A multiresolution analysis for tensor-product splines using weighted spline wavelets

We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing – as a model problem – image compression with black and white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.