Efficient dyadic wavelet transformation of images using interpolation filters

The properties of a special class of overcomplete wavelet transforms specified in terms of an interpolation filter are investigated. The decomposition is obtained by filtering the signal with a sequence of increasingly selective lowpass filters with a dyadic scale progression. The wavelet coefficients are evaluated by simple subtraction of two consecutive lowpass components. The lowpass filter bank is implemented using a standard iterative multiscale algorithm. The impulse responses of the analysis filters are shown to be interpolated versions of each other. This structure is computationally very efficient; it requires a little more than one-fourth as many operations as other comparable wavelet-based algorithms. The corresponding filter bank provides a perfect coverage of the frequency domain, which results in a trivial reconstruction procedure by summation. Extensions to the subsampled case are also presented. The decompositions associated with spline interpolation filters are considered in some detail, and some image processing examples are presented.<>