Abstract

To reveal some new results of a study concerning phase-closure imaging, first we introduce three key operators: the cophasing operator A, the phase-aberration operator B, and the phase-closure operator C. We then show that the generalized inverses of these operators are equal to their (Hilbert space) adjoints divided by the number of pupil pinholes. This remarkable property, which can be stated in terms of backprojection, plays an essential part in the understanding and the treatment of the inverse problems of aperture synthesis. The notion of backprojection is illustrated in a geometrical manner. As an example of applications we present a new self-calibration algorithm for solving the phase-restoration problem in radio imaging. The solution of this deconvolution problem is obtained without phase unwrapping by means of backprojection mechanisms. The implications of these structures in speckle imaging are also examined. Whenever possible, nonredundant configurations should be preferred. The main developments of our approach concern, in particular, the very-long-baseline array and the interferometric mode of the very large telescope.

© 1990 Optical Society of America

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References

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  1. T. J. Pearson, A. C. S. Readhead, “Image formation by selfcalibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130 (1984).
    [CrossRef]
  2. A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing; Part I: Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
    [CrossRef]
  3. A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
    [CrossRef]
  4. A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
    [CrossRef]
  5. 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).
  6. C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961).
  7. F. Schwab, “Adaptive calibration of radio interferometric data,” in 1980 International Computing Conference(Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
    [CrossRef]
  8. A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
    [CrossRef]
  9. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  10. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  11. F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
    [CrossRef]
  12. L. Michel, “Symmetry defects and broken symmetry. Configurations hidden symmetry,” Rev. Mod. Phys. 52, 617–651 (1980).
    [CrossRef]
  13. A. Lannes, “Backprojection mechanisms in phase-closure imaging. Bispectral analysis of the phase-restoration process,” Exp. Astron. 1, 47–76 (1989).
    [CrossRef]
  14. C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).
  15. J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

1989 (1)

A. Lannes, “Backprojection mechanisms in phase-closure imaging. Bispectral analysis of the phase-restoration process,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

1987 (3)

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

A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
[CrossRef]

1986 (1)

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

1984 (2)

T. J. Pearson, A. C. S. Readhead, “Image formation by selfcalibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
[CrossRef] [PubMed]

1983 (1)

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).

1980 (1)

L. Michel, “Symmetry defects and broken symmetry. Configurations hidden symmetry,” Rev. Mod. Phys. 52, 617–651 (1980).
[CrossRef]

1974 (1)

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Bartelt, H.

Casanove, M. J.

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

A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
[CrossRef]

Clark, T. A.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

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).

Counselman, C. C.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Giacovazzo, C.

C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).

Hinteregger, H. F.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Hutton, L. K.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Klemperer, W. K.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Lanczos, C.

C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961).

Lannes, A.

A. Lannes, “Backprojection mechanisms in phase-closure imaging. Bispectral analysis of the phase-restoration process,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
[CrossRef]

A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

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

Lohmann, A. W.

Marandino, G. E.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Michel, L.

L. Michel, “Symmetry defects and broken symmetry. Configurations hidden symmetry,” Rev. Mod. Phys. 52, 617–651 (1980).
[CrossRef]

Niell, A. E.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Ortega, J. M.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Pearson, T. J.

T. J. Pearson, A. C. S. Readhead, “Image formation by selfcalibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Readhead, A. C. S.

T. J. Pearson, A. C. S. Readhead, “Image formation by selfcalibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Rheinboldt, W. C.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

Roddier, F.

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

Rogers, A. E. E.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Ronnang, B. O.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Roques, S.

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

A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
[CrossRef]

Rydbeck, O. E. H.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Schwab, F.

F. Schwab, “Adaptive calibration of radio interferometric data,” in 1980 International Computing Conference(Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
[CrossRef]

Shapiro, I.I.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Warnock, W. W.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Weigelt, G.

Whitney, A. R.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

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).

Wirnitzer, B.

Wittels, J. J.

