ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filter-ing steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is well-known to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a self-contained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, non-unitary case. Like the lattice factorization, the decomposition presented here asymptotically re-duces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a wavelet-like transform that maps integers to integers. 1.

. Wavelet transforms for discrete-time periodic signals are developed. In this finite-dimensional context, key ideas from the continuous-time papers of Daubechies and of Cohen, Daubechies, and Feauveau are isolated to give a concise, rigorous derivation of the discrete-time periodic analogs of orthonormal and symmetric biorthogonal bases of compactly supported wavelets. These discrete-time periodic wavelets are expressed in terms of circular FIR filters, and thus lead to fast wavelet transforms whose complexity is order N . Keywords: Orthonormal wavelets; Biorthogonal wavelets; Symmetric biorthogonal wavelets; Pyramid algorithms; Fast wavelet transforms 1. Introduction Although there is a large and growing literature on wavelets, there is, to our knowledge, no reference that focuses on the development of wavelets for discrete-time periodic signals, and that includes a derivation of Daubechies' FIR filter coefficients in this finite-dimensional context. This paper is an attempt to fil...

Modern video applications call for computationally intensive data processing at very high data rate. In order to meet the high-performance/low-cost constraints, the state-of-the-art video processor should be a programmable design which performs various tasks in video applications without sacrificing the computational power and the manufacturing cost in exchange for such flexibility. In this paper, we present a programmable video co-processor design for numerically intensive front-end video/image communications. The proposed system is a massively parallel architecture that is capable of performing most low-level computationally intensive tasks including FIR/IIR filtering, subband filtering, discrete orthogonal transforms (DT), adaptive filtering, and motion estimation for the host processor in video applications. Since the properties of each programmed function such as parallelism and pipelinability have been fully exploited in this design, the computational speed of this co-processor i...

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