On the Relationship between Quasi-Affine Systems and the À Trous Algorithm

We seek to demonstrate a connection between refinable quasi-affine systems and the discrete wavelet transform known as the à trous algorithm. We begin with an introduction of the bracket product, which is the major tool in our analysis. Using multiresolution operators, we then proceed to reinvestigate the equivalence of the duality of refinable affine frames and their quasiaffine counterparts associated with a fairly general class of scaling functions that includes the class of compactly supported scaling functions. Our methods show that for negative scales only one of the generalized Smith-Barnwell equations is actually needed to establish the additivity property of the quasi-affine multiresolution operators. This fact is then identified with the à trous algorithm thereby illustrating the connection with quasi-affine systems. We then introduce the notion of a generalized quasi-affine (GQA) system, in which separate generating wavelets are used for non-negative and negative dilations. Sufficient conditions are described for two GQA systems to constitute dual frames, providing a means for the construction of frames from appropriate à trous systems. We conclude with a brief discussion of examples of GQA frames associated with two different biorthogonal wavelet systems. The novelty of this work is the connection established between the à trous algorithm and refinable quasi-affine systems together with the notion of GQA systems, which are introduced to exploit this connection.