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Ann. Rev. Astron. Astrophys. (1)

T. J. Pearson, A. C. S. Readhead, “Image formation by selfcalibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130 (1984).
[CrossRef]

Appl. Opt. (2)

Astrophys. J. (1)

A. E. E. Rogers, H. F. Hinteregger, A. R. Whitney, C. C. Counselman, I.I. Shapiro, J. J. Wittels, W. K. Klemperer, W. W. Warnock, T. A. Clark, L. K. Hutton, G. E. Marandino, B. O. Ronnang, O. E. H. Rydbeck, A. E. Niell, “The structure of radio sources 3C273B and 3C84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301 (1974).
[CrossRef]

Exp. Astron. (1)

A. Lannes, “Backprojection mechanisms in phase-closure imaging. Bispectral analysis of the phase-restoration process,” Exp. Astron. 1, 47–76 (1989).
[CrossRef]

J. Mod. Opt. (2)

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

A. Lannes, M. J. Casanove, S. Roques, “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

J. Opt. Soc. Am. (1)

A. Lannes, S. Roques, M. J. Casanove, “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199 (1987).
[CrossRef]

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)

F. Roddier, “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148 (1986).
[CrossRef]

Rev. Mod. Phys. (1)

L. Michel, “Symmetry defects and broken symmetry. Configurations hidden symmetry,” Rev. Mod. Phys. 52, 617–651 (1980).
[CrossRef]

Other (4)

C. Lanczos, Linear Differential Operators (Van Nostrand, New York, 1961).

F. Schwab, “Adaptive calibration of radio interferometric data,” in 1980 International Computing Conference(Book I), W. T. Rhodes, ed., Proc. Soc. Photo-Opt. Instrum. Eng.231, 18–25 (1980).
[CrossRef]

C. Giacovazzo, Direct Methods in Crystallography (Academic, New York, 1980).

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970).

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Figures (10)

Fig. 1
Fig. 1

Geometrical representation of the twofold canonical decomposition of the pinhole phase space G. The range of A, which coincides with the null space of B, is the space of constant phase functions; its dimension is equal to 1. The range of B* is the aberration pinhole phase space; its dimension is equal to n − 1. The corresponding orthogonal projections are the operators AA*/n and B*B/n.

Fig. 2
Fig. 2

Geometrical representation of the twofold canonical decomposition of the baseline phase space H. The range of B, which coincides with the null space of C, is the aberration baseline phase space; its dimension is equal to n − 1. The range of C*, which is isomorphic to the closure phase space CH, is the aberration-free baseline phase space; its dimension is equal to (n − 1)(n − 2)/2. The corresponding orthogonal projections are the operators BB*/n and C*C/n.

Fig. 3
Fig. 3

Canonical decompositions of the key spaces of phase-closure imaging. The height of each box is proportional to the dimension of the corresponding space or subspace (see Figs. 1 and 2); here n = 4. As illustrated in this diagram, the range of A is the null space of B, and the range of B is the null space of C. The aberration baseline phase space BG is isomorphic to the aberration pinholephase space B*H. Likewise, the aberration-free baseline phase space C*K is isomorphic to the closure phase space CH. The projections onto the main subspaces are specified in Figs. 1 and 2.

Fig. 4
Fig. 4

Geometrical illustration of the action of B+. Here O+ is the center of gravity of the points O1, O2 O3and O4,. For example, the position of O2 with respect to O+ is given by the backprojection operation α+(2) = [β(1, 2) + β(3, 2) + β(4, 2)]/4; β(i,j) is the vector OiOj. In interferometry, and more precisely in radio imaging (cf. Fig. 3 and Section 4), algebraic phases are to be substituted for position vectors.

Fig. 5
Fig. 5

Least-squares solutions of the equation Bα = β. In this geometrical representation, α(r) is the particular solution such that α ( r ) ( r ) = 0 ; α ( r ) is a rough approximation to α(r). In general, as illustrated here, the heuristic solution introduced by Cornwell and Wilkinson, αCW, is not a least-squares solution.

Fig. 6
Fig. 6

Geometrical illustration of the action of C+ (O+ is the center of gravity of the points O1, O2, O3, and O4). In the backprojection operation illustrated here, the area vector of triangle O+O1, O2, (for example) is given by the relation β+(1, 2) = [γ(3,1,2) + γ(4,1,2)]/4; γ(i, j, k) is the area vector of triangle Oi, Oj, Ok. For clarity, the reader may assume that all the points Oi lie in the same plane. In phaseclosure imaging, and more precisely in speckle imaging (cf. Fig. 3 and Section 5), algebraic phases are to be substituted for triangle-area vectors.

