (see Table 2).

Fig. 3
Fig. 3

Evolution of the optimal spacing of a three-telescope array with the maximum frequency (solid curve) and corresponding noise amplification parameters c (dotted curve) and c′ (dashed curve). The optimization is done with criterion c, the telescope diameter is 40 pixels, and the frequency is expressed in pixels.

Tables (3)

Tables Icon

Table 1 Optimal Configurations with Criterion ca

Tables Icon

Table 2 Optimal Configurations with Criterion c′a

Tables Icon

Table 3 Optimal Diameter of the Circle Supporting the Telescopes and Noise Amplification Parameters as a Function of the Maximum Frequency, for a Three-Telescope Arraya

Equations (29)

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

i = H o + n ,
i = h o + n .
o e = G i = G H o + G n .
= o e o = ( G H I ) o + G n ,
o + n ,
o = ( G H I ) o and n = G n .
n = G n o .
n ( G ) n
n G H = n H o .
c = G H
H = Λ s .
H 1 = ( Λ i ) 1 .
H = max | h | , and ( if H 1 exists ) H 1 = ( min | h | ) 1 ,
c = 1 / min | h | .
g ( ω ) = 1 / h ( ω ) for ( ω ) D = { ω : | ω | ω max } = 0 otherwise ,
c = G × 1 = 1 / min ω D | h ( ω ) | .
n 2 = ω | g ( ω ) | 2 | n ( ω ) | 2 ,
E ( n 2 ) = ω | g ( ω ) | 2 σ n 2 ( ω ) = ω D 1 | h ( ω ) | 2 σ n 2 ( ω ) ,
E ( n 2 ) = ( ω D 1 | h ( ω ) | 2 ) × σ n 2 1 | g ( ω ) | 2 ω D ,
c = 1 | h ( ω ) | 2 ω D .
E ( n 2 ) 1 min ω D | h ( ω ) | 2 × ω D σ n 2 ( ω ) ,
O T F = O T F e R T F ,
E [ n T ( t ) n T ( t + τ ) ] = T ( t ) T ( t + τ ) R ( τ ) .
E [ n T ( t ) n T ( t + τ ) ] T ( t ) R ( τ ) .
E [ n T ( ω 1 ) n T * ( ω 2 ) ] = + + E { n T ( t ) n T ( t + τ ) } exp [ 2 i π ( ω 1 ω 2 ) t ] exp [ 2 i π ω 2 τ ] d τ d τ + + T ( t ) exp [ 2 i π ( ω 1 ω 2 ) t ] R ( τ ) exp ( 2 i π ω 2 τ ) d τ d τ T sinc { ( ω 1 ω 2 ) T } S ( ω 2 ) ,
σ n T 2 ( ω ) = T S ( ω ) .
C U ( x ) = { U 2 : x 1 U 2 0 : x 1 + U 2 1 π arccos ( x + 1 U 2 4 x ) + U 2 π arccos [ 1 U ( x 1 U 2 4 x ) ] : 1 U 2 x 1 + U 2 2 x π 1 ( x + 1 U 2 4 x ) 2 .
O T F = P P = P + P + + P P 2 P + P ,
O T F ( x ) = 1 1 U 2 [ C 1 ( x ) + U 2 C 1 ( x / U ) 2 C U ( x ) ] ,

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