A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Morphological Shape Decomposition
IEEE Transactions on Pattern Analysis and Machine Intelligence
The analysis of morphological filters with multiple structuring elements
Computer Vision, Graphics, and Image Processing
Multirate systems and filter banks
Multirate systems and filter banks
Image Analysis and Mathematical Morphology
Image Analysis and Mathematical Morphology
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In this paper, we propose a novel signal representationbased on mathematical morphology, and with it develop representationsanalogous to the polynomial transform and the bank-of-filtersimplementation of the wavelet representation. The geometric decompositionof a signal is achieved by separating it into analysis framesand applying mathematical morphological operators with adaptivestructuring functions to each frame. The adaptation parametersare found by solving iteratively nonlinear equations that resultfrom constraining the morphological results to achieve optimalfitting. If the structuring functions are derived from real-valuedorthogonal polynomials defined on a window, the representationis analogous to the polynomial transform. Using a morphologicalinterpolation, we derive a pyramid-like structure to decomposea signal into gross and fine information components, at differentscales, just as in the wavelet transformation. Non-linear morphologicaloperators reduce the computational complexity of the proposedrepresentations. Although these representations are easily extendedto two-dimensions, one needs to consider the non–uniqueordering of the structuring functions, and the different sampling,decimation and interpolation procedures in two-dimensions. Theapplication of our procedures is mainly in image data compression,but they could also used in object identification. We illustrateour representations by means of one- and two-dimensional examples.