February Fourier Talks 2014

We consider the problem of constructing multi-D wavelets using multi-D Laurent polynomials. There have been notable advances in multi-D Laurent polynomials, but their applications to the problem of multi-D wavelet constructions have been scarce and limited. We present a new methodology for constructing multi-D wavelets using the Quillen-Suslin theorem for Laurent polynomials. Our construction method has some advantages over other existing methods including the tensor product, which is the most commonly used method, as well as most of other non-tensor-based methods. First, it works for any spatial dimension and for any sampling matrix. Second, it does not require the initial lowpass filters to satisfy any additional assumption such as interpolatory condition. Third, it provides an algorithm for constructing a wavelet filter bank from a single lowpass filter so that its vanishing moments are at least as many as the accuracy number of the lowpass filter. As a result, constructing multi-D wavelets with large number of vanishing moments (yielding high performance) can be done relatively easily. This is joint work with Hyungju Park and Fang Zheng.