Fig. 7
Fig. 7

Example of a self-calibration algorithm. In the special case under consideration, each global iteration comprises two Gauss-Seidel steps (see Fig. 8). The Gauss-Newton procedure, through which the complex visibility is corrected, corresponds to the optimization step in β. The problem is solved without phase unwrapping by means of backprojection mechanisms. The deconvolution process corresponds to the optimization step in Φ. In practice, the initial trial object Φ(0) is a low-resolution image of the object. This information provides an important constraint: a region containing the object support.

Fig. 8
Fig. 8

Geometrical illustration of the principle of self-calibration algorithms. The key operator is the projection onto the aberration baseline phase space BG. In our algorithms, this elementary operation is performed without phase unwrapping by means backprojection mechanisms. Each diagonal arrow corresponds to a deconvolution procedure (see Fig. 7).

Fig. 9
Fig. 9

Flow chart of the Gauss-Newton algorithm allowing the spectral phase factor to be restored up to a phase function lying in the null space of C. The phase-closure data are gathered in the experimental bispectral phase factor ξe. In the general case of partially redundant configurations, the bispectral operator C′ is to be substituted for C (see Section 6).

Fig. 10
Fig. 10

Minimum-norm least-squares solution. The set of leastsquares solutions, , is parallel to the null space of B;α+ is the minimum-norm least-squares solution.

Tables (1)

Tables Icon

Table 1 Matrices of the Operators B and C in the Special Case Where n = 4a

Equations (105)

