Abstract:

An algorithm for constructing the compactly supported multidimensional orthogonal symmetric wavelets with dilation factor 3 is provided based on paraunitary matrix symmetric extension. Namely, let Φ(x)∈L²(R^d)
be a d dimensional compactly supported orthogonal scaling function with dilation factor 3;  P(ξ) and (p_0, ν(ξ))_ν, ν ∈E_d, respectively, be the mask symbol and the polyphase symbol of Φ(x). Firstly, the authors propose a symmetric orthogonal transform of vectors, and take the symmetric orthogonal transform to (p_0, ν(ξ))_ν, ν ∈E_d，
then get a symmetric unit vector. Secondly, based on paraunitary matrix symmetric extension, 3^d-1 compactly supported orthogonal symmetric wavelets associated with Φ(x) are obtained. The support of ψ_ν, ν ∈E_d, is not larger than that of Φ(x). In addition, the algorithm can also be used to construct orthogonal wavelets with the dilation factor M(M ≥ 3). Finally,  an example is given.

Key words: Orthogonal symmetric transform, Paraunitary matrix symmetric extension, Orthogonal wavelets, Orthogonal scaling function, Symmetry