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A set of bi-orthogonal wavelet transforms are developed based on an extension of Sweldens; lifting method for variable support wavelet bases. The bi-orthogonal wavelet has advantages of compact support for both the transform and inverse transform process, and, like the Zernike and Legendre polynomials, can be designed to pass localized polynomial variations of orders m up to the limit, depending on the width of support of the lesser support width of either the wavelet or the scaling function. Features of this wavelet set include symmetry of the wavelet and scaling functions. The wavelet and scaling functions of a given order m are related to their pair of duals through simple relations involving position shifts and sign changes. A general method for producing transform functions is given, and results are shown for up to m equals 3, which treats up to 7th order polynomial insensitivity. The set of transforms is tested against a sample image and results show the possibilities for compression. It appears the lowest order wavelet yields the best performance for simple compression techniques on the image used. This m equals 0 transform shows the least degradation from truncation and tends to treat small regions effectively. Despite the advantages of higher order wavelets with respect to fluctuations, these tend to artificially create noise information which is passed on to the next processing stage.
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