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Abstract-The lifting scheme has been found to be a flexible method for constructing scalar wavelets with desirable properties. Here it is extended to the construction of multiwavelets. It is shown that any set of compactly supported biorthogonal multiwavelets can be obtained from the Lazy matrix filters with a finite number of lifting steps. As an illustration of the general theory, compactly supported biorthogonal multiwavelets with optimum time-frequency resolution are constructed. In addition, experimental results of applying these multiwavelets to image compression are presented.

Necessary and sufficient conditions are given for the convergence of infinite products of matrices of complex numbers. The results are applied to the solution of periodic matrix refinement equations. Conditions are given for the solutions to be in L 2 ([0, 2?) s ) and generate a multiresolution of multiplicity r. A general algorithm for constructing multidimensional periodic multiwavelets from a scaling vector which generates a multiresolution is also given. Academic PressKey Words: periodic multiresolution with multiplicity r and dilation matrix M; infinite matrix product; matrix extension; multidimensional periodic multiwavelets.

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