Abstract

Aperture synthesis allows one to measure visibilities at very high resolutions by coupling telescopes of reasonable diameters. We consider the case where visibility amplitudes and phase are measured separately. It leads to an estimation problem where the noise model yields a nonconvex data-likelihood criterion. We show how to optimally approximate the noise model while keeping the criterion convex. This approximation has been validated both on simulations and on experimental data.

© 2005 Optical Society of America

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References

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  1. L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
    [CrossRef]
  2. A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).
  3. T. J. Cornwell, P. N. Wilkinson, “A new method for making maps with unstable radio interferometers,” Mon. Not. R. Astron. Soc. 196, 1067–1086 (1981).
  4. J. W. Goodman, Statistical Optics (Wiley, 1985).
  5. D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).
  6. J. Idier, ed., Approche Bayésienne pour les Problèmes Inverses (Hermès, 2001).
  7. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust., Speech, Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  8. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).
  9. A. Lannes, “Weak-phase imaging in optical interferometry,” J. Opt. Soc. Am. A 15, 811–824 (1998).
    [CrossRef]
  10. P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
    [CrossRef]
  11. J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, G. Rousset, “Myopic deconvolution of adaptive optics images using object and point spread function power spectra,” Appl. Opt. 37, 4614–4622 (1998).
    [CrossRef]
  12. E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE4847, 174–183 (2002).
  13. G. Le Besnerais, “Méthode du maximum d’entropie sur la moyenne, critères de reconstruction d’image et synthèse d’ouverture en radio-astronomie,” Thèse de doctorat (Université de Paris-Sud, Orsay, 1993).
  14. S. C. Meimon, L. M. Mugnier, G. Le Besnerais, “A novel method of reconstruction for weak-phase optical interferometry,” in New Frontiers in Stellar Interferometry, W. A. Traub, ed., Proc. SPIE5491, 909–919 (2004).

2001 (1)

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

1999 (1)

L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
[CrossRef]

1998 (2)

1989 (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]

1985 (1)

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

1981 (1)

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

Binette, L.

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

Cabrit, S.

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

Conan, J.-M.

Cornwell, T. J.

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

Daigne, G.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Delage, L.

L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
[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]

Du Foresto, V. Coude

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Ferreira, J.

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Fusco, T.

Garcia, P. J. V.

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

Goodman, J. W.

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

Hofmann, K. H.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Hofmann, R.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Lannes, A.

Lattanzi, M.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Le Besnerais, G.

G. Le Besnerais, “Méthode du maximum d’entropie sur la moyenne, critères de reconstruction d’image et synthèse d’ouverture en radio-astronomie,” Thèse de doctorat (Université de Paris-Sud, Orsay, 1993).

S. C. Meimon, L. M. Mugnier, G. Le Besnerais, “A novel method of reconstruction for weak-phase optical interferometry,” in New Frontiers in Stellar Interferometry, W. A. Traub, ed., Proc. SPIE5491, 909–919 (2004).

Le Poole, R. S.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Meimon, S. C.

S. C. Meimon, L. M. Mugnier, G. Le Besnerais, “A novel method of reconstruction for weak-phase optical interferometry,” in New Frontiers in Stellar Interferometry, W. A. Traub, ed., Proc. SPIE5491, 909–919 (2004).

Michau, V.

Mugnier, L. M.

J.-M. Conan, L. M. Mugnier, T. Fusco, V. Michau, G. Rousset, “Myopic deconvolution of adaptive optics images using object and point spread function power spectra,” Appl. Opt. 37, 4614–4622 (1998).
[CrossRef]

S. C. Meimon, L. M. Mugnier, G. Le Besnerais, “A novel method of reconstruction for weak-phase optical interferometry,” in New Frontiers in Stellar Interferometry, W. A. Traub, ed., Proc. SPIE5491, 909–919 (2004).

Osterbart, R.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Queloz, D.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Quirrenbach, A.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Reynaud, F.

L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
[CrossRef]

Rousset, G.

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Thiébaut, E.

L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
[CrossRef]

E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE4847, 174–183 (2002).

Titterington, D. M.

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

Vakilli, F.

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

Wilkinson, P. N.

