Families of multiresolution and wavelet spaces with optimal properties |
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| Authors: | Akram Aldroubi Michael Unser |
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| Institution: | Biomedical Engineering and Instrumentation Program , National Institutes of Health , Bethesda, Maryland |
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| Abstract: | Under suitable conditions, if the scaling functions ?1 and ?2 generate the multiresolutions V (j)(?1) and V (j)(?2), then their convolution ?1*?2also generates a multiresolution V (j)(?1*?2) More over, if p is an appropriate convolution operator from l 2 into itself and if ? is a scaling function generating the multiresolution V (j)(?),then p*?is a scaling function generating the same multiresolution V (j)(?)=V (j)(p*?). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V (j)(?n) and W (j)(?n) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V (0)(?n)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W (0)(?n)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ b nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines. |
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