Chair: Cormac Herley, Hewlett Packard Laboratories (USA)
Stephen Del Marco, Aware Inc. (USA)
Peter Heller, Aware Inc. (USA)
John Weiss, Aware Inc. (USA)
This paper develops a two- dimensional $M$-band translation-invariant wavelet transform (2-d MTI). Use of the MTI overcomes the shift-variance of the wavelet transform by applying a cost function over $M$ shifts of the input signal. The new transform is proven to be translation-invariant. Use of $M$-band wavelets enables a finer frequency partitioning and greater energy compaction in the transform representation. Examples are presented which show that the translation-invariant transforms provide superior energy concentration compared to the corresponding nominal wavelet transforms. Examples are also presented comparing the energy concentration capability of $M$-band wavelets and the Modulated Lapped Transform (MLT).
I. Cohen, Technion (ISRAEL)
S. Raz, Technion (ISRAEL)
D. Malah, Technion (ISRAEL)
In this work, a shifted wavelet packet (SWP) library, containing all the time-shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a ``best basis'' is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant best basis is orthonormal and the associated expansion, characterized by the lowest information cost is shift- invariant. The shift-invariace stems from an additional degree of freedom, generated at the decomposition stage, and incorporated into the search algorithm. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. The computational complexity of SIWPD may be controlled at the expense of the attained information cost, to an extent of $O(2N log_2 N)$.
Igor Djokovic, California Institute of Technology (USA)
P. P. Vaidyanathan, California Institute of Technology (USA)
In this paper, the existing sampling theory for MRA subspaces is generalized to several more cases. We consider derivative sampling, multiband sampling and sampling of wide sense stationary (WSS) random processes. We also show that the synthesizing functions form a Riesz basis for the corresponding MRA subspace.
Sigang Qiu, University of Connecticut (USA)
Hans G. Feichtinger, University of Vienna (AUSTRIA)
of associated small block matrices. We propose an efficient algorithm, which we call the block-multiplication, and which makes explicit use of the sparsity of those Gabor-type matrices. As an interesting consequence, we show that Gabor operators corresponding to Gabor triples (g_k, a, b) commute for arbitrary signals g_k(k = 1, 2) provided that ab divides the signal length.
Nurgun Erdol, Florida Atlantic University
Feng Bao, Florida Atlantic University
Zajing Chen, Florida Atlantic University
The orthonormal wavelet transform is an efficient way for signal representation since there is no redundancy in its expression, but due to aliasing in the decimation stage it lacks the often desired property of shift invariance. On the other hand, the oversampled or nonorthogonal wavelet offers a finer resolution in translation; thus reducing the effect of shift of origin, it becomes more robust to changes in the initial phase of the signal. In some areas of signal processing, such as wideband correlation processing, sensitivity to time alignment necessitates the use of the nonorthogonal wavelet transform. The price paid for the advantage of robustness to shifting is the introduction of redundancy in the expression. In many applications, both of these two properties are needed in different stages of signal processing. Thus there is a need to know the conditions under which the redundant and nonorthonormal wavelet transform coefficients can be derived from the orthonormal wavelet transform coefficients. The answer provides us with a convenient way to switch between these two forms: the orthonormal wavelet for efficient expression, and the nonorthogonal one whenever it is necessary for feature extraction.
Steven A. Benno, Carnegie Mellon University (USA)
Jose M.F. Moura, Carnegie Mellon University (USA)
The goal of this paper is to derive an approach for designing nearly shiftable scaling functions for multiresolution analyses (MRAs). Because this method does not increase the sampling density, the sparseness and efficiency of a dyadic grid is preserved. It contrasts with other attempts to the same problem which suffer either from oversampling or from being computationally expensive and data dependent. The algorithm reshapes a starting scaling function by modifying the Zak transform of its energy spectral density (ESD). The paper shows that although the modified signal does not strictly satisfy the 2--scale equation, the approximation error is sufficiently small. The result is a wavelet representation whose subband energy is ``nearly'' invariant to translations of its input. The paper will illustrate this property with specific examples.
Douglas Nelson, U.S. Department of Defense (USA)
Most machine speech analysis and processing is based on a warped spectral represenation. The intent of this paper is to present a method by which proper warped representations can be computed efficiently. In the case of log-warping functions, the methods of this paper produce a wavelet-like transform as a linear convolution of a single log-warped wavelet basis element and a log- warped representation of the signal. In this paper, the resulting doubly warped transform is defined to be a Mellin-Wavelet transform. The majority of the paper is devoted to deriving design parameters for implementation of the transform, with speech as the primary application.
Makoto Nakashizuka, Niigata University (JAPAN)
Hisakazu Kikuchi, Niigata University (JAPAN)
Hideo Makino, Niigata University (JAPAN)
Ikuo Ishii, Niigata University (JAPAN)
This paper presents an ECG data compression technique by the multiscale peak analysis. We define the multiscale peak analysis as the wavelet maxima representation of which the basic wavelet as the second derivative of a symmetric smoothing function. The wavelet transform of an ECG shows maxima at the start, peak and stop points of five transient waves P through T. The number of wavelet maxima is expected to be less than the number of original data samples. The wavelet maxima can be enough to reconstruct original signals precisely. The wavelet maxima representation can hence lead to the ECG data compression and analysis. The compressed data still keep the peaks of QRS waves, abnormal behavior search will be feasible in practice. The result of the compression shows that a normal ECG data is compressed by a factor 10.
Mark A. Coffey, University of Colorado (USA)
Delores M. Etter, University of Colorado (USA)
The structure of arbitrary wavelet bases derived from the generalized B-spline wavelets is introduced. An arbitrary wavelet basis is constructed and then used to detect the presence of dynamic solitons in the sea surface height as measured by satellite altimetry. Comparisons are made for wavelet decompositions based on several scaling functions.
TaiChiu Hsung, Hong Kong Polytechnic (HONG KONG)
Daniel P.K. Lun, Hong Kong Polytechnic (HONG KONG)
In this paper, we present a fast algorithm for the computation of the wavelet transform in higher dimensional Euclidean space $R^n$ with arbitrary shaped wavelets. The algorithm is a direct consequence of the convolution property of the Radon transform and shows significant improvement in speed. We also present a novel approach for the computation of the Daubechies type wavelet transform under the Radon transform domain where the n-dimensional multiresolution Analysis (MRA) is reduced to one-dimensional MRA. We found applications of this approach on, for instance, multiresolution reconstruction of the tomographic image with the standard methods of denoising, where determination of wavelet coefficients is required under the Radon transform domain. Along with the possibility of reducing samples angularly with decreasing resolution, the efficiency can be further improved. Besides, extra property such as ``rotated'' wavelet can be easily implemented with this algorithm.