Wavelets: A Multiscale Analysis Tool

As a new analysis tool, the wavelet transform has had a profound impact on almost all disciplines requiring signal analysis of any significance. It provided for the first time a systematic scale-based analysis, in contrast to filters, a tractable methodology in the temporal/spatial domain. Our goal, here, is to provide an introduction to this tool with an emphasis on gaining a working knowledge for an advanced undergraduate student. We highlight the merits of the tool as well as its limitations, and present it as complementary to the Fourier analysis tool, rather than as a universal panacea. Keywords: wavelet; filter bank; multiscale; wavelet packet; basis; denoising

[1]  H. Krim,et al.  Hierarchical stochastic modeling of SAR imagery for segmentation/compression , 1999, IEEE Trans. Signal Process..

[2]  J. Rissanen,et al.  Modeling By Shortest Data Description* , 1978, Autom..

[3]  Patrick L. Combettes,et al.  Convex Multiresolution Analysis , 1998, IEEE Trans. Pattern Anal. Mach. Intell..

[4]  Georgios B. Giannakis,et al.  Principal component filter banks for optimal multiresolution analysis , 1995, IEEE Trans. Signal Process..

[5]  B. Vidakovic.  Nonlinear wavelet shrinkage with Bayes rules and Bayes factors , 1998 .

[6]  Anna Scaglione,et al.  Redundant filterbank precoders and equalizers. I. Unification and optimal designs , 1999, IEEE Trans. Signal Process..

[7]  Stéphane Mallat,et al.  A Theory for Multiresolution Signal Decomposition: The Wavelet Representation , 1989, IEEE Trans. Pattern Anal. Mach. Intell..

[8]  M. Vetterli.  Multi-dimensional sub-band coding: Some theory and algorithms , 1984 .

[9]  Anna Scaglione,et al.  Filterbank Transceivers Optimizing Information Rate in Block Transmissions over Dispersive Channels , 1999, IEEE Trans. Inf. Theory.

[10]  Eric Fabre.  New fast smoothers for multiscale systems , 1996, IEEE Trans. Signal Process..

[11]  Ronald R. Coifman,et al.  Brushlets: A Tool for Directional Image Analysis and Image Compression , 1997 .

[12]  Pamela C. Cosman,et al.  Vector quantization of image subbands: a survey , 1996, IEEE Trans. Image Process..

[13]  Alan S. Willsky,et al.  Low complexity optimal joint detection for oversaturated multiple access communications , 1997, IEEE Trans. Signal Process..

[14]  K Ramchandran,et al.  Best wavelet packet bases in a rate-distortion sense , 1993, IEEE Trans. Image Process..

[15]  Stéphane Mallat,et al.  On denoising and best signal representation , 1999, IEEE Trans. Inf. Theory.

[16]  Henrique S. Malvar.  Lapped transforms for efficient transform/subband coding , 1990, IEEE Trans. Acoust. Speech Signal Process..

[17]  Alan S. Willsky,et al.  Wavelet-packet-based multiple-access communication , 1994, Optics & Photonics.

[18]  Timothy N. Davidson,et al.  Wavelet packet division multiplexing and wavelet packet design under timing error effects , 1997, IEEE Trans. Signal Process..

[19]  Karlheinz Gröchenig,et al.  Acceleration of the frame algorithm , 1993, IEEE Trans. Signal Process..

[20]  J. Rissanen.  Stochastic Complexity and Modeling , 1986 .

[21]  John W. Woods,et al.  Subband coding of images , 1986, IEEE Trans. Acoust. Speech Signal Process..

[22]  Jerome M. Shapiro,et al.  Embedded image coding using zerotrees of wavelet coefficients , 1993, IEEE Trans. Signal Process..

[23]  Alan R. Lindsey,et al.  Wavelet packet modulation for orthogonally multiplexed communication , 1997, IEEE Trans. Signal Process..

[24]  Georgios B. Giannakis,et al.  Optimal decorrelating receivers for DS-CDMA systems: a signal processing framework , 1996, IEEE Trans. Signal Process..

[25]  David Leporini,et al.  Best Basis Representations with Prior Statistical Models , 1999 .

[26]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[27]  A. Ben Hamza,et al.  Image denoising: a nonlinear robust statistical approach , 2001, IEEE Trans. Signal Process..

[28]  Ronald R. Coifman,et al.  Entropy-based algorithms for best basis selection , 1992, IEEE Trans. Inf. Theory.

[29]  P. P. Vaidyanathan,et al.  Multirate digital filters, filter banks, polyphase networks, and applications: a tutorial , 1990, Proc. IEEE.

[30]  I. Johnstone,et al.  Adapting to Unknown Smoothness via Wavelet Shrinkage , 1995 .

[31]  W. Clem Karl,et al.  Efficient multiscale regularization with applications to the computation of optical flow , 1994, IEEE Trans. Image Process..