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

¯ Φ Φ s ¯ ¯ Ψ Ψ s ¯ μ , 0 < μ < 1 .
η = ( V x d x ) 1 / 2 ( W u d u ) 1 / 2 .
μ = 1 η 2 [ 1 2 J η 2 + O ( η 4 ) ] , [ J = 2 π 2 V W ( u x ) 2 d u d x ] .
α ( i ) = r G α r α r b ( i ) [ α r b ( i ) = δ r , i ] ;
( α ( 1 ) α ( 2 ) ) G = i = 1 n α ( 1 ) ( i ) α ( 2 ) ( i ) = r G α r ( 1 ) α r ( 2 ) .
β ( i , j ) = β ( j , i ) .
= { ( r , s ) : r , s G , r < s } .
β ( i , j ) = ( r , s ) β r , s β r , s b ( i , j ) ,
β r , s b ( i , j ) = δ r , i δ s , j δ r , j δ s , i = { 1 if i = r and j = s 1 if j = r and i = s 0 otherwise .
dim H = ( n 2 ) = n ( n 1 ) / 2 .
( β ( 1 ) | β ( 2 ) ) H = 1 2 i = 1 n i = 1 n β ( 1 ) ( i , j ) β ( 2 ) ( i , j ) = ( r , s ) β ( 1 ) ( r , s ) β ( 2 ) ( r , s ) = ( r , s ) β r , s ( 1 ) β r , s ( 2 ) .
γ ( i , j , k ) = γ ( j , i , k )
K = { ( r , s , t ) : r , s , t G , r < s < t } .
γ ( i , j , k ) = ( r , s , t ) K γ r , s , t γ r , s , t b ( i , j , k ) ,
γ r , s , t b ( i , j , k ) = δ r , i δ s , j δ t , k δ r , i δ s , k δ t , j + δ r , j δ s , k δ t , i δ r , j δ s , i δ t , k + δ r , k δ s , i δ t , j δ r , k δ s , j δ t , i .
dim K = ( n 3 ) = n ( n 1 ) ( n 2 ) / 6.
[ γ ( 1 ) | γ ( 2 ) ] K = 1 6 i = 1 n j = 1 n k = 1 n γ ( 1 ) ( i , j , k ) γ ( 2 ) ( i , j , k ) = ( r , s , t ) K γ ( 1 ) ( r , s , t ) γ ( 2 ) ( r , s , t ) = ( r , s , t ) K γ r , s , t ( 1 ) γ r , s , t ( 2 ) .
( A φ ) ( i ) = φ ( for all i G ) .
A * α = i = 1 n α ( i ) .
( B α ) ( i , j ) = α ( j ) α ( i ) ( i , j G ) .
[ ψ ( j ) exp i α ( j ) ] [ ψ ( i ) exp i α ( i ) ] * = ψ ( j ) ψ ( i ) * exp i [ α ( j ) α ( i ) ] ( i 2 = 1 ) .
( B * β ) ( j ) = i = 1 n β ( i , j ) = i j β ( i , j ) .
( C β ) ( i , j , k ) = β ( j , k ) β ( i , k ) + β ( i , j ) ( i , j , k G ) .
( C * γ ) ( j , k ) = i = 1 n γ ( i , j , k ) i j i k γ ( i , j , k ) .
B A = 0 , C B = 0 .
A F = ker B , B G = ker C .
A A * + B * B = n I G , B B * + C * C = n I H .
A + = A * / n , B + = B * / n , C + = C * / n .
α + ( j ) = 1 n i = 1 n β ( i , j ) = 1 n i j β ( i , j ) .
O 1 , 2 : α 1 : = α 1 β 1 , 2 , α 2 : = α 2 + β 1 , 2 , O 1 , 3 : α 1 : = α 1 β 1 , 3 , α 3 : = α 3 + β 1 , 3 , O 1 , 4 : α 1 : = α 1 β 1 , 4 , α 4 : = α 4 + β 1 , 4 , O 2 , 3 : α 2 : = α 2 β 2 , 3 , α 3 : = α 3 + β 2 , 3 , O 2 , 4 : α 2 : = α 2 β 2 , 4 , α 4 : = α 4 + β 2 , 4 , O 3 , 4 : α 3 : = α 3 β 3 , 4 , α 4 : = α 4 + β 3 , 4 .
O + O j = 1 n i j O i O j .
α ( j ) = α + ( j ) + φ ( for all j ) .
α ( r ) ( j ) = α + ( j ) + φ [ φ = α + ( r ) ] .
α ( r ) ( j ) = α + ( j ) α + ( r ) = 1 n i = 1 n [ β ( i , j ) β ( i , r ) ] = 1 n i = 1 n [ β ( r , i ) + β ( i , j ) ] .
