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Nonparametric estimation of the singularities of a signal from noisy measurements


Authors: A. I. Katsevich and A. G. Ramm
Journal: Proc. Amer. Math. Soc. 121 (1994), 1221-1234
MSC: Primary 62G05
MathSciNet review: 1227518
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Abstract: We study a problem of locating and estimating singularities of a signal measured with noise on a discrete set of points (fixed-design model). The signal consists of a smooth part with bounded first derivative and of finite number of singularities of the type $ (x - {t_i})_ \pm ^p{d_i},0 \leq p \leq \frac{1}{2}$. The case $ p = 0$ corresponds to a piecewise continuous function. The algorithm is based on convolving the data with a kernel having compact support. Optimal bandwidth of the kernel is calculated, the consistency of the algorithm is proved. The results of testing the proposed algorithm on model examples are presented.


References [Enhancements On Off] (What's this?)

  • [1] L. S. Davis, A survey of edge detection techniques, Comput. Graphics Image Processing 4 (1975), 248-270.
  • [2] D. Girard, From template matching to optimal approximation by piecewise smooth curves, Curves and Surfaces in Computer Vision and Graphics, vol. 1251, The Society of Photo-Optical Instrumentation Engineers, Washington, DC, 1990, pp. 174-182.
  • [3] D. Girard and P.-J. Laurent, Splines and estimation of nonlinear parameters, Mathematical methods in computer aided geometric design (Oslo, 1988), Academic Press, Boston, MA, 1989, pp. 273–298. MR 1022714
  • [4] Wolfgang Härdle, Applied nonparametric regression, Econometric Society Monographs, vol. 19, Cambridge University Press, Cambridge, 1990. MR 1161622
  • [5] W. K. Pratt, Digital image processing, 2nd ed., Wiley-Interscience, New York, 1991.
  • [6] A. G. Ramm, Multidimensional inverse scattering problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1211417
  • [7] A. G. Ramm, Random fields estimation theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 48, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 1103995
  • [8] A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 1, 109–115. MR 1168516, 10.1090/S0273-0979-1993-00350-1
  • [9] Grace Wahba, Spline models for observational data, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 59, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1990. MR 1045442

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1994-1227518-3
Keywords: Noisy data, singularity localization, kernel estimation
Article copyright: © Copyright 1994 American Mathematical Society