On restoration of noisy signals by a regularization method
Keywords:
regularization, Hilbert space, unbounded operator, weak convergence, convergence in norm, Euler inequalities, Euler identitiesAbstract
The problem for restoration of noisy signals is treated as a problem of calculation of certain values for an unbounded operator. In this connection, a Tikhonov regularization method is used. Completely theoretically justified approaches to the choice of the regularization parameter as well as pragmatic approaches based on intuitive reasons are considered. The usage of some a priori information on the structure of the required «useful» signal in the «temporary» area as well as in the «frequent» one is allowed. The discussion is given in terms of functional analysis. This permits a strict substantiation of all considerations and provides a further range of possible applications in various fields of knowledge connected with processing of experimental data. The formulation of the problem corresponds to the case of direct measurements. The effect of «large» noises is also analyzed. This work was supported by the Russian Foundation for Basic Research (project N 10-01-00297a).
References
- Tikhonov A.N., Arsenin V.Ya. Methods for solving ill-posed problems. Moscow: Nauka, 1974.
- Morozov V.A. Methods for solving incorrectly posed problems. New York: Springer, 1984.
- Morozov V.A. Regularization methods for ill-posed problems. London: CRC Press, 1993.
- Ivanov V.K., Vasin V.V., Tanana V.P. The theory of linear ill-posed problems and its applications. Moscow: Nauka, 1978.
- Vasin V.V., Ageev A.L. Ill-posed problems with a priori information. Ultrecht: VSP, 1995.
- Groetsch C.W. The theory of Tikhonov regularization for Fredholm equations of the first kind. Boston: Pitman, 1984.
- Vogel C.R. Total variation regularizations for ill-posed problems. Technical Report. Departement of Mathem. Sci. Montana State Univ., 1993.
- Malyshev V.A., Morozov V.A. Linear semigroups and differential inequalities. Moscow: Moscow Univ. Press, 1995.
- Youla D.C., Webb H. Image reconstruction by the method of convex projections // IEEE Trans. Medical Imaging. 1982. 1, N 2. 81-94.
- Wahba G. Practical approximate solutions of linear operator equations when the data are noisy // SIAM J. Numer. Anal. 1977. 14, N 4. 651-667.
- Morozov V.A. Theory of splines and problems of stable calculation of values of unbounded operators // J. Comp. Math. Math. Phys. 1971. 11, N 3. 545-558.
- Hansen P.Ch. Analysis of discrete ill-posed problems by means of the L-curve // SIAM Review. 1992. 34. 561-580.
- Morozov V.A. Some aspects of restoration of signals by a regularization method // Proc. of the Fourth Int. Conf. on Recent Advances in Numerical Methods and Applications II. Singapore: World Scientific Publ., 1999. 52-62.
- Groetsch C.W. Stable approximate evaluation of unbounded operators. Berlin: Springer, 2007.
