## Signal processing under uncertain conditions by parallel projections onto fuzzy sets

JOSA A, Vol. 16, Issue 7, pp. 1602-1611 (1999)

http://dx.doi.org/10.1364/JOSAA.16.001602

Enhanced HTML Acrobat PDF (588 KB)

### Abstract

Projection methods have been shown to be extremely powerful for signal restoration and other signal and image processing tasks. However, they break down when some system parameters cannot be exactly defined. To mitigate this problem, we propose a new algorithm that is based on the extended parallel projection method combined with fuzzy set theory. The incompleteness of the available information is taken into account by considering the constraint sets and/or the iterates as fuzzy. Then some maximum membership is searched by using parallel projections. The introduction of the fuzzy sets formalism results in a flexible technique that improves substantially the results obtained by conventional methods. The conventional projection method is shown to be a special case of this fuzzy algorithm. Moreover, whereas in the conventional algorithm the projection weights were chosen arbitrarily, in the new algorithm they are related to the degree of uncertainty involved.

© 1999 Optical Society of America

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(070.5010) Fourier optics and signal processing : Pattern recognition

(070.6020) Fourier optics and signal processing : Continuous optical signal processing

(100.3010) Image processing : Image reconstruction techniques

**History**

Original Manuscript: February 3, 1999

Manuscript Accepted: February 16, 1999

Published: July 1, 1999

**Citation**

David Lyszyk and Joseph Shamir, "Signal processing under uncertain conditions by parallel projections onto fuzzy sets," J. Opt. Soc. Am. A **16**, 1602-1611 (1999)

http://www.opticsinfobase.org/josaa/abstract.cfm?URI=josaa-16-7-1602

Sort: Year | Journal | Reset

### References

- P. L. Combettes, H. J. Trussell, “The use of noise properties in set theoretic estimation,” IEEE Trans. Signal Process. 39, 1630–1641 (1991). [CrossRef]
- M. Goldburg, R. J. Marks, “Signal synthesis in the presence of an inconsistent set of constraints,” IEEE Trans. Circuits Syst. CAS-32, 647–663 (1985). [CrossRef]
- A. Levi, H. Stark, “Signal restoration from phase by projections onto convex sets,” J. Opt. Soc. Am. 73, 810–822 (1983). [CrossRef]
- A. Levi, H. Stark, “Image restoration by the method of generalized projections with application to restoration from amplitude,” J. Opt. Soc. Am. A 1, 932–943 (1984). [CrossRef]
- M. I. Sezan, H. Stark, “Image restoration by convex projections in the presence of noise,” Appl. Opt. 22, 2781–2789 (1983). [CrossRef] [PubMed]
- D. C. Youla, V. Velasco, “Extensions of a result on the synthesis of signals in the presence of inconsistent constraints,” IEEE Trans. Circuits Syst. CAS-33, 465–468 (1986). [CrossRef]
- P. L. Combettes, H. J. Trussell, “Method of successive projections for finding a common point of sets in metric spaces,” J. Optim. Theory Appl. 67, 487–507 (1990). [CrossRef]
- T. Kotzer, N. Cohen, J. Shamir, “Generalized projection algorithms with applications to optics and signal restoration,” Opt. Commun. 156, 77–91 (1998). [CrossRef]
- T. Kotzer, N. Cohen, J. Shamir, “A projection-based algorithm for consistent and inconsistent constraints,” SIAM (Soc. Ind. Appl. Math.) J. Optim. 7, 527–546 (1997).
- R. Aharoni, Y. Censor, “Block-iterative methods for parallel computation of solutions to convex feasibility problems,” Linear Algebr. Appl. 120, 165–175 (1989). [CrossRef]
- P. L. Combettes, “Signal recovery by best feasible approximation,” IEEE Trans. Image Process. 2, 269–271 (1993). [CrossRef] [PubMed]
- P. L. Combettes, “Inconsistent signal feasibility problems: least-squares solutions in a product space,” IEEE Trans. Signal Process. 42, 2955–2966 (1994). [CrossRef]
- Y. Censor, T. Elfving, “A multiprojection algorithm us-ing Bregman projections in a product space,” Numer. Algorithms 8, 221–239 (1994). [CrossRef]
- T. Kotzer, N. Cohen, J. Shamir, “Image reconstruction by a novel parallel projection onto constraint set method,” Opt. Lett. 20, 1172–1174 (1995). [CrossRef] [PubMed]
- T. Kotzer, J. Rosen, J. Shamir, “Application of serial and parallel projection methods to correlation filter design,” Appl. Opt. 34, 3883–3895 (1995). [CrossRef] [PubMed]
- M. R. Civanlar, H. J. Trussell, “Digital signal restoration using fuzzy sets,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-34, 919–936 (1986). [CrossRef]
- S. Oh, R. J. Marks, “Alternating projection onto fuzzy convex sets,” in Proceedings of 1993 IEEE Conference on Fuzzy Systems, pp. 148–155.
- E. Cox, The Fuzzy Systems Handbook (Academic, San Diego, Calif., 1994).
- M. M. Gupta, R. K. Ragade, R. R. Yager, Advances in Fuzzy Set Theory and Applications (North-Holland, New York, 1979).
- A. Kandel, Fuzzy Mathematical Techniques with Applications (Addison-Wesley, Reading, Mass., 1986).
- A. Kaufmann, Introduction to the Theory of Fuzzy Subsets (Academic, New York, 1975), Vol. I.
- L. A. Zadeh, “From circuit theory to system theory,” Proc. IRE 50, 856–865 (1962). [CrossRef]
- L. A. Zadeh, “Fuzzy sets,” Inf. Control. 8, 338–353 (1965). [CrossRef]
- L. A. Zadeh, “Fuzzy sets and systems,” in System Theory, Microwave Research Institute Symposia Series XV, J. Fox, ed. (Polytechnic, Brooklyn, N.Y.1965), pp. 29–37.
- C. V. Negoita, “The current interest in fuzzy optimization,” Fuzzy Sets Syst. 6, 261–269 (1981). [CrossRef]
- R. Pearce, P. H. Cowley, “Use of fuzzy logic to describe constraints derived from engineering judgment in genetic algorithms,” IEEE Trans. Ind. Electron. 43, 535–540 (1996). [CrossRef]
- W. Pedrycz, Fuzzy Control and Fuzzy Systems (Wiley, New York, 1989).
- J. Ramı́k, “Extension principle in fuzzy optimization,” Fuzzy Sets Syst. 19, 29–35 (1986). [CrossRef]
- G. Pierra, “Decomposition through formalization in a product space,” Math. Program. 18, 96–115 (1984). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.