Wavelets

A Discrete-Time Wavelet Transform Based on a Continuous-Dilation Framework

Authors:

Wei Zhao,
Raghuveer M Rao,

Page (NA) Paper number 2381

Abstract:

In this paper we present a new form of wavelet transform. Unlike the continuous wavelet transform (CWT) or discrete wavelet transform (DWT), the mother wavelet is chosen to be a discrete-time signal and wavelet coefficients are computed by correlating a given discrete-time signal with continuous dilations of the mother wavelet. The results developed are based on the definition of a discrete-time scaling (dilation) operator through a mapping between the discrete and continuous frequencies. The forward and inverse wavelet transforms are formulated. The admissibility condition is derived, and examples of discrete-time wavelet construction are provided. The new form of wavelet transform is naturally suited for discrete-time signals and provides analysis and synthesis of such signals over a continuous range of scaling factors.

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Multiwavelet Systems with Disjoint Multiscaling Functions

Authors:

Felix C.A. Fernandes,
Charles Sidney Burrus,

Page (NA) Paper number 2320

Abstract:

This paper describes the first steps toward a multiwavelet system that may retain the advantages of a traditional multiwavelet system while alleviating some of its disadvantages. We attempt to achieve this through the introduction of a novel property --- the disjoint support of the multiscaling functions. We derive the conditions on the matrix filter coefficients that guarantee the disjoint support of multiscaling functions. Our preliminary results demonstrate that multiwavelet systems with this property may be arbitrarily complex. We then establish the existence of a multiwavelet system with approximation order = 2 and two multiscaling functions with disjoint support.

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Cardinal Multiwavelets and the Sampling Theorem

Authors:

Ivan W Selesnick,

Page (NA) Paper number 2034

Abstract:

This paper considers the classical Shannon sampling theorem in multiresolution spaces with scaling functions as interpolants. As discussed by Xia and Zhang, for an orthogonal scaling function to support such a sampling theorem, the scaling function must be cardinal. They also showed that the only orthogonal scaling function that is both cardinal and of compact support is the Haar function, which has only 1 vanishing moment and is not continuous. This paper addresses the same question, but in the multiwavelet context, where the situation is different. This paper presents the construction of orthogonal multiscaling functions that are simultaneously cardinal, of compact support, and have more than one vanishing moment. The scaling functions thereby support a Shannon-like sampling theorem. Such wavelet bases are appealing because the initialization of the discrete wavelet transform (prefiltering) is the identity operator --- the projection of a function onto the scaling space is given by its samples.

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Theory of Wavelet Transform Over Finite Fields

Authors:

Faramarz Fekri,
Russell M Mersereau,
Ronald W Schafer,

Page (NA) Paper number 1572

Abstract:

In this paper, we develop the theory of the wavelet transform over Galois fields. To avoid the limitations inherent in the number theoretic Fourier transform over finite fields, our wavelet transform relies on a basis decomposition in the time domain rather than in the frequency domain. First, we characterize the infinite dimensional vector spaces for which an orthonormal basis expansion of any sequence in the space can be obtained using a symmetric bilinear form. Then, by employing a symmetric, non-degenerate, canonical bilinear form we derive the necessary and sufficient condition that basis functions over finite fields must satisfy in order to construct an orthogonal wavelet transform. Finally, we give a design methodology to generate the mother wavelet and scaling function over Galois fields by relating the wavelet transform to a two channel paraunitary filter bank. Online relevant information can be found at http://www.ee.gatech.edu/users/fekri.

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Least and Most Disjoint Root Sets for Daubechies Wavelets

Authors:

Carl Taswell,

Page (NA) Paper number 1164

Abstract:

A new set of wavelet filter families has been added to the systematized collection of Daubechies wavelets. This new set includes complex and real, orthogonal and biorthogonal, least and most disjoint families defined using constraints derived from the principle of separably disjoint root sets in the complex z-domain. All of the new families are considered to be "constraint selected" without a search and without any evaluation of filter properties such as time-domain regularity or frequency-domain selectivity. In contrast, the older families in the collection are considered to be "search optimized" for extremal properties. Some of the new families are demonstrated to be equivalent to some of the older families, thereby obviating the necessity for any search in their computation. A library that displays images of all filter families in the collection is available at www.toolsmiths.com.

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Shift Invariant Properties of the Dual-Tree Complex Wavelet Transform

Authors:

Nick G Kingsbury, Dept. of Engineering, University of Cambridge, UK (U.K.)

Page (NA) Paper number 1238

Abstract:

We discuss the shift invariant properties of a new implementation of the Discrete Wavelet Transform, which employs a dual tree of wavelet filters to obtain the real and imaginary parts of complex wavelet coefficients. This introduces limited redundancy (2^m:1 for m-dimensional signals) and allows the transform to provide approximate shift invariance and directionally selective filters (properties lacking in the traditional wavelet transform) while preserving the usual properties of perfect reconstruction and computational efficiency with good well-balanced frequency responses.

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A Special Class of Orthonormal Wavelets: Theory, Implementations, and Applications

Authors:

Lixin Shen, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Jo Yew Tham, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Seng Luan Lee, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Hwee Huat Tan, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)

Page (NA) Paper number 1384

Abstract:

This paper introduces a novel class of length-4N orthonormal scalar wavelets, and presents the theory, implementational issues, and their applications to image compression. We first give the necessary and sufficient conditions for the existence of this class. The parameterized representation of filters with different lengths are then given. Next, we derive new and efficient decomposition and reconstruction algorithms specifically tailored to this class of wavelets. We will show that the proposed discrete wavelet transformations are orthogonal and have lower computational complexity than conventional octave-bandwidth transforms using Daubechies' orthogonal filters of equal length. In addition, we also verify that symmetric boundary extensions can be applied. Finally, our image compression results further confirm that improved performance can be achieved with lower computational cost.

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A New Multifilter Design Property for Multiwavelet Image Compression

Authors:

Jo Yew Tham, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Lixin Shen, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Seng Luan Lee, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)
Hwee Huat Tan, Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore 119260 (Singapore)

Page (NA) Paper number 1381

Abstract:

Approximation order, linear phase symmetry, time-frequency localization, regularity, and stopband attenuation are some criteria that are widely used in wavelet filter design. In this paper, we propose a new criterion called "good multifilter properties" (GMPs) for the design and construction of multiwavelet filters targeting image compression applications. We first provide the definition of GMPs, followed by a necessary and sufficient condition for an orthonormal multiwavelet system to have a GMP order of at least 1. We then present an algorithm to construct orthogonal multiwavelets possessing GMPs, starting from any length-2N scalar CQFs. Image compression experiments are performed to evaluate the importance of GMPs for image compression, as compared to other common filter design criteria. Our results confirmed that multiwavelets that possess GMPs not only yield superior PSNR performances, but also require much lower computations in their transforms.

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