ABSTRACT
Semantically, nonlinear filtering concerns all image-to-image operators that are nonlinear, and since digital images do not form a vector space, all image filtering. Nonetheless, insofar as classical linear techniques are adapted to image filtering, linear methods do compose a large segment of image filtering. The algebraic structure of nonlinear filtering is now complete, insofar as nonlinear filtering is taken to mean filtering by increasing, translation-invariant filters. This chapter considers median filters, and morphological filters. As intuitively conceived, median filters are numerically based, and morphological filters are shape based. Median filters arise from classical maximum-likelihood estimation and from certain operations on logical variables; morphological filters arise from fitting shape probes within larger shapes. The one-dimensional median filter is implemented by sliding a window of odd length over the input signal one sample at a time.
