ORTHONORMAL MRA WAVELETS: SPECTRAL FORMULAS AND ALGORITHMS

This paper investigates the mathematical background of multiresolution analysis in the specific context where the signal is represented by irregularly sampled data at known locations. The study is related to the construction of nested piecewise polynomial multiresolution spaces represented by their corresponding orthonormal bases. Using simple spline basis orthonormalization procedures involves the construction of a large family of orthonormal spline scaling bases defined on consecutive bounded intervals. However, if no more additional conditions than those coming from multiresolution are imposed on each bounded interval, the orthonormal basis is represented by a set of discontinuous scaling functions. The spline wavelet basis also has the same problem. Moreover, the dimension of the corresponding wavelet basis increases with the spline degree. An appropriate orthonormalization procedure of the basic spline space basis, whatever the degree of the spline, allows us to (i) provide continuous scaling and wavelet functions, (ii) reduce the number of wavelets to only one, and (iii) reduce the complexity of the filter bank. Examples of the multiresolution implementations illustrate that the main important features of the traditional multiresolution are also satisfied.

In this contribution we introduce the concept of multiresolution analysis (MRA) and give an explanation of the relationship between MRA and orthonormal wavelet basis. The construction of the orthonormal B-spline wavelet bases is described in detail. We extend the B-splines to `non-integral order' cases and thus obtain a new family of orthonormal wavelet bases for the space L2(R). Some good properties of the new wavelets are demonstrated. The new wavelet family gives satisfactory performances in our research projects including seismic signal compression and gravity tide data processing.

Studies the autocorrelation and cross-correlation structure of the scaling and wavelet functions associated with compactly supported orthonormal wavelet bases. These correlation structures play an important role in both wavelet-based interpolation and in answering the question of existence of scale-limited signals. It is shown how to efficiently compute the correlation functions approximately at the M-adic rationals. Furthermore, when the tight frame is, in particular, an orthonormal wavelet basis, it is shown that the approximations involved in the computation of the samples of the correlations cancel in a manner to make the computations exact. For the case of orthonormal wavelet bases, an attempt is made to give a complete description of the zeros at the M-adic rationals. An interesting fact that arises from the analysis is that all the correlation functions possible have infinitely many zeros in their support. >

Under suitable conditions, if the scaling functions ϕ1 and ϕ2 generate the multiresolutions V (j)(ϕ1) and V (j)(ϕ2), then their convolution ϕ1*ϕ2also generates a multiresolution V (j)(ϕ1*ϕ2) More over, if p is an appropriate convolution operator from l 2 into itself and if ϕ is a scaling function generating the multiresolution V (j)(ϕ),then p*ϕis a scaling function generating the same multiresolution V (j)(ϕ)=V (j)(p*ϕ). Using these two properties, we group the scaling and wavelet functions into equivalent classes and consider various equivalent basis functions of the associated function spaces We use the n-fold convolution product to construct sequences of multiresolution and wavelet spaces V (j)(ϕn) and W (j)(ϕn) with increasing regularity. We discuss the link between multiresolution analysis and Shannon's sampling theory. We then show that the interpolating and orthogonal pre- and post-filters associated with the multiresolution sequence V (0)(ϕn)asymptotically converge to the ideal lowpass filter of Shannon. We also prove that the filters associated with the sequence of wavelet spaces W (0)(ϕn)convergeto the ideal bandpass filter. Finally, we construct the basic wavelet sequences ψ b nand show that they tend to Gabor functions. Thisprovides wavelets that are nearly time-frequency optimal. The theory is illustrated with the example of polynomial splines.

The wavelet representation using orthonormal wavelet bases has received widespread attention. Recently M-band orthonormal wavelet bases have been constructed and compactly supported M-band wavelets have been parameterized. This paper gives the theory and algorithms for obtaining the optimal wavelet multiresolution analysis for the representation of a given signal at a predetermined scale in a variety of error norms. Moreover, for classes of signals, this paper gives the theory and algorithms for designing the robust wavelet multiresolution analysis that minimizes the worst case approximation error among all signals in the class. All results are derived for the general M-band multiresolution analysis. An efficient numerical scheme is also described for the design of the optimal wavelet multiresolution analysis when the least-squared error criterion is used. Wavelet theory introduces the concept of scale which is analogous to the concept of frequency in Fourier analysis. This paper introduces essentially scale limited signals and shows that band limited signals are essentially scale limited, and gives the wavelet sampling theorem, which states that the scaling function expansion coefficients of a function with respect to an M-band wavelet basis, at a certain scale (and above) completely specify a band limited signal (i.e., behave like Nyquist (or higher) rate samples).< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Gabardo and Nashed have studied nonuniform multiresolution analysis based on the theory of spectral pairs in a series of papers, see Refs. 4 and 5. Farkov,3 has extended the notion of multiresolution analysis on locally compact Abelian groups and constructed the compactly supported orthogonal p-wavelets on L2(ℝ+). We have considered the nonuniform multiresolution analysis on positive half-line. The associated subspace V0 of L2(ℝ+) has an orthonormal basis, a collection of translates of the scaling function φ of the form {φ(x ⊖ λ)}λ∈Λ+ where Λ+ = {0, r/N} + ℤ+, N &gt; 1 (an integer) and r is an odd integer with 1 ≤ r ≤ 2N - 1 such that r and N are relatively prime and ℤ+ is the set of non-negative integers. We find the necessary and sufficient condition for the existence of associated wavelets and derive the analogue of Cohen's condition for the nonuniform multiresolution analysis on the positive half-line.

