Signal reconstruction by the method of regularization

Authors

  • V.A. Morozov

Keywords:

восстановление зашумленных сигналов
численный анализ
численные методы
метод регуляризации
неограниченные операторы
некорректные задачи
сходимость

Abstract

The problem on reconstruction of signals with noise is considered as that of calculating the values of an unbounded operator by Tikhonov’s method of regularization. Several techniques (theoretically or pragmatically justified) for choosing the regularization parameter are discussed. The formulation and usage of a priori knowledge on a structure of the sought-for useful signal in time and frequency regions are allowed. The conceptions of functional analysis are used; this permits us to strictly justify our theoretical conclusions as well as to provide the breadth of possible applications in various fields of science related to the processing of experimental data. The problem statement considered in this paper corresponds to the case of direct measurements. The influence of high-level noise is studied.

New computational technologies

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Published

2001-05-20

Issue

Vol. 2 (2001): Issue 1.

Section

Section 1. Numerical methods and applications

Author Biography

V.A. Morozov

Lomonosov Moscow State University,
Research Computing Center

References

  1. Тихонов А.Н., Арсенин В.Я. Методы решения некорректных задач. М.: Наука, 1974.
  2. Morozov V.A. Methods for solving incorrectly posed problems. New York: Springer-Verlag, 1984.
  3. Morozov V.A. Regularization methods for ill-posed problems. London: CRC Press, 1993.
  4. Groetsch C.W. The theory of Tikhonov regularization for Fredholm equations of the first kind. Boston: Pitman, 1984.
  5. Vogel C.R. Total variation regularizations for ill-posed problems. Report. Dept. of Mathem. Sci., Montana State Univ., USA, 1993.
  6. Malyshev V.A., Morozov V.A. Linear semigroups and differential inequalities. Moscow: Moscow University Press, 1995.
  7. Youla D.C., Webb H. Image reconstruction by the method of convex projections // IEEE Trans. on Medical Imaging. 1982, T. MI-1, N 2. 81-94.
  8. Wahba G. Practical approximate solutions of linear operator equations when the data are noisy // SIAM Journal of Numerical Analysis. 1977. 14, N 4. 651-667.
  9. Morozov V.A. Theory of splines and problems of stable calculation of values of unbounded operators // J. Comp. Math. and Math. Phys. 1971. 11, N 3. 545-558.
  10. Hansen P.Ch. Analysis of discrete ill-posed problems by means of the L-curve // SIAM Review. 1992. 34. 561-580.
  11. Морозов В.А., Поспелов В.В. Цифровая обработка сигналов. М.: Изд-во Моск. ун-та, 1986.
  12. Fitzpabrick B.G., Keeling S.L. On approximation in total variation penalization for image reconstruction and inverse problems // J. Numer. Funct. Anal. and Optimiz. 1997. 18, N 9. 941-950.
  13. Nashed M.Z., Scherzer O. Least squares and bounded variation regularization with nondifferentiable functions // J. Numer. Funct. Anal. and Optimiz. 1998. 19, N 7. 873-901.
  14. Морозов В.А. Об устойчивых численных методах решения совместных систем линейных алгебраических уравнений // Журн. вычисл. матем. и матем. физики. 1984. 24, № 2. 179-186.