A tutorial on wavelets from an electrical engineering perspective.2. The continuous case

The wavelet transform is described from the perspective of a Fourier transform. The relationships among the Fourier transform, the Gabor (1946) transform (windowed Fourier transform), and the wavelet transform are described. The differences are also outlined, to bring out the characteristics of the wavelet transform. The limitations of the wavelets in localizing responses in various domains are also delineated. Finally, an adaptive window is presented that may be optimally tailored to suit one's needs, and hence, possibly, the scaling functions and the wavelets     

Family of scaling chirp functions, diffraction, and holography

It is observed that diffraction is a convolution operation with a chirp kernel whose argument is scaled. Family of functions obtained from a prototype by shifting and argument scaling form the essential ground for wavelet framework. Therefore, a connection between diffraction and wavelet transform is developed. However, wavelet transform is essentially prescribed for time-frequency and/or multiresolution analysis which is irrelevant in our case. Instead, the proposed framework is useful in various location-depth type of analysis in imaging. The linear transform when the analyzing functions are the chirps is called the scaling chirp transform. The scaled chirp functions do not satisfy the commonly used admissibility condition for wavelets. However, it is formally shown that these neither band nor time limited signals can be used as wavelet functions and the inversion is still possible. Diffraction and in-line holography are revisited within the scaling chirp transform context. It is formally proven that a volume in-line hologram gives perfect reconstruction. The developed framework for wave propagation based phenomena has the potential of advancing both signal processing and optical applications     

Isotropic polyharmonic B-splines: scaling functions and wavelets.

In this paper, we use polyharmonic B-splines to build multidimensional wavelet bases. These functions are nonseparable, multidimensional basis functions that are localized versions of radial basis functions. We show that Rabut's elementary polyharmonic B-splines do not converge to a Gaussian as the order parameter increases, as opposed to their separable B-spline counterparts. Therefore, we introduce a more isotropic localization operator that guarantees this convergence, resulting into the isotropic polyharmonic B-splines. Next, we focus on the two-dimensional quincunx subsampling scheme. This configuration is of particular interest for image processing because it yields a finer scale progression than the standard dyadic approach. However, up until now, the design of appropriate filters for the quincunx scheme has mainly been done using the McClellan transform. In our approach, we start from the scaling functions, which are the polyharmonic B-splines and, as such, explicitly known, and we derive a family of polyharmonic spline wavelets corresponding to different flavors of the semi-orthogonal wavelet transform; e.g., orthonormal, B-spline, and dual. The filters are automatically specified by the scaling relations satisfied by these functions. We prove that the isotropic polyharmonic B-spline wavelet converges to a combination of four Gabor atoms, which are well separated in the frequency domain. We also show that these wavelets are nearly isotropic and that they behave as an iterated Laplacian operator at low frequencies. We describe an efficient fast Fourier transform-based implementation of the discrete wavelet transform based on polyharmonic B-splines.     

Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-Like Transforms

We propose a novel method for constructing Hilbert transform (HT) pairs of wavelet bases based on a fundamental approximation-theoretic characterization of scaling functions—the B-spline factorization theorem. In particular, starting from well-localized scaling functions, we construct HT pairs of biorthogonal wavelet bases of ${rm L}^2({BBR})$ by relating the corresponding wavelet filters via a discrete form of the continuous HT filter. As a concrete application of this methodology, we identify HT pairs of spline wavelets of a specific flavor, which are then combined to realize a family of complex wavelets that resemble the optimally-localized Gabor function for sufficiently large orders. Analytic wavelets, derived from the complexification of HT wavelet pairs, exhibit a one-sided spectrum. Based on the tensor-product of such analytic wavelets, and, in effect, by appropriately combining four separable biorthogonal wavelet bases of ${rm L}^2({BBR}^2)$, we then discuss a methodology for constructing 2-D directional-selective complex wavelets. In particular, analogous to the HT correspondence between the components of the 1-D counterpart, we relate the real and imaginary components of these complex wavelets using a multidimensional extension of the HT—the directional HT. Next, we construct a family of complex spline wavelets that resemble the directional Gabor functions proposed by Daugman. Finally, we present an efficient fast Fourier transform (FFT)-based filterbank algorithm for implementing the associated complex wavelet transform.      

