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

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 $τ$.