Form-invariant linear filtering: Theory and applications

The form-invariant filters are, by definition, those shift-variant filters such that their output, for any given input, turns out to be linearly scaled (implying that its "form" does not change) whenever the input is linearly scaled. In this paper the most general classes of 1- D and 2-D linear form-invariant filters are derived and their properties are discussed, together with their implementation techniques. Two main implementation approaches are considered: one based on the Mellin transform, the other on a combination of coordinate mappings and shift-invariant filtering. The paper also discusses the related works of other authors covering quite different fields such as optical pattern recognition, image restoration and image reconstruction from projections, radar signal processing, etc. It is shown that the mathematics of form-invariant filtering provides a common framework, if not a powerful unified approach, to the many signal processing techniques spread in the above-mentioned works and apparently different application areas. The paper ends with a processing example showing the usefulness of form-invariant filtering in a pattern recognition problem, that is, in the area where the most promising applications of such a filtering are foreseen.