Abstract

Optical interferometers provide multiple wavelength measurements. In order to fully exploit the spectral and spatial resolution of these instruments, new algorithms for image reconstruction have to be developed. Early attempts to deal with multichromatic interferometric data have consisted in recovering a gray image of the object or independent monochromatic images in some spectral bandwidths. The main challenge is now to recover the full three-dimensional (spatiospectral) brightness distribution of the astronomical target given all the available data. We describe an approach to implement multiwavelength image reconstruction in the case where the observed scene is a collection of point-like sources. We show the gain in image quality (both spatially and spectrally) achieved by globally taking into account all the data instead of dealing with independent spectral slices. This is achieved thanks to a regularization that favors spatial sparsity and spectral grouping of the sources. Since the objective function is not differentiable, we had to develop a specialized optimization algorithm that also accounts for non-negativity of the brightness distribution.

© 2013 Optical Society of America

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    [CrossRef]
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  6. J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
    [CrossRef]
  7. F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).
  8. S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
    [CrossRef]
  9. S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
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  10. S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
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  12. U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).
  13. F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
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  15. M. Fornasier and H. Rauhut, “Recovery algorithms for vector valued data with joint sparsity constraints,” SIAM J. Numer. Anal. 46, 577–613 (2008).
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  23. R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).
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  27. M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
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  28. P. L. Combettes and J.-C. Pesquet, Proximal Splitting Methods in Signal Processing (Springer, 2011).
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  30. Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
    [CrossRef]
  31. W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
    [CrossRef]
  32. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
    [CrossRef]
  33. M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
    [CrossRef]
  34. M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bureau Standards 49, 409–436 (1952).
    [CrossRef]
  35. J. Eckstein and D. Bertsekas, “On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program. 55, 293–318 (1992).
    [CrossRef]
  36. M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
    [CrossRef]
  37. S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
    [CrossRef]
  38. G. H. Jacoby, D. A. Hunter, and C. A. Christian, “A library of stellar spectra,” Astrophys. J. Suppl. 56, 257–281 (1984).
    [CrossRef]
  39. M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
    [CrossRef]
  40. F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
    [CrossRef]

2012

É. Thiébaut and F. Soulez, “Multi-wavelength imaging algorithm for optical interferometry,” Proc. SPIE 8445, 84451C (2012).
[CrossRef]

2011

S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
[CrossRef]

S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
[CrossRef]

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
[CrossRef]

2010

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

É. Thiébaut and J.-F. Giovannelli, “Image reconstruction in optical interferometry,” IEEE Signal Process. Mag. 27, 97–109 (2010).
[CrossRef]

2009

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

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

M. Kowalski, “Sparse regression using mixed norms,” Appl. Comput. Harmon. Anal. 27, 303–324 (2009).
[CrossRef]

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

2008

É. Thiébaut, “MiRA: an effective imaging algorithm for optical interferometry,” Proc. SPIE 7013, 70131I (2008).
[CrossRef]

F. Baron and J. S. Young, “Image reconstruction at Cambridge University,” Proc. SPIE 7013, 70133X (2008).
[CrossRef]

M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
[CrossRef]

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[CrossRef]

W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

M. Fornasier and H. Rauhut, “Recovery algorithms for vector valued data with joint sparsity constraints,” SIAM J. Numer. Anal. 46, 577–613 (2008).
[CrossRef]

2007

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

2006

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. B 68, 49–67 (2006).
[CrossRef]

D. Donoho, “For most large underdetermined systems of linear equations, the minimal ℓ-1 norm near-solution approximates the sparsest near-solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

2005

2003

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51, 560–574 (2003).
[CrossRef]

2002

É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, 174–183 (2002).
[CrossRef]

1993

K.-H. Hofmann and G. Weigelt, “Iterative image reconstruction from the bispectrum,” Astron. Astrophys. 278, 328–339 (1993).

1992

J. Eckstein and D. Bertsekas, “On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program. 55, 293–318 (1992).
[CrossRef]

1987

K. A. Marsh and J. M. Richardson, “The objective function implicit in the CLEAN algorithm,” Astron. Astrophys. 182, 174–178 (1987).

