Abstract
This paper presents an extension to the recently introduced class of nonlinear filters known as Aperture Filters. By taking a multiresolution approach, it can be shown that more accurate filtering results (in terms of mean absolute error) may be achieved compared to the standard aperture filter given the same size of training set. Most optimisation techniques for nonlinear filters require a knowledge of the conditional probabilities of the output. These probabilities are estimated from observations of a representative training set. As the size of the training set is related to the number of input combinations of the filter, it increases very rapidly as the number of input variables increases. It can be impossibly large for all but the simplest binary filters. In order to design nonlinear filters of practical use, it is necessary to limit the size of the search space i.e. the number of possible filters (and hence the training set size) by the application of filter constraints. Filter constraints take several different forms, the most general of which is the window constraint where the output filter value is estimated from only a limited range of input variables.
Aperture filters comprise a special case of nonlinear filters in which the input window is limited not only in its domain (or duration) but also in its amplitude. The reduced range of input signal leads directly to a reduction in the size of training set required to produce accurate output estimates. However in order to solve complex filtering problems, it is necessary for the aperture to be sufficiently large so as to observe enough of the signal to estimate its output accurately.
In this paper it is shown how the input range of the aperture may be expanded without increasing the size of the search space by adopting a multiresolution approach. The constraint applied in this case is the resolution constraint. This paper presents both theoretical and practical results to demonstrate and quantify the improvement.
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Green, A.C., Dougherty, E.R., Marshall, S. et al. Optimal Filters with Multiresolution Apertures. Journal of Mathematical Imaging and Vision 20, 237–250 (2004). https://doi.org/10.1023/B:JMIV.0000024041.44525.e6
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DOI: https://doi.org/10.1023/B:JMIV.0000024041.44525.e6