Wavelets generated by using discrete singular convolution kernels

Abstract

This paper explores the connection between wavelet methods and an efficient computational algorithm - the discrete singular convolution (DSC). Many new DSC kernels are constructed and they are identified as wavelet scaling functions. Two approaches are proposed to generate wavelets from DSC kernels. Two well known examples, the Canny filter and the Mexican hat wavelet, are found to be special cases of the present DSC kernel-generated wavelets. A family of wavelet generators proposed in this paper are found to form an infinite-dimensional Lie group which has an invariant subgroup of translation and dilation. If DSC kernels form an orthogonal system, they are found to span a wavelet subspace in a multiresolution analysis.