1984

G. H. Jacoby, D. A. Hunter, and C. A. Christian, “A library of stellar spectra,” Astrophys. J. Suppl. 56, 257–281 (1984).
[CrossRef]

1978

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).

1974

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

1952

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bureau Standards 49, 409–436 (1952).
[CrossRef]

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
[CrossRef]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

Amorim, A.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Araujo-Hauck, C.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Audibert, J.

R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).

Bach, F.

R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).

Bacon, R.

F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).

Baron, F.

F. Baron and J. S. Young, “Image reconstruction at Cambridge University,” Proc. SPIE 7013, 70133X (2008).
[CrossRef]

Bartko, H.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Bertsekas, D.

J. Eckstein and D. Bertsekas, “On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program. 55, 293–318 (1992).
[CrossRef]

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
[CrossRef]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

Bongard, S.

S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
[CrossRef]

F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).

Bourguignon, S.

S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
[CrossRef]

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Brandner, W.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Buscher, D. F.

D. F. Buscher, “Direct maximum-entropy image reconstruction from the bispectrum,” in Proceedings of IAU Symposium 158: Very High Angular Resolution Imaging, J. G. Robertson and W. J. Tango, eds. (University of Sydney, 1994), p. 91–93.

Christian, C. A.

G. H. Jacoby, D. A. Hunter, and C. A. Christian, “A library of stellar spectra,” Astrophys. J. Suppl. 56, 257–281 (1984).
[CrossRef]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Combettes, P. L.

P. L. Combettes and J.-C. Pesquet, Proximal Splitting Methods in Signal Processing (Springer, 2011).

Darbon, J.

W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

Delplancke, F.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Derie, F.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Donoho, D.

D. Donoho, “For most large underdetermined systems of linear equations, the minimal ℓ-1 norm near-solution approximates the sparsest near-solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

J. Eckstein and D. Bertsekas, “On the Douglas–Rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program. 55, 293–318 (1992).
[CrossRef]

Eisenhauer, F.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Fessler, J. A.

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51, 560–574 (2003).
[CrossRef]

Figueiredo, M.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
[CrossRef]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[CrossRef]

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

Fornasier, M.

M. Fornasier and H. Rauhut, “Recovery algorithms for vector valued data with joint sparsity constraints,” SIAM J. Numer. Anal. 46, 577–613 (2008).
[CrossRef]

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

Gillessen, S.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Giovannelli, J.-F.

É. Thiébaut and J.-F. Giovannelli, “Image reconstruction in optical interferometry,” IEEE Signal Process. Mag. 27, 97–109 (2010).
[CrossRef]

Glindemann, A.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Goldfarb, D.

W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bureau Standards 49, 409–436 (1952).
[CrossRef]

Hofmann, K.-H.

K.-H. Hofmann and G. Weigelt, “Iterative image reconstruction from the bispectrum,” Astron. Astrophys. 278, 328–339 (1993).

Högbom, J. A.

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

Hunter, D. A.

G. H. Jacoby, D. A. Hunter, and C. A. Christian, “A library of stellar spectra,” Astrophys. J. Suppl. 56, 257–281 (1984).
[CrossRef]

Ireland, M.

M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
[CrossRef]

Jacoby, G. H.

G. H. Jacoby, D. A. Hunter, and C. A. Christian, “A library of stellar spectra,” Astrophys. J. Suppl. 56, 257–281 (1984).
[CrossRef]

Jenatton, R.

R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93, 216–231 (2005).
[CrossRef]

Kowalski, M.

M. Kowalski, “Sparse regression using mixed norms,” Appl. Comput. Harmon. Anal. 27, 303–324 (2009).
[CrossRef]

Lacour, S.

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

le Besnerais, G.

le Bouquin, J.-B.

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

Lévêque, S.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Lévy, F.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Lin, Y.

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. B 68, 49–67 (2006).
[CrossRef]

Marsh, K. A.

