Scalable filter banks

Abstract

A finite frame is said to be scalable if its vectors can be rescaled so that the resulting set of vectors is a tight frame. The theory of scalable frame has been extended to the setting of Laplacian pyramids which are based on (rectangular) paraunitary matrices whose column vectors are Laurent polynomial vectors. This is equivalent to scaling the polyphase matrices of the associated filter banks. Consequently, tight wavelet frames can be constructed by appropriately scaling the columns of these paraunitary matrices by diagonal matrices whose diagonal entries are square magnitude of Laurent polynomials. In this paper we present examples of tight wavelet frames constructed in this manner and discuss some of their properties in comparison to the (non tight) wavelet frames they arise from.