Wavelets centered on a knot sequence: theory, construction, and applications

TL;DR

This paper introduces a new framework for constructing orthogonal wavelets centered on irregular knot sequences, providing efficient algorithms and demonstrating applications in image analysis and quasi-crystal lattices.

Contribution

It develops a general theory for wavelets on irregular knots, introduces two new families of continuous, piecewise polynomial orthogonal wavelets, and applies them to practical data and mathematical structures.

Findings

Efficient algorithms for wavelet implementation.

Successful application to ocelot image data.

Construction of wavelet bases on quasi-crystal lattices.

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 $\tau$.