1 Introduction to Wavelets 2 A Multiresolution Formulation of Wavelet Systems 3 Filter Banks and the Discrete Wavelet Transform 4 Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Tight Frames, and Unconditional Bases 5 The Scaling Function and Scaling Coefficients, Wavelet and Wavelet Coefficients 6 Regularity, Moments, and Wavelet System Design 7 Generalizations of the Basic Multiresolution Wavelet System 8 Filter Banks and Transmultiplexers 9 Calculation of the Discrete Wavelet Transform 10 Wavelet-Based Signal Processing and Applications 11 Summary Overview 12 References Bibliography Appendix A Derivations for Chapter 5 on Scaling Functions Appendix B Derivations for Section on Properties Appendix C Matlab Programs Index

Introduction to Wavelets and Wavelet Transforms: A Primer

Digital Coding of Waveforms

Several authors have demonstrated that significant improvements can be obtained in wavelet-based signal processing by utilizing a pair of wavelet transforms where the wavelets form a Hilbert transform pair. This paper describes design procedures, based on spectral factorization, for the design of pairs of dyadic wavelet bases where the two wavelets form an approximate Hilbert transform pair. Both orthogonal and biorthogonal FIR solutions are presented, as well as IIR solutions. In each case, the solution depends on an allpass filter having a flat delay response. The design procedure allows for an arbitrary number of vanishing wavelet moments to be specified. A Matlab program for the procedure is given, and examples are also given to illustrate the results.

The design of approximate Hilbert transform pairs of wavelet bases

This paper introduces a new approach to deblocking of JPEG compressed images using overcomplete wavelet representations. By exploiting cross-scale correlations among wavelet coefficients, edge information in the JPEG compressed images is extracted and protected, while blocky noise in the smooth background regions is smoothed out in the wavelet domain. Compared with the iterative methods reported in the literature, our simple wavelet-based method has much lower computational complexity, yet it is capable of achieving the same peak signal-to-noise ratio (PSNR) improvement as the best iterative method and giving visually very pleasing images as well.

A deblocking algorithm for JPEG compressed images using overcomplete wavelet representations

[1] The first near-global high-resolution digital elevation model (DEM) of the Earth has recently been released following the successful Shuttle Radar Topography Mission (SRTM) of 2000. This data set will have applications in a wide range of fields and will be especially valuable in the Earth sciences. Prior to widespread dissemination and use, it is important to acquire knowledge regarding the accuracy characteristics. In this work a comprehensive analysis of the vertical errors present in the data set and the assessment of their effects on different hydrogeomorphic products is performed. In particular, the work consisted of (1) measuring the vertical accuracy of the data set in two areas with different topographic characteristics; (2) characterizing the error structure by comparing elevation residuals with terrain attributes; (3) assessing a wavelet-based filter for removing speckle; and (4) assessing the effects of vertical errors on hydrogeomorphic products and on slope stability modeling. The results indicate that in the two sites, relief has a strong effect on the vertical accuracy of the SRTM DEM. In the high-relief terrain, large errors and data voids are frequent, and their location is strongly influenced by topography, while in the low- to medium-relief site, errors are smaller, although the hilly terrain still produces an effect on the sign of the errors. Speckling generates deviations in the drainage network in one of the investigated areas, but the application of a wavelet filter proved to be an effective tool for removing vertical noise, although further fine tuning is necessary. Vertical errors cause differences in automatically extracted hydrogeomorphic products that range between 4 and 1090.

https://agupubs.onlinelibrary.wiley.com/doi/pdf/10.1029/2003JF000113

Analysis and characterization of the vertical accuracy of digital elevation models from the Shuttle Radar Topography Mission

We construct general biorthogonal Coifman wavelet systems, a new class of compactly supported biorthogonal wavelet systems with vanishing moments equally distributed for a scaling function and wavelet pair. A time-domain design method is employed and closed-form expressions for the impulse responses and the frequency responses of the corresponding dual filters are derived. The resulting filter coefficients are all dyadic fractions, which is an attractive feature in the realization of multiplication-free discrete wavelet transform. Even-ordered systems in this family are symmetric, which correspond to linear-phase dual filters. In particular, three filterbanks (FBs) in this family are systematically verified to have competitive compression potential to the 9-7 tap biorthogonal wavelet FB by Cohen et al. (1992), which is currently the most widely used one in the field of wavelet transform coding. In addition, the proposed FB's have much smaller computational complexity in terms of floating-point operations required in transformation, and therefore indicate a better tradeoff between compression performance and computational complexity.

A new class of biorthogonal wavelet systems for image transform coding

We show that imposing a certain number of vanishing moments on a scaling function (e.g., coiflets) leads to fairly small phase distortion on its associated filter bank in the neighborhood of DC. However, the phase distortion at the other frequencies can be much larger. We design a new class of real-valued, compactly supported, orthonormal, and nearly symmetric wavelets (we call them generalized coiflets) with a number of nonzero-centered vanishing moments equally distributed on scaling function and wavelet. Such a generalization of the original coiflets offers one more free parameter, the mean of the scaling function, in designing filter banks. Since this parameter uniquely characterizes the first several moments of the scaling function, it is related to the phase response of the lowpass filter at low frequencies. We search for the optimal parameter to minimize the maximum phase distortion of the filter bank over the lowpass half-band. Also, we are able to construct nearly odd-symmetric generalized coiflets, whose associated lowpass filters are surprisingly similar to those of some biorthogonal spline wavelets. These new wavelets can be useful in a broad range of signal and image processing applications because they provide a better tradeoff between the two desirable but conflicting properties of the compactly supported and real-valued wavelets, i.e., orthonormality versus symmetry, than the original coiflets.

Generalized coiflets with nonzero-centered vanishing moments

Simultaneous Speckle Reduction and Data Compression using Best Wavelet Packet Bases with Applications to SAR based ATD/R

We study the problem of estimating signal parameters from a noisy data sequence containing superimposed damped sinusoids. We propose three novel methods by combining the reduced-rank Hankel approximation and the matrix pencil method. We demonstrate that two of the proposed methods significantly outperform both the original matrix pencil method and the modified Kumaresan-Tufts (1982) method, especially at low signal-to-noise ratio.

/pdf/improved-matrix-pencil-methods-4ewqx15pdm.pdf

Improved matrix pencil methods

We propose a nonlinear, universal method based on wavelet thresholding to efficiently improve the performance of various coding schemes. Coarse quantization of the transform coefficients often results in some undesirable artifacts, such as ringing effect, contouring effect and blocking effect, especially at very low bit rate. We perform the wavelet-domain thresholding on the decompressed image to attenuate the quantization noise effect while maintaining the relatively sharp features (e.g. edges) of the original image. Both the objective quality and the subjective quality of the reconstructed image are significantly improved with the reduction of coding artifacts. Experimental results show that de-noising using the undecimated discrete wavelet transform (DWT) achieves better performance than using the orthonormal DWT, with an acceptable computational complexity (O(MN log/sub 2/ (MN)) for an image of size M/spl times/N).

Dong Wei

Papers

A new class of biorthogonal wavelet systems for image transform coding

Generalized coiflets with nonzero-centered vanishing moments

Simultaneous Speckle Reduction and Data Compression using Best Wavelet Packet Bases with Applications to SAR based ATD/R

Improved matrix pencil methods

Optimal wavelet thresholding for various coding schemes