International Journal of Wavelets, Multiresolution and Information Processing

A method of construction of the local approximations in the case of functions defined on n-dimensional (n ≥ 1) smooth manifold with boundary is proposed. In particular, spline and finite-element methods on manifold are discussed. Nondegenerate simplicial subdivision of the manifold is introduced and a simple method for evaluations of approach is examined (the evaluations are optimal as to N-width of corresponding compact set).

This paper deals with engineering application of wavelet transform for processing of real seismological signals. Methodology for processing of these slight signals using wavelet transform is presented in this paper. Briefly, three basic aims are connected with this procedure:.

1. Selection of optimal wavelet and optimal wavelet basis B^opt for selected data set based on minimal entropy: B^opt = arg min_B E(X,B). The best results were reached by symmetric complex wavelets with scaling coefficients SCD-6.

2. Wavelet packet decomposition and filtration of data using universal criterion of thresholding of the form , where σ is minimal variance of the sum of packet decomposition of chosen level.

3. Cluster analysis of decomposed data.

All programs were elaborated using program MATLAB 5.

Polynomial frames based on orthogonal decompositions with respect to weights of Laguerre and Hermite type are considered. Asymptotic formulas for the coefficients from expansions on such frames are presented. The ability to detect singularities of all order is studied in detail.

In old movies, scratches are common damages that mostly result from a mechanical stress during the projection. A method for removing vertical scratches is proposed and suitable to be automatically applied to sequences of images. The method uses a wavelet decomposition of the original digital image, I, in order to separate the high frequency components and to elaborate corrupted data in the regular matrix, A, and in the vertical details matrix, V, only. For A, approximating functions are constructed in suitable spline spaces, which depend on the morphological quality of the image near the scratch. Monochromatic old images and images with simulated scratches have been considered to validate the method.

In this paper, the wavelet Riesz bases in the space ℓ²(ℤ) are studied. The main tool employed is discrete splines, which have received a considerable attention by many authors. The discrete splines are used for multiresolution analysis of the space ℓ²(ℤ).

Refinement equations of the type play an exceptional role in the theory of wavelets, subdivision algorithms and computer design. It is known that the regularity of their compactly supported solutions (refinable functions) depends on the spectral properties of special N-dimensional linear operators T₀, T₁ constructed by the coefficients of the equation. In particular, the structure of kernels and of common invariant subspaces of these operators have been intensively studied in the literature. In this paper, we give a complete classification of the kernels and of all the root subspaces of T₀ and T₁, as well as of their common invariant subspaces. This result answers several open questions stated in the literature and clarifies the structure of the space spanned by the integer translates of refinable functions. This also leads to some results on the moduli of continuity of refinable functions and wavelets in various functional spaces. In particular, it is proved that the Hölder exponent of those functions is sharp whenever it is not an integer.

In this paper, we deal with two different problems. First, we provide the convergence rates of multiresolution approximations, with respect to the supremum norm, for the class of elliptic splines defined in Ref. 10, and in particular for polyharmonic splines. Secondly, we consider the problem of recovering a function from a sample of noisy data. To this end, we define a linear and smooth estimator obtained from a multiresolution process based on polyharmonic splines. We discuss its asymptotic properties and we prove that it converges to the unknown function almost surely.

The aim of this paper is to describe a new five-dimensional, variable degree polynomial space. This space has the same properties of P₄ and possesses generalized tension properties, in the sense that it is close to P₂ for large values of the degrees. Its applications in the construction of spline functions will also be discussed.

In recent years, many papers have been devoted to the topic of balanced multiwavelets, namely, multiwavelet bases which are especially designed to avoid the prefiltering step in the implementation of the multiwavelet transform. In this work, we give a simple algebraic proof of how scalar wavelets can be reinterpreted as the most natural balanced multiwavelets, which maintain the good properties of the wavelet bases they come from. We then show how these new bases can be successfully used to apply matrix thresholding for the denoising of images corrupted by Gaussian noise. In fact, this new approach discovers a balanced matrix nature in Daubechies bases, hence obtaining better numerical results with respect to those achieved via scalar thresholding. In particular, this reinterpretation of scalar wavelets as balanced multiwavelets allows us to successfully use the thresholding filters, previously introduced in the scalar case, in a matrix setting.

In the present work, a new multilevel backprojection procedure for both parallel and fan beam geometries based on 1D MRA wavelet is derived and analyzed. It is based on the characterization property of orthonormal wavelets via their autocorrelation functions. The algorithm does not involve any wavelet-based decomposition in the normal sense, yet it uses a multiresolution formula as a crucial element in a CBP type procedure valid for both the parallel and complex fan beam geometries. The algorithm can be used for wavelet based image analysis.

In this paper, Journe wavelet function is introduced as a wavelet generating function. The expression of reproducing kernel function for the image space of this wavelet transform is obtained based on the fact that the image space of the wavelet transform is a reproducing kernel Hilbert space. Then the isometric identity of Journe wavelet transform is obtained. The connections between the image space of the wavelet transform and the image space of the known reproducing kernel space are established by the theories of reproducing kernel. The properties and the structures of the image space of the wavelet transform can be characterized by the properties and the structures of the image space of the known reproducing kernel space. Using the ideas of reproducing kernel, we consider there are relations between the wavelet transform and the sampling theorem. Meanwhile, the approximations in sampling theorems is shown and the truncation error is given. This provides a theoretical basis for us to study the image space of the general wavelet transform and broadens the scope of application of theories of the reproducing kernel space.

A necessary and sufficient condition for the perturbation of a Banach frame by a non-zero functional to be a Banach frame has been obtained. Also a sufficient condition for the perturbation of a Banach frame by a sequence in E* to be a Banach frame has been given. Finally, a necessary condition for the perturbation of a Banach frame by a finite linear combination of linearly independent functionals in E* to be a Banach frame has been given.

In computer vision and image processing, edge detection concerns the localization of significant variations of the grey level image and the identification of the physical phenomena that originated them. It is difficult to design a general edge detection algorithm which performs well in many contexts and captures the requirements of subsequent processing stages. Consequently, over the history of digital image processing, a variety of edge detectors have been devised which differ in their mathematical and algorithmic properties. In this paper, we briefly describe some existing edge detectors. We investigate the wavelet transform with direction which is related to the Canny and Marr–Hildreth's edge detectors. This wavelet transform has a remarkable feature that plays an important role in image processing applications. We discuss its applications to edge and target detections. The direction effect enables the wavelets to analyze the directional features of images. Algorithms and experiments for edge and target detections have been developed based on the theory. And, very promising results have been shown in image processing.