A morphological wavelet transform

Using adaptive structuring functions, we develop a morphological interpolation that allows a signal representation similar to the given by the wavelet bank of filters. With morphological operators, the interpolation problem is reduced to solving non-linear equations iteratively to get an approximate expansion of a sampled signal in terms of the structuring functions. We obtain a pyramid-like structure to decompose the signal into smoothed and detail components at different scales, just as in the wavelet representation. The use of non-linear filters in our algorithm reduces the computational complexity associated with the decomposition and synthesis. Our representation is valid for one- and two-dimensional signals. In the two-dimensional case, we consider the non-unique ordering of the structuring functions, and the variety of possible sampling, decimation and interpolation procedures. We illustrate our one- and two-dimensional representations by means of examples