Convolution Using The Undecimated Discrete Wavelet Transform

Convolution is one of the most widely used digital signal processing operations. It can be implemented using the fast Fourier transform (FFT), with a computational complexity of O(N log N). The undecimated discrete wavelet transform (UDWT) is linear and shift invariant, so it can also be used to implement convolution. In this paper, we propose a scheme to implement the convolution using the UDWT, and study its advantages and limitations. 1. INTRODUCTION Convolution is the fundamental operation of linear system theory, and discrete convolution is one of the most widely used digital signal processing operation. Finite impulse response (FIR) digital filters are designed to be convolved with input signals to achieve certain effects, and the fast Fourier transform (FFT) is mostly used to implement convolution. Therefore any scheme that can speedup the convolution process is theoretically interesting and practically important. The Fourier transform, Laplace transform and Z-transform all hav..

Digital filter design

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Phase-preserving Compression of Seismic Data using the Self-adjusting Wavelet Transform

Phase information is crucial for seismic data processing. However, traditional compression schemes do not pay special attention in preserving the phase of the seismic data, resulting in the loss of critical information. In this paper, we propose a lossy compression method that preserves the phase as much as possible. The method is based on the self-adjusting wavelet transform that adapts to the locations of the significant signal components. The elegant method of embedded zero-tree wavelet compression is modified and incorporated into our compression scheme. Our method can be applied to both one dimensional seismic signals and two dimensional seismic images. 1 Introduction The seismic method plays a prominent role in the search for hydrocarbons. Seismic exploration consists of three main stages: data acquisition, processing, and interpretation. Due to the massive data acquisition activities, the need to compress the seismic data is paramount. Unlike other images, the seismic data are ..

On Cosine-Modulated Wavelet Orthonormal Bases

Recently multiplicity M , K-regular, orthonormal wavelet bases (that have implications in transform coding applications) have been constructed by several authors [1, 2, 3]. This paper describes and parameterizes the cosine-modulated class of multiplicity M wavelet tight frames (WTFs). In these WTFs (like in the general multiplicity 2 case in [4]), the scaling function uniquely determines the wavelets. This is in contrast to the general multiplicity M case, where one has to, for any given application, design the scaling function and the wavelets. Several design techniques for the design of K regular cosine-modulated WTFs are described and their relative merits discussed. Wavelets in K-regular WTFs may or may not be smooth. Since coding applications use WTFs with short length scaling and wavelet vectors (since long filters produce ringing artifacts which is undesirable in, say, image coding), many smooth designs of K regular WTFs of short lengths are presented. In some cases, analytical ..