The double-density dual-tree DWT

This paper introduces the double-density dual-tree discrete wavelet transform (DWT), which is a DWT that combines the double-density DWT and the dual-tree DWT, each of which has its own characteristics and advantages. The transform corresponds to a new family of dyadic wavelet tight frames based on two scaling functions and four distinct wavelets. One pair of the four wavelets are designed to be offset from the other pair of wavelets so that the integer translates of one wavelet pair fall midway between the integer translates of the other pair. Simultaneously, one pair of wavelets are designed to be approximate Hilbert transforms of the other pair of wavelets so that two complex (approximately analytic) wavelets can be formed. Therefore, they can be used to implement complex and directional wavelet transforms. The paper develops a design procedure to obtain finite impulse response (FIR) filters that satisfy the numerous constraints imposed. This design procedure employs a fractional-delay allpass filter, spectral factorization, and filterbank completion. The solutions have vanishing moments, compact support, a high degree of smoothness, and are nearly shift-invariant.     

The discrete wavelet transform: wedding the a trous and Mallatalgorithms

Two separately motivated implementations of the wavelet transform are brought together. It is observed that these algorithms are both special cases of a single filter bank structure, the discrete wavelet transform, the behavior of which is governed by the choice of filters. In fact, the a trous algorithm is more properly viewed as a nonorthonormal multiresolution algorithm for which the discrete wavelet transform is exact. Moreover, it is shown that the commonly used Lagrange a trous filters are in one-to-one correspondence with the convolutional squares of the Daubechies filters for orthonormal wavelets of compact support. A systematic framework for the discrete wavelet transform is provided, and conditions are derived under which it computes the continuous wavelet transform exactly. Suitable filter constraints for finite energy and boundedness of the discrete transform are also derived. Relevant signal processing parameters are examined, and it is observed that orthonormality is balanced by restrictions on resolution     

Theory of Dual-Tree Complex Wavelets

We study analyticity of the complex wavelets in Kingsbury's dual-tree wavelet transform. A notion of scaling transformation function that defines the relationship between the primal and dual scaling functions is introduced and studied in detail. The analyticity property is examined and dealt with via the transformation function. We separate analyticity from other properties of the wavelet such as orthogonality or biorthogonality. This separation allows a unified treatment of analyticity for general setting of the wavelet system, which can be dyadic or M-band; orthogonal or biorthogonal; scalar or multiple; bases or frames. We show that analyticity of the complex wavelets can be characterized by scaling filter relationship and wavelet filter relationship via the scaling transformation function. For general orthonormal wavelets and dyadic biorthogonal scalar wavelets, the transformation function is shown to be paraunitary and has a linear phase delay of $ omega /2$ in $[{0},2pi )$.      

Efficient wavelet prefilters with optimal time-shifts

A wavelet prefilter maps sample values of an analyzed signal to the scaling function coefficient input of standard discrete wavelet transform (DWT) algorithms. The prefilter is the inverse of a certain postfilter convolution matrix consisting of integer sample values of a noninteger-shifted wavelet scaling function. For the prefilter and the DWT algorithms to have similar computational complexity, it is often necessary to use a "short enough" approximation of the prefilter. In addition to well-known quadrature formula and identity matrix prefilter approximations, we propose a Neumann series approximation, which is a band matrix truncation of the optimal prefilter, and derive simple formulas for the operator norm approximation error. This error shows a dramatic dependence on how the postfilter noninteger shift is chosen. We explain the meaning of this shift in practical applications, describe how to choose it, and plot optimally shifted prefilter approximation errors for 95 different Daubechies, Symlet, and B-spline wavelets. Whereas the truncated inverse is overall superior, the Neumann filters are by far the easiest ones to compute, and for some short support wavelets, they also give the smallest approximation error. For example, for Daubechies 1-5 wavelets, the simplest Neumann prefilter provide an approximation error reduction corresponding to 100-10 000 times oversampling in a nonprefiltered system.     

On solving first-kind integral equations using wavelets on abounded interval

The conventional method of moments (MoM), when applied directly to integral equations, leads to a dense matrix which often becomes computationally intractable. To overcome the difficulties, wavelet-bases have been used previously which lead to a sparse matrix. The authors refer to “MoM with wavelet bases” as “wavelet MoM”. There have been three different ways of applying the wavelet techniques to boundary integral equations: 1) wavelets on the entire real line which requires the boundary conditions to be enforced explicitly, 2) wavelet bases for the bounded interval obtained by periodizing the wavelets on the real line, and 3) “wavelet-like” basis functions. Furthermore, only orthonormal (ON) bases have been considered. The present authors propose the use of compactly supported semi-orthogonal (SO) spline wavelets specially constructed for the bounded interval in solving first-kind integral equations. They apply this technique to analyze a problem involving 2D EM scattering from metallic cylinders. It is shown that the number of unknowns in the case of wavelet MoM increases by m-1 as compared to conventional MoM, where m is the order of the spline function. Results for linear (m=2) and cubic (m=4) splines are presented along with their comparisons to conventional MoM results. It is observed that the use of cubic spline wavelets almost “diagonalizes” the matrix while maintaining less than 1.5% of relative normed error. The authors also present the explicit closed-form polynomial representation of the scaling functions and wavelets     

