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Applications Of Multi-resolution Analysis To Images

The concept of a multiresolution decomposition is a powerful tool in single and multidimensional signal processing applications. The basic idea behind them is to examine signals “under a microscope” with different zoom factors. In their current form proposed in the literature, the shape of the underlying analyzing filter/lens remains the same from .one resolution to the next. While this results in a compact representation of signals that “look” the same across different scales, such as fractal type signals, it is not necessarily an optimal way of analyzing signal which have widely different characteristics at different scales and therefore need to be processed with different shape filters at different resolutions. In this presentation, we investigate applications of multi-resolution analysis to images from two complementary viewpoints. On the one hand we are interested in wavelet decompositions, in which the image is analyzed by functions which are formed by dilating and shifting a basic wavelet function. Such a representation can be advantageous in the recognition and exploitation of self-similarity in fractal type images. On the other hand, taking a more general view of multi-resolution decomposition, we abandon the traditional requirement that each successive resolution subspace, , be generated by a basis of dilated basis vectors of K+1. Such an approach, being less constrained than the traditional concept of a multi-resolution decomposition, is better suited to some adaptive coding approaches. The main characteristic of fractal objects is that a small number of parameters should be sufficient for their representation, and the main goal of a coding or image representation scheme would be that of finding these parameters. It has been shown that the redundancy of a fractal signal with respect to scale variation is mirrored by its wavelet decomposition, in that the wavelet transform at a finer scale is obtainable from the wavelet transform at coarser scales according to a causal prescription. In this presentation