Abstract
The minimum-norm least-squares solution of the phase-closure equations of an interferometric array is very stable. Furthermore, this canonical solution can be obtained by simple backprojection, each closure phase leaving its “algebraic” imprint on the corresponding baselines. More precisely, the generalized inverse of the phase-closure operator C of an n-point array is equal to its adjoint (its Hermitian transpose) divided by n: C + =C * /n. Likewise, the generalized inverse of the phase-aberration operator B is equal to B * /n. These remarkable properties, which have so far remained unnoticed, play an essential part in the algebraic analysis of phase-closure imaging, and thereby in the understanding and the treatment of the inverse problems of aperture synthesis.
The applications presented in this paper concern the phase-factor restoration problem in optical interferometry and speckle imaging. We first propose a new iterative procedure for obtaining a particular least-squares solution. In the framework of this nonlinear technique, we then show how to initialize at each iteration the inner process of linear optimization. The backprojection method, which is the obvious choice in the case of weakly-redundant devices, is compared with the recursive techniques used in bispectral analysis for highly-redundant configurations. At the end of the restoration step under consideration, the phase indetermination reduces to a vector lying in the null space of the bispectral operator. The global reconstruction process is closely related to the regularization methods used for band-limited extrapolation. In this context, we outline the final hybrid procedure to be implemented, indicating how certain regularizing constraints can raise the intrinsic indeterminations related to the existence of the null space.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Arnot, N.R., Atherton, P.D., Greenaway, A.L. and Noordam, J.E. (1985), “Phase closure in optical astronomy,” Traitement du signal, Recherches 2, 129–135.
Bartelt, H., Lohmann, A.W. and Wirnitzer, B. (1984), “Phase and amplitude recovery from bispectra,” Appl. Optics 23, 3121–3129.
Cornwell, T.J. and Wilkinson, P.N. (1981), “A new method for making maps with unstable radio interferometers,” Mon. Not. R. astr. Soc. 196, 1067–1086.
Giacovazzo, C. (1980), Direct methods in crystallography, Academic Press, London, New York, Toronto, Sydney, San Francisco.
Golay, M.J.E. (1971), “Points arrays having compact, nonredundant autocorrelations,” J. Opt. Soc. Am. 61, 272–273.
Hofmann, K.H. and Weigelt, G. (1986), “High angular resolution shearing spectroscopy and triple shearing interferometry,” Appl. Optics 25, 4280–4287.
Lannes, A. and Pérez, J.Ph. (1983), Optique de Fourier en microscopie électronique, Masson, Paris.
Lannes, A., Roques, S. and Casanove, M.J. (1987a), “Stabilized reconstruction in signal and image processing; Part I: Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Optics 34, 161–226.
Lannes, A., Casanove, M.J. and Roques, S. (1987b), “Stabilized reconstruction in signal and image processing; Part II: Iterative reconstruction with and without constraint. Interactive implementation,” J. Mod. Optics 34, 321–370.
Lannes, A., Roques, S. and Casanove, M.J. (1987c), “Resolution and robustness in image processing: a new regularization principle,” J. Opt. Soc. Am. 4, 189–199.
Lannes, A., and Roques, S. (1987), “On the concept of resolution ellipse in aperture synthesis. Matched deconvolution with error analysis,” Proc. ESA workshop on Optical Interferometry in space, Granada, Spain, 45–48.
Lannes, A. (1988), “On a new class of iterative algorithms for phase-closure imaging and bispectral analysis,” Proc. NOAO-ESO Conference on High-resolution Imaging by Interferometry, Garching bei München, F.R.G., 169–180.
Lohmann, A.W., Weigelt, G. and Wirnitzer, B. (1983), “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Optics 22, 4028–4037.
Ortega, J.M. and Rheinboldt, W.C. (1970), Iterative solution of nonlinear equations in several variables, Academic Press, New York.
Pearson, T.J., and Readhead, A.C.S. (1984), “Image formation by self-calibration in radio astronomy,” Ann. Rev. Astron. Astrophys. 22, 97–130.
Readhead, A.C.S., Nakajima, T.S., Pearson, T.J., Oke, J.B. and Sargent, W.L.W. (1988), “Diffraction-limited imaging with ground-based optical telescopes,” Astron. J. 95, 1278–1296.
Rhodes, W.T. and Goodman, J.W. (1973), “Interferometric technique for recording and restoring images degraded by unknown aberrations,” J. Opt. Soc. Am. A 63, 647–657.
Roddier, F. (1986), “Triple correlation as a phase closure technique,” Opt. Commun. 60, 145–148.
Roddier, C. and Roddier, F. (1988), “Phase recovery from pupil-plane interferometry” Proc. NOAO-ESO Conference on High-resolution Imaging by Interferometry, Garching bei München, F.R.G., 257–265.
Rogers, A.E.E., Hinteregger, H.F., Whitney, A.R., Counselman, C.C., Shapiro, I.I., Wittels, J.J., Klemperer, W.K., Warnock, W.W., Clark, T.A., Hutton, L.K., Marandino, G.E., Ronnang, B.O., Rydbeck, O.E.H. and Niell, A.E. (1974), “The structure of radio sources 3C 273B and 3C 84 deduced from the closure phases and visibility amplitudes observed with three-element interferometers,” Astrophys. J. 193, 293–301
Rohlfs, K. (1986), Tools for Radio Astronomy, Springer-Verlag, Heidelberg, New York, Paris, Tokyo.
Schwab, F.R. and Cotton, W.D. (1983), “Global fringe search techniques for VLBI,” Astron. J. 88, 688–694.
Zirker, J.B. and Brown, T.M. (1986), “Phase recovery with dual nonredundant arrays,” J. Opt. Soc. Am. A 3, 2077–2081.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lannes, A. Backprojection mechanisms in phase-closure imaging. Bispectral analysis of the phase-restoration process. Exp Astron 1, 47–76 (1989). https://doi.org/10.1007/BF00414795
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00414795