Families of Orthogonal Two-Dimensional Wavelets
Peter Maaß · SIAM Journal on Mathematical Analysis · 1996
We construct orthonormal wavelet bases of $L^2 (\mathbb{R}^2 )$ with compact support for dilation matrices of determinant 2. The key idea is to describe the set $\mathcal{H}_2 $ of all two-dimensional (2D) scaling coefficients satisfying the orthogonality condition as an implicit function. This set includes the scaling coefficients for induced 1D wavelets. We compute the tangent space of $\mathcal{H}_2 $ at $H_N $, the scaling coefficients for induced 1D Daubechies wavelets. The structure of the tangent space allows us to build nonseparable wavelets by starting at $H_N $ and tracing $\mathcal{H}$ along its tangent lines. Various families of compactly supported orthogonal 2D wavelets for the quincunx grid are explicitly given.