β ( r , i ) + β ( i , j ) [ α ( r ) ( i ) α ( r ) ( r ) ] + [ α ( r ) ( j ) α ( r ) ( i ) ] α ( r ) ( j ) .
α ( CW ) ( j ) = 1 n 1 i j [ β ( r , i ) + β ( i , j ) ] = 1 n 1 [ i j β ( i , j ) i r i j β ( i , r ) ] ,
ϕ j = 1 N 1 ( k j θ j , k k ref k j θ ref , k ) .
α ( CW ) α ( r ) = 1 n 1 [ α ( r ) α ( r ) ] ,
α ( r ) ( j ) = β ( r , j ) .
β + ( j , k ) = 1 n i = 1 n γ ( i , j , k ) = 1 n i j i k n γ ( i , j , k ) .
O 1 , 2 , 3 : β 1 , 2 : = β 1 , 2 + γ 1 , 2 , 3 , β 1 , 3 : = β 1 , 3 γ 1 , 2 , 3 , β 2 , 3 : = β 2 , 3 + γ 1 , 2 , 3 , O 1 , 2 , 4 : β 1 , 2 : = β 1 , 2 + γ 1 , 2 , 4 , β 1 , 4 : = β 1 , 4 γ 1 , 2 , 4 , β 2 , 4 : = β 2 , 4 + γ 1 , 2 , 4 , O 1 , 3 , 4 : β 1 , 3 : = β 1 , 3 + γ 1 , 3 , 4 , β 1 , 4 : = β 1 , 4 γ 1 , 3 , 4 , β 3 , 4 : = β 3 , 4 + γ 1 , 3 , 4 , O 2 , 3 , 4 : β 2 , 3 : = β 2 , 3 + γ 2 , 3 , 4 , β 2 , 4 : = β 2 , 4 γ 2 , 3 , 4 , β 3 , 4 : = β 3 , 4 + γ 2 , 3 , 4 .
[ O + O j Λ O + O k ] / 2 = 1 n i j i k [ O i O j Λ O i O k ] / 2 .
q ( Φ ; μ , β ) = ¯ Ψ ̂ g Φ ̂ ¯ 2 ,
Ψ ̂ = ψ ̂ exp ( μ + i β ) ( ψ ̂ = g s t V e ) .
β Φ ( u ) = β Φ ( u ) , i . e . , β Φ ( i , j ) = β Φ ( i , j ) .
¯ Ψ ̂ g Φ ̂ ¯ 2 = ( i , j ) | Ψ ̂ ( i , j ) g ( i , j ) Φ ̂ ( i , j ) | 2 .
E = E Φ ( E μ E β ) .
β ( i , j ) = ( B α ) ( i , j ) = α ( j ) α ( i ) .
q Φ ( β ) = ¯ Ψ ̂ g Φ ̂ ¯ 2 ( Ψ ̂ = ψ ̂ exp i β ) .
P Im { g Φ ̂ Ψ ̂ * } = 0 ,
¯ ρ × ( κ i ϑ ) ¯ [ κ i = Im ( κ ) ] ,
κ = { g Φ ̂ Ψ ̂ * } / ρ 2 ρ = | Ψ ̂ | = | ψ ̂ | .
P ρ 2 ϑ = P ρ 2 κ i ( P = B B * / n ) .
ϑ = P κ i , i . e . , ϑ ( i , j ) = 1 n s = 1 n [ κ i ( s , j ) κ i ( s , i ) ] .
κ i sin ( β Φ β Ψ ) .
i j ρ Φ ( i , j ) ρ Ψ ( i , j ) sin { β Φ ( i , j ) β Ψ ( i , j ) } = 0 ( j G ) .
i j ρ Φ ( i , j ) ρ Ψ ( i , j ) sin { [ β Φ ( i , j ) β Ψ ( i , j ) ] β ( i , j ) } = 0 ( j G ) ,
ξ = exp i γ ( γ = C β ; β H ) .
q e ( β ) = ¯ g e × ( ξ e ξ ) ¯ 2 ,
¯ g e × ( ξ e ξ ) ¯ 2 ( i , j , k ) K g e 2 ( i , j , k ) | ξ e ( i , j , k ) ξ ( i , j , k ) | 2 .
ζ ζ exp i ϑ .
d ( ϑ ) = ¯ g e × [ γ e ( γ + C ϑ ) ] ¯ ¯ g e × [ ( γ e γ ) C ϑ ) ] ¯ .
d ( ϑ ) e ( ϑ ) ,
e ( ϑ ) = ¯ g e × ( κ i C ϑ ) ¯ ,
κ i = sin ( γ e γ ) = Im [ exp i ( γ e γ ) ] = Im { ξ e ξ * } .
C * g e 2 C ϑ = C * g e 2 κ i .
ϑ = ϑ + = C * κ i / n .
exp ( j + i α j ) * exp ( k + i α k ) = exp ( j + k ) exp i ( α k α j ) .
( y | A x ) = ( A * y | x ) x , y ,
A * α = i = 1 n α ( i ) .
( α | A φ ) G = i = 1 n α ( i ) φ [ i = 1 n α ( i ) ] φ = ( A α | φ ) F .
( B * β ) ( j ) = i = 1 n β ( i , j ) .
