Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 1, Pages 1-14
doi:10.1155/S1048953300000010
Controlling the Gibbs phenomenon in noisy deconvolution of irregular multivariable input signals
1Texas Tech University, Department of Mathematics and Statistics, Lubbock 79409, TX, USA
2Katholieke Universiteit Nijmegen, Department of Mathematics, Nijmegen 6525 ED, The Netherlands
Received 1 January 1998; Revised 1 November 1998
Copyright © 2000 Kumari Chandrawansa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
An example of inverse estimation of irregular multivariable signals is provided by picture restoration. Pictures typically have sharp edges and therefore will be modeled by functions with discontinuities, and they could be
blurred by motion. Mathematically, this means that we actually observe
the convolution of the irregular function representing the picture with a
spread function. Since these observations will contain measurement errors,
statistical aspects will be pertinent. Traditional recovery is corrupted by
the Gibbs phenomenon (i.e., overshooting) near the edges, just as in the
case of direct approximations. In order to eliminate these undesirable
effects, we introduce an integral Cesàro mean in the inversion procedure,
leading to multivariable Fejér kernels. Integral metrics are not sufficiently
sensitive to properly assess the quality of the resulting estimators. Therefore, the performance of the estimators is studied in the Hausdorff metric,
and a speed of convergence of the Hausdorff distance between the graph of
the input signal and its estimator is obtained.