Two-Dimensional Phase Restoration
The constraints laid on the phase of a Fourier transform by its intensity are reviewed in the contexts of well known phase problems. The considerable differences between phase problems involving one-dimensional and multi-dimensional images, and finite-sized (as arise in astronomy, for instance) and periodic (as occur in crystallography) images, are explained. The crucial importance, for uniqueness questions, of the concept of the image-form (and also its most compact manifestation) is emphasised, as is the almost always unique connection between the image-form of a positive multi-dimensional image and the intensity of its Fourier transform. The current status of phase recovery algorithms, as regards Fourier transforms of finite-sized images, is assessed. The necessity for composite algorithms, incorporating simple but powerful constructions, is pleaded and reinforced by computational examples illustrating our previously reported defogging routine and a new procedure called fringe magnification.
KeywordsImage Space Fourier Space Phase Retrieval Phase Problem Central Lobe
Unable to display preview. Download preview PDF.
- Bates, R.H.T., 1982a, Fourier phase problems are uniquely solvable in more than one dimension. I: underlying theory, Optik, 61: 247–262.Google Scholar
- Bates, R.H.T., and Fright, W.R., 1984, Reconstructing images from their Fourier “intensities”, in: “Advances in Computer Vision and Image Processing Vol. 1” T.S. Huang, ed., J.A.I. Press, in press.Google Scholar
- Bates, R.H.T., Fright, W.R., and Norton, W.A., 1984, Phase restoration is successful in the optical as well as the computational laboratory, in “Indirect Imaging”, J.A. Roberts, ed. Cambridge Univ. Press, Cambridge, pp.119–124.Google Scholar
- Bracewell, R.N., 1978, “Fourier Transform and its Applications”, McGraw-Hill (2nd edn), New York.Google Scholar
- Brigham, E.D., 1974, “Fast Fourier Transform”, Prentice-Hall, New Jersey.Google Scholar
- Fienup, J.R., 1984, Experimental evidence of the uniqueness of phase retrieval from intensity data, in: “Indirect Imaging”, J.A. Roberts, ed., Cambridge Univ. Press, Cambridge, pp. 99–109.Google Scholar
- Fright, W.R., Bates, R.H.T., 1982, Fourier phase problems are uniquely solvable in more than one dimension. III: Computational examples for two dimensions, Optik, 62: 219–230.Google Scholar
- Kiedron, P., 1981, On the 2-D solution ambiguity of the phase recovery problem, Optik, 59: 303–309.Google Scholar
- Labeyrie, A ., 1970, Attainment of diffraction-limited resolution in large telescopes by Fourier analysing speckle patterns in star images, Astronomy&Astrophysics, 6: 85–87.Google Scholar
- Napier, PJ., Bates, RHT., 1974, Inferring phase information from modulus information in two-dimensional aperture synthesis, Astronomy&Astrophysics Supplement Series, 15: 427–430.Google Scholar
- Ramachandran, G., and Srinivasan, R., 1970, “Fourier Methods in Crystallography”, Wiley, New York.Google Scholar