Wavelets centered on a knot sequence: theory, construction, and applications
This work provides a theoretical and practical framework for wavelet construction on irregular domains, which is incremental for signal processing and approximation theory.
The paper develops a general notion of orthogonal wavelets centered on irregular knot sequences, constructing two families of continuous, piecewise polynomial wavelets with efficient algorithms. Applications include denoising an ocelot image and building wavelet bases on the quasi-crystal lattice of τ-integers, yielding a multiresolution analysis with scaling factor τ.
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 $τ$-integers where $τ$ is the golden ratio. The resulting spaces then generate a multiresolution analysis of $L^2(\mathbf{R})$ with scaling factor $τ$.