Design of regular nonseparable bidimensional wavelets using Grobner basis techniques

The design of two-dimensional (2-D) filter banks yielding orthogonality and linear-phase filters and generating regular wavelet bases is a difficult task involving the algebraic properties of multivariate polynomials. Using cascade forms implies dealing with nonlinear optimization. We turn the issue of optimizing the orthogonal linear-phase cascade from Kovacevic and Vetterli (1992) into a polynomial problem and solve it using Grobner basis techniques and computer algebra. This leads to a complete description of maximally flat wavelets among the orthogonal linear-phase family proposed by Kovacevic and Vetterli. We obtain up to five degrees of flatness for a 16×16 filter bank, whose Sobolev exponent is 2.11, making this wavelet the most regular orthogonal linear-phase nonseparable wavelet to the authors´ knowledge,