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

Appl. Opt. (1)

Astron. Astrophys. (2)

P. J. V. Garcia, S. Cabrit, J. Ferreira, L. Binette, “Atomic T Tauri disk winds heated by ambipolar diffusion. II. Observational tests,” Astron. Astrophys. 377, 609–616 (2001).
[CrossRef]

D. M. Titterington, “General structure of regularization procedures in image reconstruction,” Astron. Astrophys. 144, 381–387 (1985).

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]

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

Mon. Not. R. Astron. Soc. (1)

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

Opt. Commun. (1)

L. Delage, F. Reynaud, E. Thiébaut, “Imaging laboratory test on a fiber linked telescope array,” Opt. Commun. 160, 27–32 (1999).
[CrossRef]

Other (7)

A. Quirrenbach, V. Coude Du Foresto, G. Daigne, K. H. Hofmann, R. Hofmann, M. Lattanzi, R. Osterbart, R. S. Le Poole, D. Queloz, F. Vakilli, “PRIMA: study for a dual-beam instrument for the VLT interferometer,” in Astronomical Interferometry, R. D. Reasenberg, ed., Proc. SPIE3350, 807–817 (1998).

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

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1988).

J. Idier, ed., Approche Bayésienne pour les Problèmes Inverses (Hermès, 2001).

E. Thiébaut, “Optimization issues in blind deconvolution algorithms,” in Astronomical Data Analysis II, J.-L. Starck and F. D. Murtagh, eds., Proc. SPIE4847, 174–183 (2002).

G. Le Besnerais, “Méthode du maximum d’entropie sur la moyenne, critères de reconstruction d’image et synthèse d’ouverture en radio-astronomie,” Thèse de doctorat (Université de Paris-Sud, Orsay, 1993).

S. C. Meimon, L. M. Mugnier, G. Le Besnerais, “A novel method of reconstruction for weak-phase optical interferometry,” in New Frontiers in Stellar Interferometry, W. A. Traub, ed., Proc. SPIE5491, 909–919 (2004).

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Figures (8)

Fig. 1
Fig. 1

Polar and Cartesian coordinate systems in C.

Fig. 2
Fig. 2

Noise distribution contour lines, for σ r z 0 < σ φ .

Fig. 3
Fig. 3

Noise distribution contour lines, for σ r z 0 > σ φ .

Fig. 4
Fig. 4

IRMSE repartition histogram.

Fig. 5
Fig. 5

(Color online) Simulation results. (a) True object, (b) u–v coverage—360 frequencies, (c) reconstruction with elliptic approximation, (d) reconstruction with circular approximation. 256 × 256   pixels . Pixel size 0.2 mas .

Fig. 6
Fig. 6

Experimental frequency coverage.

Fig. 7
Fig. 7

True object (left) and restored one (right). Contour levels: 10%, 20%,…,100% of the maximum.

Fig. 8
Fig. 8

(Color online) True object (left) and restored one (right). D is the diameter of the main star used in Table 2.

Tables (2)

Tables Icon

Table 1 Influence of Regularization Parameter on IRMSE

Tables Icon

Table 2 Relative Positions and Flux of the Three Faintest Stars w.r.t. the Brightest One

Equations (57)

