Convergence of symmetric weighted median filters (Q2725025)

This paper is devoted to the study of a special sequence of real numbers, \(x^{(p)}= \{x^{(p)}(n)\}_{n\in\mathbb{Z}}\), resulting from an arbitrary sequence (of real numbers) \(x= \{x(n)\}_{n\in\mathbb{Z}}\) through \(p\) times symmetric weighted median filters, with window \(2k+1\). The paper provides an example showing that previous results by which, for \(p\) large enough, \(x^{(p)}\) is a root or \(x^{(p)}\) is a recurrent sequence of period 2, are not true for general starting sequences \(x\). The authors give a complete answer to the problem of the \(x^{(p)}\) nature and behaviour, proving that, when \(p\to\infty\), both \(x^{(2p)}\) and \(x^{(2p- 1)}\) are convergent (sub)sequences. For the case when the sequence \(x^{(p)}\) is no longer weighted (the weights are all equal to 1), this result represents generalizations of other issues related to well-known median filters.