Adaptive Morphological Representation of Signals: Polynomial and Wavelet Methods

In this paper, we propose a novel signal representation based on mathematical morphology, and with it develop representations analogous to the polynomial transform and the bank-of-filters implementation of the wavelet representation. The geometric decomposition of a signal is achieved by separating it into analysis frames and applying mathematical morphological operators with adaptive structuring functions to each frame. The adaptation parameters are found by solving iteratively nonlinear equations that result from constraining the morphological results to achieve optimal fitting. If the structuring functions are derived from real-valued orthogonal polynomials defined on a window, the representation is analogous to the polynomial transform. Using a morphological interpolation, we derive a pyramid-like structure to decompose a signal into gross and fine information components, at different scales, just as in the wavelet transformation. Non-linear morphological operators reduce the computational complexity of the proposed representations. Although these representations are easily extended to two-dimensions, one needs to consider the non–unique ordering of the structuring functions, and the different sampling, decimation and interpolation procedures in two-dimensions. The application of our procedures is mainly in image data compression, but they could also used in object identification. We illustrate our representations by means of one- and two-dimensional examples.