Differential Inclusions, Control and Optimization 20 (2000) 27-40
doi: 10.7151/dmdico.1002

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Dieter Schott

Hochschule Wismar, Fachbereich Elektrotechnik und Informatik
Philipp-Müller-Straβe, D-23952 Wismar, Germany
e-mail: d.schott@et.hs-wismar.de


The aim is to reconstruct a signal function x ∈ L2 if the phase of the Fourier transform [^x] and some additional a-priori information of convex type are known. The problem can be described as a convex feasibility problem. We solve this problem by different Fejér monotone iterative methods comparing the results and discussing the choice of relaxation parameters. Since the a-priori information is partly related to the spectral space the Fourier transform and its inverse have to be applied in each iterative step numerically realized by FFT techniques. The computation uses MATLAB routines.

Keywords: signal reconstruction, convex feasibility problem, projection onto convex sets, Fejér monotone iterative methods, Fourier transforms.

1991 Mathematics Subject Classification: 65J15, 49D20.


[1] H.H. Bauschke and J.M. Borwein, On projection algorithms for solving convex feasibility problems, SIAM Rev. 38 (1996), 367-426.
[2] P.L. Combettes, Fejér-monotonicity in convex optimization, in: C.A. Floudas and P.M. Pardalos (eds.), Encyclopedia of Optimization, Kluwer Acad. Publ., Dordrecht 2000.
[3] L.G. Gubin, B.T. Polyak and E.V. Raik, The method of projections for finding the common point of convex sets, USSR Comput. Math. Math. Phys. 7 (1967), 1-24.
[4] M.H. Hayes, J.S. Lim and A.V. Oppenheim, Signal reconstruction from phase or magnitude, IEEE Trans. Acoust. Speech and Signal Process. ASSP-28 (1980), 672-680.
[5] M.H. Hayes, The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform, IEEE Trans. Acoust. Speech and Signal Process. ASSP-30 (1982), 140-154.
[6] A. Levi and H. Stark, Restoration from Phase and Magnitude by Generalized Projections, in: [], Chapter 8, 277-320.
[7] D. Schott, Iterative solution of convex problems by Fejér monotone methods, Numer. Funct. Anal. Optimiz. 16 (1995), 1323-1357.
[8] D. Schott, Basic properties of Fejér monotone mappings, Rostock. Math. Kolloq. 50 (1997), 71-84.
[9] D. Schott, Weak convergence of iterative methods generated by strongly Fejér monotone methods, Rostock. Math. Kolloq. 51 (1997), 83-96.
[10] D. Schott, About strongly Fejér monotone mappings and their relaxations, Zeitschr. Anal. Anw. 16 (1997), 709-726.
[11] H. Stark (ed.), Image recovery: Theory and applications, Academic Press, New York 1987.
[12] D.C. Youla, Mathematical Theory of Image Restoration by the Method of Convex Projections, in: [], Chapter 2, 29-76.

Received 15 November 1999
Revised 15 February 2000