K. A. Marsh and J. M. Richardson, “The objective function implicit in the CLEAN algorithm,” Astron. Astrophys. 182, 174–178 (1987).

Mary, D.

S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
[CrossRef]

Meimon, S.

Ménardi, S.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Merand, A.

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

Monnier, J.

M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
[CrossRef]

Mugnier, L.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

É. Thiébaut and L. Mugnier, “Maximum a posteriori planet detection and characterization with a nulling interferometer,” in Proceedings of IAU Colloquium 200: Direct Imaging of Exoplanets: Science & Techniques, C. Aime and F. Vakili, eds. (Cambridge University, 2006), pp. 547–552.

Mugnier, L. M.

Nocedal, J.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

Nowak, R.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

Nowak, R. D.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

Osher, S.

W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

Paresce, F.

F. Delplancke, F. Derie, F. Paresce, A. Glindemann, F. Lévy, S. Lévêque, and S. Ménardi, “PRIMA for the VLTI—Science,” Astrophys. Space Sci. 286, 99–104 (2003).
[CrossRef]

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Paumard, T.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

Pécontal, E.

S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
[CrossRef]

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

Perraut, K.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Perrin, G.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Pesquet, J.-C.

P. L. Combettes and J.-C. Pesquet, Proximal Splitting Methods in Signal Processing (Springer, 2011).

Rauhut, H.

M. Fornasier and H. Rauhut, “Recovery algorithms for vector valued data with joint sparsity constraints,” SIAM J. Numer. Anal. 46, 577–613 (2008).
[CrossRef]

Renard, S.

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

Richardson, J. M.

K. A. Marsh and J. M. Richardson, “The objective function implicit in the CLEAN algorithm,” Astron. Astrophys. 182, 174–178 (1987).

Schöller, M.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

Schwarz, U. J.

U. J. Schwarz, “Mathematical-statistical description of the iterative beam removing technique (Method CLEAN),” Astron. Astrophys. 65, 345–356 (1978).

Slezak, É.

S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
[CrossRef]

Soulez, F.

É. Thiébaut and F. Soulez, “Multi-wavelength imaging algorithm for optical interferometry,” Proc. SPIE 8445, 84451C (2012).
[CrossRef]

S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
[CrossRef]

F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).

Stiefel, E.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bureau Standards 49, 409–436 (1952).
[CrossRef]

Straubmeier, C.

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

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J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51, 560–574 (2003).
[CrossRef]

Thiébaut, E.

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
[CrossRef]

Thiébaut, É.

É. Thiébaut and F. Soulez, “Multi-wavelength imaging algorithm for optical interferometry,” Proc. SPIE 8445, 84451C (2012).
[CrossRef]

S. Bongard, F. Soulez, É. Thiébaut, and E. Pécontal, “3-D deconvolution of hyper-spectral astronomical data,” Mon. Not. R. Astron. Soc. 418, 258–270 (2011).
[CrossRef]

É. Thiébaut and J.-F. Giovannelli, “Image reconstruction in optical interferometry,” IEEE Signal Process. Mag. 27, 97–109 (2010).
[CrossRef]

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

É. Thiébaut, “MiRA: an effective imaging algorithm for optical interferometry,” Proc. SPIE 7013, 70131I (2008).
[CrossRef]

É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, 174–183 (2002).
[CrossRef]

É. Thiébaut and L. Mugnier, “Maximum a posteriori planet detection and characterization with a nulling interferometer,” in Proceedings of IAU Colloquium 200: Direct Imaging of Exoplanets: Science & Techniques, C. Aime and F. Vakili, eds. (Cambridge University, 2006), pp. 547–552.

F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).

Thureau, N.

M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
[CrossRef]

Vincent, F.

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
[CrossRef]

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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[CrossRef]

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K.-H. Hofmann and G. Weigelt, “Iterative image reconstruction from the bispectrum,” Astron. Astrophys. 278, 328–339 (1993).