M带离散时间子波变换

二带连续时间子波变换可由无限级树形正交镜像滤波器(QMF)组产生,类似地,二带离散时间子波变换可表示为有限级树形QMF组。该文对二带离散时间子波变换进行了推广,给出了M带离散时间于波变换,并研究了M带离散时间子波变换与M带仿酉滤波器组之间的关系。结果表明,在L级M带树形滤波器组中,如果每级滤波器组是仿酉滤波器组,则该树形滤波器组所产生的离散时间子波基是正交基。     

经典规范正交子波的一种简单广义化方法及其应用

从最简单的Haar尺度函数入手，提出一种简单而又快捷的方法，将每一个经典的规范正交子波基进行拓展得到一类新的规范正交子波基。新子波类中的每一个子波均继承了原始子波的许多基本性质，比如规范正交性，正则阶，时、频局域化特性等，同时也得到某些性能的改善，文中重点探讨广义Haar子波、广义Shannon子波和Meyer子波、Daubechies子波等的简单广义化；最后讨论新子波系统的一个直接应用：实（序列）信号解析子波变换的快速算法问题。     

用于信号逼近的最佳正交子波选择方法

离散子波变换将离散时间信号分解为一系列分辨率下的离散逼近和离散细节。紧支的正交规范子波与完全重建正交镜象滤波器组相对应。本文提出一种用于信号最佳逼近的正交子波选择方法,即选择满足一定条件的滤波器的方法。通过对滤波器参数化,可以将带约束的最优化问题转化为无约束最优化问题,通过对参数在一定范围内的搜索,得到最优解。文中给出了计算机模拟的结果。     

Algorithms for designing wavelets to match a specified signal

Algorithms for designing a mother wavelet /spl psi/(x) such that it matches a signal of interest and such that the family of wavelets {2/sup -(j/2)//spl psi/(2/sup -j/x-k)} forms an orthonormal Riesz basis of L/sup 2/(/spl Rscr/) are developed. The algorithms are based on a closed form solution for finding the scaling function spectrum from the wavelet spectrum. Many applications require wavelets that are matched to a signal of interest. Most current design techniques, however, do not design the wavelet directly. They either build a composite wavelet from a library of previously designed wavelets, modify the bases in an existing multiresolution analysis or design a scaling function that generates a multiresolution analysis with some desired properties. In this paper, two sets of equations are developed that allow us to design the wavelet directly from the signal of interest. Both sets impose bandlimitedness, resulting in closed form solutions. The first set derives expressions for continuous matched wavelet spectrum amplitudes. The second set of equations provides a direct discrete algorithm for calculating close approximations to the optimal complex wavelet spectrum. The discrete solution for the matched wavelet spectrum amplitude is identical to that of the continuous solution at the sampled frequencies. An interesting byproduct of this work is the result that Meyer's spectrum amplitude construction for an orthonormal bandlimited wavelet is not only sufficient but necessary. Specific examples are given which demonstrate the performance of the wavelet matching algorithms for both known orthonormal wavelets and arbitrary signals.     

Two-band wavelets and filterbanks over finite fields with connections to error control coding

Recently, we have developed a new framework to study error-control coding using finite-field wavelets and filterbanks (FBs). This framework reveals a rich set of signal processing techniques that can be exploited to investigate new error correcting codes and to simplify encoding and decoding techniques for some existing ones. The paper introduces the theory of wavelet decompositions of signals in vector spaces defined over Galois fields. To avoid the limitations of the number theoretic Fourier transform, our wavelet transform relies on a basis decomposition in the time rather than the frequency domain. First, by employing a symmetric, nondegenerate canonical bilinear form, we obtain a necessary and sufficient condition that the basis functions defined over finite fields must satisfy in order to construct an orthogonal wavelet transform. Then, we present a design methodology to generate the mother wavelet and scaling function over finite fields by relating the wavelet transform to two-channel paraunitary (PU) FBs. Finally, we describe the application of this transform to the construction of error correcting codes. In particular, we give examples of double circulant codes that are generated by wavelets.     

