Spline Interpolation and Wavelet Construction

The method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages from B-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of ‘‘good’’ filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression.