Signal representation using fuzzy morphology

In this paper we propose a signal representation based on fuzzy morphology and adaptive structuring functions which is analogous to polynomial transforms. For completeness, we use a set of normalized and ordered structuring functions derived from orthogonal polynomials defined on a window. After windowing the given signal, the geometric decomposition of a windowed signal is achieved by applying fuzzy morphological opening sequentially using each of the structuring functions to fit the signal. The resulting representation is made to resemble an orthogonal expansion by constraining the result of opening to equal the adapted structuring function. Properties of the geometric decomposition permit us to develop an iterative procedure to calculate the adaptation parameters. The analysis and synthesis procedures are illustrated for images.