Wavelets and filter banks: theory and design

The wavelet transform is compared with the more classical short-time Fourier transform approach to signal analysis. Then the relations between wavelets, filter banks, and multiresolution signal processing are explored. A brief review is given of perfect reconstruction filter banks, which can be used both for computing the discrete wavelet transform, and for deriving continuous wavelet bases, provided that the filters meet a constraint known as regularity. Given a low-pass filter, necessary and sufficient conditions for the existence of a complementary high-pass filter that will permit perfect reconstruction are derived. The perfect reconstruction condition is posed as a Bezout identity, and it is shown how it is possible to find all higher-degree complementary filters based on an analogy with the theory of Diophantine equations. An alternative approach based on the theory of continued fractions is also given. These results are used to design highly regular filter banks, which generate biorthogonal continuous wavelet bases with symmetries     

Application of the dual-tree wavelet transform for digital filtering of noisy audio signals

The problem of refinement of the quality of filtering of noisy audio signals with the help of the methods based on a discrete wavelet transform with real bases and a dual-tree (complex) wavelet transform using analytical wavelets as basis functions is considered. Test examples and processing of experimental data have shown that, in the case of the optimum selection of the threshold level, the approach using the dual-tree wavelet transform ensures the minimum signal reconstruction error after correction of wavelet coefficients.     

Wavelets and recursive filter banks

It is shown that infinite impulse response (IIR) filters lead to more general wavelets of infinite support than finite impulse response (FIR) filters. A complete constructive method that yields all orthogonal two channel filter banks, where the filters have rational transfer functions, is given, and it is shown how these can be used to generate orthonormal wavelet bases. A family of orthonormal wavelets that have a maximum number of disappearing moments is shown to be generated by the halfband Butterworth filters. When there is an odd number of zeros at π it is shown that closed forms for the filters are available without need for factorization. A still larger class of orthonormal wavelet bases having the same moment properties and containing the Daubechies and Butterworth filters as the limiting cases is presented. It is shown that it is possible to have both linear phase and orthogonality in the infinite impulse response case, and a constructive method is given. It is also shown how compactly supported bases may be orthogonalized, and bases for the spline function spaces are constructed     

Wavelets and wideband correlation processing

This tutorial presents the application of wavelet transforms to wideband correlation processing. One major difference between most applications of wavelets and the work presented is the choice of mother wavelet. It focuses on nonorthogonal, continuous mother wavelets, whereas most applications use the orthogonal mother wavelets that were advanced by Daubechies (1988). The continuous wavelet transform then provides an additional tool for studying and gaining insight into wideband correlation processing. In order to understand when wideband processing may be required, its counterpart, narrowband processing, is presented and its limitations are discussed. Identifying those applications requiring wideband processing and presenting techniques to implement the processing are two of the goals of this tutorial article. The underlying tool is the wavelet transform     

Generalized Daubechies Wavelet Families

We present a generalization of the orthonormal Daubechies wavelets and of their related biorthogonal flavors (Cohen-Daubechies-Feauveau, 9/7). Our fundamental constraint is that the scaling functions should reproduce a predefined set of exponential polynomials. This allows one to tune the corresponding wavelet transform to a specific class of signals, thereby ensuring good approximation and sparsity properties. The main difference with the classical construction of Daubechies is that the multiresolution spaces are derived from scale-dependent generating functions. However, from an algorithmic standpoint, Mallat's fast wavelet transform algorithm can still be applied; the only adaptation consists in using scale-dependent filter banks. Finite support ensures the same computational efficiency as in the classical case. We characterize the scaling and wavelet filters, construct them and show several examples of the associated functions. We prove that these functions are square-integrable and that they converge to their classical counterparts of the corresponding order.     

Wavelets and differential-dilation equations

It is shown how differential-dilation equations can be constructed using iterations, similar to the iterations with which wavelets and dilation equations are constructed. A continuous-time wavelet is constructed starting from a differential-dilation equation. It has compact support and excellent time domain and frequency domain localization properties. The wavelet is infinitely differentiable and therefore cannot be obtained using digital filter banks. In addition, the wavelet has excellent approximation properties. New sampling and differentiation techniques are also introduced. Results on image interpolation using the solution of the differential-dilation equation are presented. Examples are given, demonstrating the suitability of the new wavelet function for signal analysis