Hostname: page-component-586b7cd67f-2brh9 Total loading time: 0 Render date: 2024-11-25T12:03:27.228Z Has data issue: false hasContentIssue false
  • English
  • Français

On Gerchberg's method for the Fourier inverse problem

Published online by Cambridge University Press:  17 February 2009

T. J. Connolly ,
K. A. Landman  and
L. R. White
L. R. White
Affiliation:
Department of Mathematics, University of Melbourne, Parkville Victoria 3052, Australia.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

If a finite segment of a spectrum is known, the determination of the finite object function in image space (or the full spectrum in frequency space) is a fundamental problem in image analysis. Gerchberg's method, which solves this problem, can be formulated as a fixed point iteration. This and other related algorithms are shown to be equivalent to a steepest descent method applied to the minimization of an appropriate functional for the Fourier Inversion Problem. Optimal steepest descent and conjugate gradient methods are derived. Numerical results from the application of these techniques are presented. The regularization of the problem and control of noise growth in the iteration are also discussed.

Type
Research Article
Information
The ANZIAM Journal , Volume 37 , Issue 1 , July 1995 , pp. 26 - 44
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Adams, R. A., Sobolev spaces (Academic, New York, 1975).Google Scholar
[2]Bates, R. H. T. and McDonnell, M. J., Image restoration and reconstruction (Clarendon Press, Oxford, 1986).Google Scholar
[3]Bertero, M., “Regularization methods for linear inverse problems”, Inverse problems. Lecture notes in mathematics 1225 (1986) 52–112.CrossRefGoogle Scholar
[4]Bertero, M., De Mol, C. and Viano, G. A., “The stability of inverse problems”, in Inverse scattering problems in optics (ed. Baltes, H. P.), (Springer Verlag, Berlin, 1979) 161–213.Google Scholar
[5]Brakhage, Helmut, “On ill-posed problems and the method of conjugate gradients”, in Inverse and ill-posed problems (eds Engl, Heinz W. and Groetsch, C. W.), (1987) 165–175.CrossRefGoogle Scholar
[6]Bregman, L. M., “The method of successive projection for finding a common point in convex sets”, Sov. Math. Dokl. 6 (1965) 688–692.Google Scholar
[7]Connolly, T. J., “Nonlinear methods for inverse problems”, Ph. D. Thesis, University of Canterbury, Christchurch, New Zealand, 1990.Google Scholar
[8]Davies, A. R., Iqbal, M., Maleknejad, K. and Redshaw, T. C., “A comparison of statistical regularization and Fourier extrapolation methods for numerical convolution”, in Numerical treatment of inverse problems in differential and integral equations (eds Deufhlard, P. and Hairer, E.), (Birkhauser-Verlag, 1983) 320–334.CrossRefGoogle Scholar
[9]Eicke, B., Louis, A. K. and Plato, R., “The instability of some gradient methods for ill-posed problems”, Numer. Math. 58 (1990) 129–134.CrossRefGoogle Scholar
[10]Fienup, J. R., “Reconstruction of an object from the modulus of its Fourier transform”, Optics Letters 3 (1978) 27–29.CrossRefGoogle ScholarPubMed
[11]Fienup, J. R., “Phase retrieval algorithms: a comparison”, Appl. Optics 21 (1982) 2758–2769.CrossRefGoogle ScholarPubMed
[12]Gerchberg, R. W., “Super-resolution through error energy reduction”, Optica Acta 21 (1974) 709–720.CrossRefGoogle Scholar
[13]Grünbaum, F. A., “A Study of Fourier space methods for limited angle image reconstruction”,Numer. Funct. Anal. and Optimiz. 2 (1980) 31–42.CrossRefGoogle Scholar
[14]Hanke, M., “Accelerated Landweber iterations for the solution of ill-posed problems”, Numerische Matematik 60 (1991) 341–373.CrossRefGoogle Scholar
[15]Hansen, Per Christian, “Analysis of discrete ill-posed problems by means of the L -curve”, SIAM Review 34 (1992) 561–580.CrossRefGoogle Scholar
[16]Kammerer, W. J. and Nashed, M. Z., “Steepest descent of singular linear operators with non-closed range”, Applic. Anal. 1 (1971) 143–159.CrossRefGoogle Scholar
[17]Kammerer, W. J. and Nashed, M. Z., “On the convergence of the conjugate gradient method for singular linear operator equations”, SIAM J. Numer. Anal. 9 (1972) 165–181.CrossRefGoogle Scholar
[18]Landweber, L., “An iteration formula for Fredholm integral equations of the first kind”, Amer. J. Math. 73 (1951) 615–624.CrossRefGoogle Scholar
[19]Lent, A. and Tuy, H., “An iterative method for the extrapolation of band-limited functions”, J. of Math. Anal. and Applic. 83 (1981) 554–565.CrossRefGoogle Scholar
[20]Louis, Alfred K., “Convergence of the conjugate gradient method for compact operators”, in Inverse and ill-posed problems (eds Engl, Heinz W. and Groetsch, C. W.), (1987) 177–183.CrossRefGoogle Scholar
[21]Luenberger, D. G., Introduction to linear and nonlinear programming (Addison-Wesley, Reading, 1965).Google Scholar
[22]Wan, J. A. Lum and White, L. R., “A novel technique for the calculation of site energy density from gas absorption isotherm data”, J. Chem. Soc. Faraday Trans. 87 (1991) 3051–3062.Google Scholar
[23]Wan, J. A. Lum and White, L. R., “A numerical solution of inverse problems of Fourier convolution type”, App. Math. Modelling 15 (1991) 359–366.Google Scholar
[24]Miller, K., “Least squares method for illposed problems with a prescibed bound”, SIAM J. Math. Analy. 1 (1970) 52–74.CrossRefGoogle Scholar
[25]Nemirovskii, A. S., “The regularizing properties of the adjoint gradient method in ill-posed problems”, USSR Comput. Math. Math. Phys. 26 (2) (1986) 7–16.CrossRefGoogle Scholar
[26]Papoulis, A., “A new algorithm in spectral analysis and band-limited extrapolation”, IEEE Trans on Circuits and Systems CAS22 (1975) 735–742.CrossRefGoogle Scholar
[27]Schock, E., “Comparison principles for iterative methods”, in Inverse and ill-posed problems (eds Engl, Heinz W. and Groetsch, C. W.), (1987) 185–193.CrossRefGoogle Scholar
[28]Stark, H. (ed.), Image recovery: Theory and application (Academic Press, Orlando, 1981).Google Scholar
[29]Tikhonov, A. N. and Arsenin, V. Y., Solutions of ill-posed problems (V. H. Winston, Washington D C, 1977).Google Scholar
[30]Vasin, V. V., “Iterative methods for the approximate solution of ill-posed problems with a priori information and their applications”, in Inverse and ill-posed problems (eds Engl, Heinz W. and Groetsch, C. W.), (1987) 211–229.CrossRefGoogle Scholar
[31]Wahba, Grace, “Three topics in ill-posed problems”, in Inverse and ill-posed problems (eds Engl, Heinz W. and Groetsch, C. W.), (1987) 37–51.CrossRefGoogle Scholar