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M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

Wright, S. J.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Trans. Signal Process. 57, 2479–2493 (2009).
[CrossRef]

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[CrossRef]

Yin, W.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
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W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

Young, J. S.

F. Baron and J. S. Young, “Image reconstruction at Cambridge University,” Proc. SPIE 7013, 70133X (2008).
[CrossRef]

Yuan, M.

M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. B 68, 49–67 (2006).
[CrossRef]

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Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
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M. Kowalski, “Sparse regression using mixed norms,” Appl. Comput. Harmon. Anal. 27, 303–324 (2009).
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K. A. Marsh and J. M. Richardson, “The objective function implicit in the CLEAN algorithm,” Astron. Astrophys. 182, 174–178 (1987).

K.-H. Hofmann and G. Weigelt, “Iterative image reconstruction from the bispectrum,” Astron. Astrophys. 278, 328–339 (1993).

J.-B. le Bouquin, S. Lacour, S. Renard, E. Thiébaut, and A. Merand, “Pre-maximum spectro-imaging of the Mira star T Lep with AMBER/VLTI,” Astron. Astrophys. 496, L1–L4 (2009).
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D. Donoho, “For most large underdetermined systems of linear equations, the minimal ℓ-1 norm near-solution approximates the sparsest near-solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
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S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2010).
[CrossRef]

IEEE J. Select. Topics Signal Process.

M. Figueiredo, R. Nowak, and S. Wright, “Gradient projection for sparse reconstruction: application to compressed sensing and other inverse problems,” IEEE J. Select. Topics Signal Process. 1, 586–597 (2007).
[CrossRef]

S. Bourguignon, D. Mary, and É. Slezak, “Restoration of astrophysical spectra with sparsity constraints: models and algorithms,” IEEE J. Select. Topics Signal Process. 5, 1002–1013 (2011).
[CrossRef]

IEEE Signal Process. Mag.

É. Thiébaut and J.-F. Giovannelli, “Image reconstruction in optical interferometry,” IEEE Signal Process. Mag. 27, 97–109 (2010).
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M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “An augmented Lagrangian approach to the constrained optimization formulation of imaging inverse problems,” IEEE Trans. Image Process. 20, 681–695 (2011).
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M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
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IEEE Trans. Signal Process.

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51, 560–574 (2003).
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R. Jenatton, J. Audibert, and F. Bach, “Structured variable selection with sparsity-inducing norms,” J. Mach. Learn. Res. 12, 2777–2824 (2011).

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M. Yuan and Y. Lin, “Model selection and estimation in regression with grouped variables,” J. R. Stat. Soc. Ser. B 68, 49–67 (2006).
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[CrossRef]

F. Vincent, T. Paumard, G. Perrin, L. Mugnier, F. Eisenhauer, and S. Gillessen, “Performance of astrometric detection of a hotspot orbiting on the innermost stable circular orbit of the galactic centre black hole,” Mon. Not. R. Astron. Soc. 412, 2653–2664 (2011).
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É. Thiébaut, “Image reconstruction with optical interferometers,” New Astron. Rev. 53, 312–328 (2009).
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[CrossRef]

É. Thiébaut, “Optimization issues in blind deconvolution algorithms,” Proc. SPIE 4847, 174–183 (2002).
[CrossRef]

M. Ireland, J. Monnier, and N. Thureau, “Monte-Carlo imaging for optical interferometry,” Proc. SPIE 6268, 62681T (2008).
[CrossRef]

S. Gillessen, F. Eisenhauer, G. Perrin, W. Brandner, C. Straubmeier, K. Perraut, A. Amorim, M. Schöller, C. Araujo-Hauck, H. Bartko, and , “Gravity: a four-telescope beam combiner instrument for the VLTI,” Proc. SPIE 7734, 77340Y (2010).
[CrossRef]

É. Thiébaut and F. Soulez, “Multi-wavelength imaging algorithm for optical interferometry,” Proc. SPIE 8445, 84451C (2012).
[CrossRef]

F. Baron and J. S. Young, “Image reconstruction at Cambridge University,” Proc. SPIE 7013, 70133X (2008).
[CrossRef]

SIAM J. Imaging Sci.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[CrossRef]

W. Yin, S. Osher, D. Goldfarb, and J. Darbon, “Bregman iterative algorithms for ℓ1-minimization with applications to compressed sensing,” SIAM J. Imaging Sci. 1, 143–168 (2008).
[CrossRef]

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M. Fornasier and H. Rauhut, “Recovery algorithms for vector valued data with joint sparsity constraints,” SIAM J. Numer. Anal. 46, 577–613 (2008).
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J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer-Verlag, 2006).

