Explicit construction of compactly supported biorthogonal multiwavelets via matrix extension

Abstract

Polyphase matrix extension of the scaling vector functions plays an important role in the construction of compactly supported biorthogonal multiwavelets. However, the involved computations are very complicated, and there is no unified, direct formula available so far. In this paper, abstract algebraic methods are used to investigate the canonical forms of polyphase matrices of the scaling vector functions. According to the related properties of canonical forms of polyphase matrices, it is proved that the matrix extension equations are always solvable so that two explicit formula groups of the solution set corresponding to different canonical forms can be derived. All the explicit formulas are represented via the submatrices of polyphase matrices directly. Furthermore, for a given matrix extension problem, any solution can be obtained from these explicit formulas via product-preserving transformations, which means that, the proposed algorithm provides a complete solution set. Computational examples demonstrated that by using the explicit formulas, our matrix extension algorithm is direct and effective. Finally, a simple application by using multiwavelets for denoising is presented and the experimental results showed that the multiwavelets outperformed the scalar wavelets under different test signals.