Scaling Laplacian Pyramids

Abstract: Laplacian pyramid based Laurent polynomial (LP$^2$) matrices are generated by Laurent polynomial column vectors and have long been studied in connection with Laplacian pyramidal algorithms in Signal Processing. In this paper, we investigate when such matrices are scalable, that is when right multiplication by Laurent polynomial diagonal matrices results in paraunitary matrices. The notion of scalability has recently been introduced in the context of finite frame theory and can be considered as a preconditioning method for frames. This paper significantly extends the current research on scalable frames to the setting of polyphase representations of filter banks. Furthermore, as applications of our main results we propose new construction methods for tight wavelet filter banks and tight wavelet frames.