F. Soulez, É. Thiébaut, S. Bongard, and R. Bacon, “Restoration of hyperspectral astronomical data from integral field spectrograph,” in Evolution in Remote Sensing (WHISPERS), 2011 3rd Workshop on Hyperspectral Image and Signal Processing (IEEE,2011).

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P. L. Combettes and J.-C. Pesquet, Proximal Splitting Methods in Signal Processing (Springer, 2011).

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Figures (5)

Fig. 1.
Fig. 1.

Integrated flux for the star cluster. From top to bottom, left to right: true object; reconstruction with fully separable sparsity prior fsparse(x) defined in Eq. (9); reconstruction assuming a gray object and with spatial sparsity fsparse(g) defined in Eq. (14); and reconstruction with joint-sparsity prior fjoint(x) defined in Eq. (10). The spectra of the sources encircled by the boxes are shown in Fig. 2. Axes units are in milliarcseconds.

Fig. 2.
Fig. 2.

Spectra of two selected sources. Each panel shows the spectra of one of the sources encircled by the boxes in Fig. 1. Thick curves are for the true spectra and thin curves with markers are for the restored spectra. The open squares indicate the reconstruction with fully separable sparsity prior fsparse(x), the open triangles indicate the reconstruction with joint sparsity prior fjoint(x), and the filled triangles indicate the restored spectra after debiasing (which are virtually indistinguishable from the true ones).

Fig. 3.
Fig. 3.

Histograms of the mean fluxes of the sources for the true object (in black), for the 3D images restored with fully separable sparsity (in white) and with joint-sparsity (in dark gray) priors, and for the two-dimensional gray image restored with sparsity prior (in light gray). The vertical scale has been truncated to focus on the distributions of the brightest sources.

Fig. 4.
Fig. 4.

Histograms of the mean flux of the true and false positive detection in the reconstructions under joint-sparsity and gray-sparsity priors. A positive detection is defined as a pixel with nonzero mean flux in the reconstruction. The vertical scale has been truncated to focus on the distributions of the true positive sources.

Fig. 5.
Fig. 5.

Evolution of the convergence criterion ϕ(t), defined in Eq. (41), for different strategies to choose the augmented penalty parameter. Dashed curves are for a constant ρ. Solid curves are for ρ automatically set to have ηalt(t)1.

Equations (72)

