Digital signal restoration using fuzzy sets

A new signal restoration method with considerable flexibility in incorporating a priori information is developed. The method defines a fuzzy set for each piece of information to restrict the set of acceptable solutions. Using fuzzy sets makes it possible to model partially defined information as well as exact knowledge. The intersection of all the fuzzy sets is the feasibility set. The original signal is a member of this set with a high membership value, and any high membership valued element of this set is a nonrejectable solution. Such solutions can be computed by using optimization techniques. Ideally, the feasibility set contains only the original signal. The chance of recovering the original signal decreases as the feasibility set gets larger. Thus, the size of the feasibility set gives a quality measure for the solution. The method generated successful results in many restorations for which the conventional techniques have failed, and may be applied in image coding and tomography.

[1]  A. Caramazza,et al.  A Fuzzy Set Approach to Modifiers and Vagueness in Natural Language , 2005 .

[2]  Albert N. Badre,et al.  On the Precision of Adjectives Which Denote Fuzzy Sets , 1974 .

[3]  S. P. Han,et al.  Variable metric methods for minimizing a class of nondifferentiable functions , 1977, Math. Program..

[4]  M. Sezan,et al.  Image Restoration by the Method of Convex Projections: Part 2-Applications and Numerical Results , 1982, IEEE Transactions on Medical Imaging.

[5]  Azriel Rosenfeld,et al.  Digital Picture Processing , 1976 .

[6]  A. Lent,et al.  An iterative method for the extrapolation of band-limited functions , 1981 .

[7]  H. Trussell,et al.  Convergence criteria for iterative restoration methods , 1983 .

[8]  W. Zangwill.  An Algorithm for the Chebyshev Problem---With an Application to Concave Programming , 1967 .

[9]  Witold Pedrycz,et al.  Contribution to application of energy measure of fuzzy sets , 1982 .

[10]  D. Youla,et al.  Image Restoration by the Method of Convex Projections: Part 1ߞTheory , 1982, IEEE Transactions on Medical Imaging.

[11]  O. Strand.  Theory and methods related to the singular-function expansion and Landweber's iteration for integral equations of the first kind , 1974 .

[12]  H. Trussell,et al.  Constructing membership functions using statistical data , 1986 .

[13]  M. Reha Civanlar,et al.  The initial estimate in constrained iterative restoration , 1983, ICASSP.

[14]  A. Conn,et al.  An Efficient Method to Solve the Minimax Problem Directly , 1978 .

[15]  Witold Pedrycz,et al.  SOME ASPECTS OF FUZZY DECISION‐MAKING , 1982 .

[16]  Mehmet Reha Civanlar,et al.  Digital signal restoration using projection and fuzzy set techniques , 1985 .

[17]  Hiroaki Takajo,et al.  Improved Methods of Maximum a Posteriori Restoration , 1980 .

[18]  Bobby R. Hunt,et al.  Bayesian Methods in Nonlinear Digital Image Restoration , 1977, IEEE Transactions on Computers.

[19]  L. Landweber.  An iteration formula for Fredholm integral equations of the first kind , 1951 .

[20]  P. Neumann.  W. Freiberger und U. Grenander, A Short Course in Computational Probability and Statistics. (Applied Mathematical Sciences, Vol. 6). XII + 155 S. m. 35 Fig. Berlin/Heidelberg/New York 1971. Springer-Verlag. Preis brosch. DM 24,— , 1973 .

[21]  Kaj Madsen,et al.  Combined lp and quasi-Newton methods for minimax optimization , 1981, Math. Program..

[22]  H. Trussell.  Maximum power signal restoration , 1981 .

[23]  H. Trussell,et al.  The feasible solution in signal restoration , 1984 .

[24]  Jr. Edwin Randolf Cole.  The removal of unknown image blurs by homomorphic filtering , 1973 .

[25]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[26]  M. Nowakowska.  Methodological problems of measurement of fuzzy concepts in the social sciences , 1977 .

[27]  D. G. Watts,et al.  Spectral analysis and its applications , 1968 .

[28]  A. Kandel,et al.  Fuzzy sets, fuzzy algebra, and fuzzy statistics , 1978, Proceedings of the IEEE.

[29]  R.W. Schafer,et al.  Constrained iterative restoration algorithms , 1981, Proceedings of the IEEE.

[30]  A. Papoulis.  A new algorithm in spectral analysis and band-limited extrapolation. , 1975 .

[31]  Ulf Grenander,et al.  A Short Course In Computational Probability And Statistics , 1971 .

[32]  P. H. Cittert.  Zum Einfluß der Spaltbreite auf die Intensitätsverteilung in Spektrallinien. II , 1930 .

[33]  G. Zoutendijk,et al.  Mathematical Programming Methods , 1976 .

[34]  B. R. Hunt,et al.  The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer , 1973, IEEE Transactions on Computers.

[35]  Boris Polyak,et al.  The method of projections for finding the common point of convex sets , 1967 .

[36]  R. Gerchberg.  Super-resolution through Error Energy Reduction , 1974 .

[37]  R. A. Fox,et al.  Introduction to Mathematical Statistics , 1947 .