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
(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
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)