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

x+=argminxX{fdata(x)+μfprior(x)},
X={xRn:x0},
xn,Iλ(θn),
yp,m,=(H·x)p,m,+ep,m,=nHp,m,n,xn,+ep,m,,
Hp,m,n,={+cos(θn·Bm/λ)forp=1sin(θn·Bm/λ)forp=2
y=H·x+e,
HR·F·S,
fdata(x)=12(H·xy)·W·(H·xy),
fsparse(x)=x1=defk,|xk,|=sgn(x)·x,
fjoint(x)=n(xn,2)1/2,
fspectral(x)=nμn(xn,+1xn,)2
xn,=gn,(n,),
yp,m,=nHp,m,n,gn+ep,m,,
fsparse(g)=g1=n|gn|=sgn(g)·g.
minxX,z{fdata(z)+μfprior(x)}s.t.x=z.
Lρ(x,z,u)=fdata(z)+μfprior(x)+u·(xz)+ρ2xz22
x(t)=argminxXLρ(t)(x,z(t1),u(t1))=argminxX{fprior(x)+ρ(t)2μxx˜(t)22}
x˜(t)=z(t1)u(t1)/ρ(t),
z(t)=argminzLρ(t)(x(t),z,u(t1))=argminz{fdata(z)+ρ(t)2zz˜(t)22}
z˜(t)=x(t)+u(t1)/ρ(t),
u(t)=u(t1)+ρ(t)(x(t)z(t)).
A(t)·z(t)=b(t)
A(t)=H·W·H+ρ(t)I,
b(t)=H·W·y+ρ(t)x(t)+u(t1)
A(t)·z(t)=b(t),
A(t)=H·W·H+ρ(t)I,
b(t)=H·W·y+ρ(t)x(t)+u(t1),
t=1z(t)zexact(t)2<,
A(t)·z(t)b(t)2ϵCGA(t)·z(t1)b(t)2
x*=z*,
0xL0(x*,z*,u*)=μfprior(x*)+u*,
0zL0(x*,z*,u*)=fdata(z*)u*,
u*=fdata(z*).
fdata(z(t))u(t1)+ρ(t)(z(t)x(t))=0fdata(z(t))=u(t),
0μfprior(x(t))+u(t1)+ρ(t)(x(t)z(t1))μfprior(x(t))+u(t)+ρ(t)(z(t)z(t1))ρ(t)(z(t)z(t1))μfprior(x(t))+u(t),
s(t)=ρ(t)(z(t)z(t1)),
r(t)=x(t)z(t),
r(t)2τprim(t)ands(t)2τdual(t),
τprim(t)=defNϵabs+ϵrelmax(x(t)2,z(t)2),
τdual(t)=defNϵabs+ϵrelu(t)2,
u(0)=fdata(z(0))=H·W·(H·z(0)y).
ϕ(t)1withϕ(t)=defmax(r(t)2τprim(t),s(t)2τdual(t)).
ρ(t)ρ*(t)=defargminρϕt(ρ),
ρ(t)=ρ*(t)r(t)2τprim(t)=s(t)2τdual(t).
η(t)1withη(t)=defr(t)2τdual(t)s(t)2τprim(t),
ηalt(t)1withηalt(t)=defr(t)2τdual(t1)s(t)2τprim(t1).
ρ{ρminρmaxifρmin>0ρmax/γotherwise,
ρ{ρminρmaxifρmax<γρminotherwise,
ρ(1)=argminρfdata(z(1)u(1)/ρ)=u(1)·H·W·H·u(1)u(1)·u(1),
ρ(t+1)={γρ(t)ifηt>τ,ρ(t)/γifηt<1/τ,ρ(t)else;
minxX{αf(x)+12xx˜22}
proxαf(x˜)=defargminxRN{αf(x)+12xx˜22}.
proxαf+(x˜)=defargminxR+N{αf(x)+12xx˜22}
proxαfsparse(x˜)n,={x˜n,αifx˜n,>α;x˜n,+αifx˜n,<α;0else.
proxαfsparse+(x˜)n,={x˜n,αifx˜n,>α;0else.
c(x)=αn(xn,2)1/2fjoint(x)+12xx˜22,
c(x)xn,=0αβnxn,+(xn,x˜n,)=0xn,=x˜n,1+α/βn
βn=def(xn,2)1/2,
βn=β˜n1+α/βn,
β˜n=def(x˜n,2)1/2.
βn=β˜nα.
proxαfjoint(x˜)n,={(1αβ˜n)x˜n,ifβ˜n>α;0else,
xn,=max(0,x˜n,)1+α/βn,
proxαfjoint+(x˜)=proxαfjoint(max(0,x˜)).
A(t)·zexact(t)b(t)=0,
A(t)·z(t)b(t)=v(t),
z(t)zexact(t)=[A(t)]1·v(t).
z(t)zexact(t)2v(t)2/ρ(t).
t=1z(t)zexact(t)2<,
t=1v(t)2/ρ(t)<.
v(t)2γCGρ(t)ξCGt,
t=1z(t)zexact(t)2t=1v(t)2/ρ(t)γCGt=1ξCGt=γCGξCG1ξCG.

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