( β | B α ) H = 1 2 i = 1 n j = 1 n β ( i , j ) ( B α ) ( i , j ) = 1 2 i = 1 n j = 1 n β ( i , j ) [ α ( j ) α ( i ) ] = 1 2 j = 1 n i = 1 n β ( i , j ) α ( j ) + 1 2 i = 1 n j = 1 n β ( j , i ) α ( i ) = j = 1 n i = 1 n β ( i , j ) α ( j ) = j = 1 n ( B β ) ( j ) α ( j ) = ( B β | α ) G .
( C * γ ) ( j , k ) = i = 1 n γ ( i , j , k ) .
( γ | C β ) K = 1 6 i = 1 n j = 1 n k = 1 n γ ( i , j , k ) ( C β ) ( i , j , k ) = 1 6 i = 1 n j = 1 n k = 1 n γ ( i , j , k ) [ β ( j , k ) β ( i , k ) + β ( i , j ) ] = 1 2 j = 1 n k = 1 n i = 1 n γ ( i , j , k ) β ( j , k ) = 1 2 j = 1 n k = 1 n ( C γ ) ( j , k ) β ( j , k ) = ( C γ | β ) H .
( B * B α ) ( j ) = i = 1 n ( B α ) ( i , j ) = i = 1 n [ α ( j ) α ( i ) ] = n α ( j ) i = 1 n α ( i ) = n α ( j ) A * α .
( C * C β ) ( j , k ) = i = 1 n ( C β ) ( i , j , k ) = i = 1 n [ β ( j , k ) β ( i , k ) + β ( i , j ) ] = n β ( j , k ) ( B * β ) ( k ) + ( B * β ) ( j ) = n β ( j , k ) ( B B * β ) ( j , k ) .
( α | B * β ) G = ( B α | β ) H .
α + = B + β .
α + = lim μ 0 [ ( B * B + μ I ) 1 B * β ] .
α ( CW ) ( j ) = 1 n 1 { i = 1 n [ β ( r , i ) + β ( i , j ) β ( r , j ) ] } .
α ( CW ) = 1 n 1 [ n α ( r ) α ( r ) ] = α ( r ) + 1 n 1 [ α ( r ) α ( r ) ] .
α ( CW ) α ( r ) = 1 n 1 [ α ( r ) α ( r ) ] .
q ( X ) = ¯ f ( X ) ¯ 2 .
X X + h .
h ¯ k A h ¯ 2 [ k = f ( X ) ; A = f ( X ) ] .
lim n ¯ X X n + 1 ¯ ¯ X X n ¯ = 0 .
f ( β ) = Ψ ̂ g Φ ̂ ( Ψ ̂ = ψ ̂ exp i β ; β B G ) .
( ψ ̂ ( 1 ) | ψ ̂ ( 2 ) ) E i = ( i , j ) Re { ψ ̂ ( 1 ) ( i , j ) ψ ̂ ( 2 ) ( i , j ) * } .
A h = f ( β ) ϑ = i Ψ ̂ ϑ .
[ ψ ̂ i | A ϑ ] E i = [ ψ ̂ i | i Ψ ̂ ϑ ] E i = [ Re { i Ψ ̂ * ψ ̂ i } | ϑ ] E i = [ P Re { i Ψ ̂ * ψ ̂ i } | ϑ ] E i = [ P Im { Ψ ̂ * ψ ̂ i } | ϑ ] E 0 .
A * ψ ̂ i = P Im { Ψ ̂ * ψ ̂ i } .
A * A h = P Im { i Ψ ̂ * Ψ ̂ } ϑ = P ρ 2 ϑ ( ρ = | Ψ ̂ | = | ψ ̂ | ) .
A * k = P Im { Ψ ̂ * × ( Ψ ̂ g Φ ̂ ) } = P Im { g Φ ̂ Ψ ̂ * } .
P ρ 2 P ϑ = P ρ 2 κ i or ( ρ P ) * ( ρ P ) ϑ = ( ρ P ) * ( ρ κ i ) .
ϑ ¯ ( ρ κ i ) ( ρ P ) ϑ ¯ = ¯ ρ × ( κ i ϑ ) ¯ .
f ( β ) = g e × ( ξ e ξ ) ( ξ = exp i γ , γ = C β ) .
[ ξ ( 1 ) | ξ ( 2 ) ] E i = ( i , j , k ) K Re { ξ ( 1 ) ( i , j , k ) ξ ( 2 ) ( i , j , k ) * } .
A h = f ( β ) ϑ = i g e ξ C ϑ .
[ ξ i | A ϑ ] = [ ξ i | i g e ξ C ϑ ] E i = [ Re { i g e ξ * ξ i } | C ϑ ] E i = [ C * ( Re { i g e ξ * ξ i } | ϑ ] E 0 = [ C * g e Im { ξ * ξ i } | ϑ ] E 0 .
A * ξ i = C * g e Im { ξ * ξ i } .
A * A h = C * g e Im { ξ * i g e ξ } C ϑ = C * g e 2 C ϑ
A * k = C * g e Im { ξ * g e ( ξ e ξ ) } = C * g e 2 Im { ξ e ξ * } .
C * g e 2 C ϑ = C * g e 2 κ i or ( g e C ) * ( g e C ) ϑ = ( g e C ) * ( g e κ i ) .
ϑ ¯ ( g e κ i ) ( g e C ) ϑ ¯ .

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