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

V ( ν = [ u v ] ) = x ( a , b ) exp ( 2 π i ( u a + v b ) ) d a d b ,
V ( ν ) = j N p h ( j , ν ) X j ,
V ( X ) = H X ,
V i = V ( ν i ) ,
h i , j = h ( j , ν i ) ,
V i meas = V i ( X ) + b , i ,
arg V i meas = arg V i ( X ) + b arg , i ,
p ( X data ) p ( data X ) p ( X )
log p ( X data ) = log p ( data X ) log p ( X ) + const .
J = J data + λ J prior ,
J data ( X ) log ( p ( V meas X ) ) .
J 1 ( X ) = i = 0 N b 1 ( V i meas V i ( X ) σ , i ) 2 + i = 0 N b 1 ( arg V i meas arg V i ( X ) σ arg , i ) 2 ,
J ̃ 1 ( z ) = ( z z 0 ) 2 σ 2 + ( arg z arg z 0 ) 2 σ arg 2 .
J ̃ 1 ( z 1 + z 2 2 ) = ( z 0 2 ) 2 σ 2 + π 2 σ arg 2 ,
1 2 ( J ̃ 1 ( z 1 ) + J ̃ 1 ( z 2 ) ) = 0 + 4 π 2 9 σ arg 2 .
z = z 0 + r ,
arg z = arg z 0 + φ .
p ( z = ( z 0 + r ) exp [ i ( arg z 0 + φ ) ] ) = f ( r , φ ) ,
f ( r , φ ) exp [ 1 2 ( r 2 σ r 2 + φ 2 σ φ 2 ) ] .
z = z 0 + B ,
B = ( x + i y ) exp ( i arg z 0 ) .
x = ( z 0 + r ) cos φ z 0 ,
y = ( z 0 + r ) sin φ .
R ( ψ ) = [ cos ψ sin ψ sin ψ cos ψ ] .
f g ( x , y ) = 1 2 π det Σ exp { 1 2 [ x x ¯ y y ¯ ] t Σ 1 [ x x ¯ y y ¯ ] }
δ ( f 1 , f 2 ) = f 1 log ( f 1 f 2 ) .
x ¯ = E f { x } = z 0 [ exp ( σ φ 2 2 ) 1 ] ,
y ¯ = E f { y } = 0 ,
Σ = Diag { σ 1 2 , σ 2 2 } ,
σ 1 2 = E f { ( x x ¯ ) 2 } = z 0 2 + σ r 2 2 [ 1 + exp ( 2 σ φ 2 ) ] z 0 2 exp ( σ φ 2 ) ,
σ 2 2 = E f { ( y y ¯ ) 2 } = z 0 2 + σ r 2 2 [ 1 exp ( 2 σ φ 2 ) ] .
x ¯ = 0 ,
y ¯ = E f { y } = 0 ,
σ 1 2 = σ r 2 ,
σ 2 2 = z 0 2 σ φ 2 .
J g ( X ) = 2 log f g ( V 0 meas V 0 ( X ) ) .
J g ( X ) = V 0 meas V 0 ( X ) m 0 Σ 0 , R 2
Σ 0 , R = R ( arg V 0 meas ) Σ R ( arg V 0 meas )
Σ 0 , R 2 l = [ R ( ) I ( ) ] t Σ 0 , R 1 [ R ( ) I ( ) ] .
J g ( X ) = i = 0 N 1 V i meas V i ( X ) m i Σ i , R 2
= V meas V ( X ) m Σ 2 ,
J prior ( X ) = ν X ̃ ( ν ) X ̃ m ( ν ) 2 PSD ( ν ) .
PSD ( ν ) = K ( ν ρ 0 ) p + 1 .
g ( X ) = 1 2 π det Σ exp [ 1 2 P ( X ) ] ,
P ( X ) = ( X X ¯ ) t Σ 1 ( X X ¯ ) ,
X ¯ = E f { X } ,
Σ = E f { ( X E f { X } ) ( X E f { X } ) t } = Var ( X ) .
δ ( f , g ) = f ( X ) log f ( X ) g ( X ) d X = E f { log g } + const.
δ ( f , g ) = 1 2 ( E f { P ( X ) } + log det Σ ) + const.
δ ( f , g ) X ¯ = 0 E f { P ( X ) } X ¯ = 0 E f { P ( X ) X ¯ } = 0 E f { 2 Σ 1 ( X X ¯ ) } = 0 E f { ( X X ¯ ) } = 0 E f { X } = X ¯ .
δ ( f , g ) Σ = 0 Σ [ E f { P } + log det Σ ] = 0 E f { P Σ } + log det Σ Σ = 0 E f { Σ t ( X X ¯ ) ( X X ¯ ) t Σ t } + Σ t = 0 .
δ ( f , g ) Σ = 0 Σ t E f { ( X X ¯ ) ( X X ¯ ) t } Σ t = Σ t E f { ( X X ¯ ) ( X X ¯ ) t } = Σ Σ t Σ Σ = E f { ( X X ¯ ) ( X X ¯ ) t } Σ = Var ( X ) ,
g ( x , y ) = 1 2 π det Σ exp 1 2 P ,
P ( x ¯ , y ¯ ) = [ x x ¯ y y ¯ ] t Σ 1 [ x x ¯ y y ¯ ]
x ¯ = E f { x } ,
y ¯ = E f { y } ,
Σ = E f { [ x x ¯ y y ¯ ] [ x x ¯ y y ¯ ] t } .

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