Foundations of Physics

, Volume 33, Issue 11, pp 1691–1698 | Cite as

The Mermin Fixed Point

  • Veit Elser
Article

Abstract

The most efficient known method for solving certain computational problems is to construct an iterated map whose fixed points are by design the problem's solution. Although the origins of this idea go back at least to Newton, the clearest expression of its logical basis is an example due to Mermin. A contemporary application in image recovery demonstrates the power of the method.

Mermin fixed point 

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References

  1. 1.
    N. D. Mermin, private communication and old Physics 209 homework problem (unpublished).Google Scholar
  2. 2.
    R. P. Millane, “Phase retrieval in crystallography and optics”, J. Opt. Soc. Am. A 7, 394–411 (1990).Google Scholar
  3. 3.
    J. R. Fienup, T. R. Crimmins, and W. Holsztynski, “Reconstruction of the support of an object from the support of its autocorrelation”, J. Opt. Soc. Am. 72, 610–624 (1982).Google Scholar
  4. 4.
    H. Stark and Y. Yang, Vector Space Projections (Wiley, 1998).Google Scholar
  5. 5.
    J. Miao, D. Sayre, and H. N. Chapman, “Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects”, J. Opt. Soc. Am. A 15, 1662–1669 (1998).Google Scholar
  6. 6.
    V. Elser, “Phase retrieval by iterated projections”, J. Opt. Soc. Am. A 20, 40–55 (2003).Google Scholar
  7. 7.
    J. R. Fienup, “Phase retrieval algorithms: A comparison”, Appl. Opt. 21, 2758–2769 (1982).Google Scholar
  8. 8.
    H. H Bauschke, P. L. Combettes, and D. R. Luke, “Phase retrieval, Gerchberg–Saxton algorithm, and Fienup variants: A view from convex optimization”, J. Opt. Soc. Am. A 19, 1334–1345 (2002).Google Scholar
  9. 9.
    M. Zwick, B. Lovell, and J. Marsh, “Global optimization studies on the 1-D phase problem”, Internat. J. Gen. Syst. 25, 47–59 (1996).Google Scholar

Copyright information

© Plenum Publishing Corporation 2003

Authors and Affiliations

  • Veit Elser
    • 1
  1. 1.Department of PhysicsCornell UniversityIthaca

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