Adaptive wavelet-based image compression methods are introduced. In contrast to the classical wavelet decomposition scheme one can use different wavelet and scaling functions at every scale-this leads to nonstationary multiresolution analysis. An inhomogeneous multiresolution analysis is obtained by using different functions for the two directions in the tensor product of the bidimensional multiresolution analysis, furthermore these two methods are combined. The freedom in using different functions is exploited for image compression in an adaptive way. Special non-stationary and/or inhomogeneous multiresolution analysis is built out of the functions in a given library that is best suited for compression of a given image. This is done by minimizing cost-functions at every scale-level.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Orthonormal wavelets have been successfully used as basis and testing functions for the integral equations and extremely sparse impedance matrices have been obtained. However, in many practical problems, the solution domain is confined in a bounded interval, while the wavelets are originally defined on the entire real line. To overcome this problem, periodic wavelets have been described in the literature. Nonetheless, the unknown functions must take on equal values at the endpoints of the bounded interval in order to apply periodic wavelets as the basis functions. We present the intervallic Coifman wavelets (coiflets) for the method of moments (MoM). The intervallic wavelets release the endpoints restrictions imposed on the periodic wavelets. The intervallic wavelets form an orthonormal basis and preserve the same multiresolution analysis (MRA) of other usual unbounded wavelets. The coiflets possesses a special property that their scaling functions have many vanishing moments. As a result, the zero entries of the matrices are identified directly, without using a truncation scheme with an artificially established threshold. Further, the majority of matrix elements are evaluated directly without performing numerical integration procedures such as Gaussian quadrature. For an n/spl times/n matrix, the number of actual numerical integrations is reduced from n/sup 2/ to the order of 3n(2L-1), when the coiflets of order L is employed. The construction of intervallic wavelets is presented. Numerical examples of scattering problems are discussed and the relative error of this method is studied analytically.

Application of the Scaling Functions to Nonparametric Regression

: Images are decomposed and reconstructed by applying recently developed multiresolution techniques to a particular class of orthonormal wavelets. An original image is decomposed into a series of sub-images in which the low- and high-frequency information of the original image has been segregated. The image is then reconstructed after any redundant high-frequency information is discarded. This procedure allows a perceptually equivalent image to be generated using less information. Further, the resulting reduced-data image can be stored, processed, transmitted and displayed more efficiently. The advantages of the technique are that the wavelets used are continuous and well- localized in both space and spatial-frequency, the reconstruction can be performed by simply reversing the multiresolution decomposition process, and the reconstruction is error-free. The C source code required to implement the present technique on a PC-compatible computer is also presented. Data compression, Multiresolution analysis, Data reduction, Orthonormal bases, Image processing, Wavelets.

On the spectrums of frame multiresolution analyses

Wavelet analysis is used in direct thrust control system of linear motor drive in this paper. The change of the primary resistance of linear motor will greatly affect the performances of direct thrust control system at low speeds. The observation of flux linkage is improved by accurate identification of primary resistance, and the direct thrust control performance of linear motor is further improved. Because of the time frequency location characteristic and multi-scale ability of wavelet transform and multi-resolution analysis, wavelet transform research on direct thrust control system is proposed. This paper selects different wavelets to analyze the extract data in different scaling function, and obtains the trace data from basic data. Then realized signal wavelet identification. The simulation results show these identifiers can precisely measure the value of resistance and efficiently improve the low-speed performances of direct thrust control system.

The aim of this paper is to present a modification to the Chui and Quak (1992) spline multiresolution analysis for the finite interval. Boundary scaling functions with multipole nodes at interval endpoints are rejected, in favor of the classical B-spline scaling function restricted to the interval. This necessitates derivation of revised boundary wavelets. In addition, a direct method of decomposition results in bandwidth reduction on solving some associated linear systems, and image distortion is reduced by employing natural spline projection. Finally, a hybrid projection scheme is proposed, which particularly for large systems further lowers operation count. Numerical experiments proving the algorithm are indicated.

We characterize the scaling functions of a multiresolution analysis in a general context, where instead of the dyadic dilation one considers the dilation given by a fixed linear map A : R n → R n A: \mathbb {R}^n\rightarrow \mathbb {R}^n such that A ( Z n ) ⊂ Z n A(\mathbb {Z}^n) \subset \mathbb {Z}^n and all (complex) eigenvalues of A A have absolute value greater than 1. 1. In the general case the conditions depend on the map A . A. We identify some maps for which the obtained condition is equivalent to the dyadic case, i.e., when A A is a diagonal matrix with all numbers in the diagonal equal to 2. 2. There are also easy examples of expanding maps for which the obtained condition is not compatible with the dyadic case. The complete characterization of the maps for which the obtained conditions are equivalent is out of the scope of the present note.

Generalized multiple scale reproducing kernel particle methods