A morphological polynomial transform

It is shown that, using grayscale morphology and a set of functions related to orthogonal polynomials-defined on a window-as structuring elements, a signal representation analogous to the polynomial transform can be developed. The geometric decomposition of a signal is achieved by windowing and using a morphological opening operation to approximate the windowed signal with each of the structuring elements. The resulting representation is made to resemble an orthogonal expansion by constraining the result of opening to be equal to the structuring function. Properties of the recursive decomposition algorithm make it possible to develop an algorithm for calculating a set of coefficients used in the reconstruction. The analysis and synthesis of one-dimensional signals are illustrated with an image data compression example.<>