Abstract
The standard median filter based on a symmetric moving window has only one tuning parameter: the window width. Despite this limitation, this filter has proven extremely useful and has motivated a number of extensions: weighted median filters, recursive median filters, and various cascade structures. The Hampel filter is a member of the class of decsion filters that replaces the central value in the data window with the median if it lies far enough from the median to be deemed an outlier. This filter depends on both the window width and an additional tuning parameter t, reducing to the median filter when t=0, so it may be regarded as another median filter extension. This paper adopts this view, defining and exploring the class of generalized Hampel filters obtained by applying the median filter extensions listed above: weighted Hampel filters, recursive Hampel filters, and their cascades. An important concept introduced here is that of an implosion sequence, a signal for which generalized Hampel filter performance is independent of the threshold parameter t. These sequences are important because the added flexibility of the generalized Hampel filters offers no practical advantage for implosion sequences. Partial characterization results are presented for these sequences, as are useful relationships between root sequences for generalized Hampel filters and their median-based counterparts. To illustrate the performance of this filter class, two examples are considered: one is simulation-based, providing a basis for quantitative evaluation of signal recovery performance as a function of t, while the other is a sequence of monthly Italian industrial production index values that exhibits glaring outliers.
Highlights
In their paper, “On a class of nonlinear filters,” Sicuranza and Carini begin by noting [1]:“The set of nonlinear filters is extremely large since their definition excludes the applicability of the linear superposition property on which the theory of linear filters is based
A more detailed comparison of the recursive and nonrecursive Hampel filter mean absolute signal recovery error (MAE) performance is shown in Fig. 7: for small t, the recursive filter performance is much worse than the standard filter, for t values between 1.0 and 3.0, the recursive filter performs slightly better; for larger t values, both filters exhibit essentially identical performance
This determination is based on a threshold parameter t chosen by the user and the MAD scale estimate for the moving window, and the filter reduces to the standard median filter if t = 0
Summary
“On a class of nonlinear filters,” Sicuranza and Carini begin by noting [1]:“The set of nonlinear filters is extremely large since their definition excludes the applicability of the linear superposition property on which the theory of linear filters is based. “On a class of nonlinear filters,” Sicuranza and Carini begin by noting [1]:. From the very beginning, attempts have been done to suitably classify nonlinear filters on the basis of some peculiar properties, leading to the identification of certain classes of nonlinear filters.”. This paper adopts a similar philosophy, restricting consideration to a class of nonlinear filters obtained by combining two previously studied filter classes: the Hampel filter described, and the median filter extensions described in Sections 4 and 7. The result is a class of nonlinear filters we believe to be new, that includes all of these previously studied filters as special cases, but which exhibits a greater degree of design flexibility. The standard median filter MK was introduced by J.W. Tukey in 1974 [2] and is obtained by computing the median of the moving data window WKk : mk = median{xk−K , .
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