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Journal of the Optical Society of America A

Journal of the Optical Society of America A

| OPTICS, IMAGE SCIENCE, AND VISION

  • Vol. 4, Iss. 1 — Jan. 1, 1987
  • pp: 171–179

Ill-posedness and precision in object-field reconstruction problems

William L. Root  »View Author Affiliations


JOSA A, Vol. 4, Issue 1, pp. 171-179 (1987)
http://dx.doi.org/10.1364/JOSAA.4.000171


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Abstract

The inherent stability or instability in reconstructing an object field, in the presence of observation noise, for a class of ill-posed problems is investigated for situations in which constraints are imposed on the object fields. The class of ill-posed problems includes inversion of truncated Fourier transforms. Two kinds of constraint are considered. It is shown that if the object field is restricted to a subset of L2 space over Rn that is bounded, closed, convex, and has nonempty interior, then a (nonlinear) least-squares estimate always exists but is unstable. It is also shown that if one is primarily concerned with the situation in which the object field belongs to a compact parallelepiped in L2, aligned in a natural way, there is a satisfactory, stable linear estimate that is optimal according to a min–max criterion. This also leads to a nonlinear modification for the case in which the object field is actually restricted to the parallelepiped. A summary of some relevant mathematical background is included.

© 1987 Optical Society of America

History
Original Manuscript: April 18, 1986
Manuscript Accepted: September 9, 1986
Published: January 1, 1987

Citation
William L. Root, "Ill-posedness and precision in object-field reconstruction problems," J. Opt. Soc. Am. A 4, 171-179 (1987)
http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-4-1-171


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References

  1. A proper definition of white noise in Hilbert space cannot be made with a (conventional) countably additive probability measure. One approach is to use finitely additive measures. See Ref. 5, Appendix B and further references given there. All this does not really matter here, because when one projects on any finite-dimensional subspace, the induced white-noise vector is well defined in the usual way.
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  12. A. Albert, Regression and the Moore–Penrose Pseudoinverse (Academic, New York, 1972).
  13. D. C. Youla, H. Webb, “Image restoration by the method of convex projections: Part 1—theory,”IEEE Trans Med. Imaging MI-1, 81–84 (1982). [CrossRef]
  14. W. L. Root, “Estimation in identification theory,” in Proceedings of the Ninth Allerton Conference on Circuit and System Theory (U. Illinois Press, Urbana, Ill., 1971), pp. 1–10.
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