Two-dimensional orthogonal and symmetrical wavelets and filter-banks

Two-dimensional compactly supported, symmetric, orthogonal wavelets and filter-banks are presented. Two cases are discussed, filters with two-fold symmetry (centrosymmetric), and filters with four-fold symmetry that are symmetric (or anti-symmetric) about the vertical and horizontal axes. We show that imposing the requirement of symmetry (linear phase) in the case of factorable wavelets and filter-banks, imposes a simple constraint on each of its polynomial degree-1 factors. We thus obtain a simple and complete method of constructing orthogonal factorable filter-banks with linear phase. This method is exemplified by design in the case of four-band separable sampling. An interesting result is obtained, similar to the one well known in the case of 1D, orthogonal factorable wavelets can not be both continuous and have four-fold symmetry