Wavelets Centered on a Knot Sequence: Theory, Construction, and Applications

Abstract

We develop a general notion of orthogonal wavelets ‘centered’ on an irregular knot sequence. We present two families of orthogonal wavelets that are continuous and piecewise polynomial. We develop efficient algorithms to implement these schemes and apply them to a data set extracted from an ocelot image. As another application, we construct continuous, piecewise quadratic, orthogonal wavelet bases on the quasi-crystal lattice consisting of the $$\tau $$ -integers where $$\tau $$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $$L^2(\mathbf {R})$$ with scaling factor $$\tau $$ .