Gabardo and Nashed [‘Nonuniform multiresolution analyses and spectral pairs’, J. Funct. Anal. 158(1) (1998), 209–241] have introduced the concept of nonuniform multiresolution analysis (NUMRA), based on the theory of spectral pairs, in which the associated translated set $\Lambda =\{0,{r}/{N}\}+2\mathbb Z$ is not necessarily a discrete subgroup of $\mathbb{R}$ , and the translation factor is $2\textrm{N}$ . Here r is an odd integer with $1\leq r\leq 2N-1$ such that r and N are relatively prime. The nonuniform wavelets associated with NUMRA can be used in signal processing, sampling theory, speech recognition and various other areas, where instead of integer shifts nonuniform shifts are needed. In order to further generalize this useful NUMRA, we consider the set $\widetilde {\Lambda }=\{0,{r_1}/{N},{r_2}/{N},\ldots ,{r_q}/{N}\}+s\mathbb Z$ , where s is an even integer, $q\in \mathbb {N}$ , $r_i$ is an integer such that $1\leq r_i\leq sN-1,\,(r_i,N)=1$ for all i and $N\geq 2$ . In this paper, we prove that the concept of NUMRA with the translation set $\widetilde {\Lambda }$ is possible only if $\widetilde {\Lambda }$ is of the form $\{0,{r}/{N}\}+s\mathbb Z$ . Next we introduce $\Lambda _s$ -nonuniform multiresolution analysis ( $\Lambda _s$ -NUMRA) for which the translation set is $\Lambda _s=\{0,{r}/{N}\}+s\mathbb Z$ and the dilation factor is $sN$ , where s is an even integer. Also, we characterize the scaling functions associated with $\Lambda _s$ -NUMRA and we give necessary and sufficient conditions for wavelet filters associated with $\Lambda _s$ -NUMRA.

This paper establishes the mapping properties of pseudo-differential operators and the Fourier integral operators on the weighted Morrey spaces with variable exponents and the weighted Triebel–Lizorkin–Morrey spaces with variable exponents. We obtain these results by extending the extrapolation theory to the weighted Morrey spaces with variable exponents. This extension also gives the mapping properties of Calderón–Zygmund operators on the weighted Hardy–Morrey spaces with variable exponents and the wavelet characterizations of the weighted Hardy–Morrey spaces with variable exponents.

Recent advances in high-resolution fluorescence microscopy have enabled the systematicstudy of morphological changes in large populations of cells induced by chemical andgenetic perturbations, facilitating the discovery of signaling pathways underlyingdiseases and the development of new pharmacological treatments. In these studies, though,due to the complexity of the data, quantification and analysis of morphological featuresare for the vast majority handled manually, slowing significantly data processing andlimiting often the information gained to a descriptive level. Thus, there is an urgentneed for developing highly efficient automated analysis and processing tools forfluorescent images. In this paper, we present the application of a method based on theshearlet representation for confocal image analysis of neurons. The shearletrepresentation is a newly emerged method designed to combine multiscale data analysis withsuperior directional sensitivity, making this approach particularly effective for therepresentation of objects defined over a wide range of scales and with highly anisotropicfeatures. Here, we apply the shearlet representation to problems of soma detection ofneurons in culture and extraction of geometrical features of neuronal processes in braintissue, and propose it as a new framework for large-scale fluorescent image analysis ofbiomedical data.

In this paper we consider a smoothness parameter estimation problem for a density function. The smoothness parameter of a function is defined in terms of Besov spaces. This paper is an extension of recent results (K. Dziedziul, M. Kucharska, B. Wolnik, Estimation of the smoothness parameter). The construction of the estimator is based on wavelets coefficients. Although we believe that the effective estimation of the smoothness parameter is impossible in general case, we can show that it becomes possible for some classes of the density functions.

This article describes the implementation of a simple wavelet-based optical-flow motion estimator dedicated to continuous motions such as fluid flows. The wavelet representation of the unknown velocity field is considered. This scale-space representation, associated to a simple gradient-based optimization algorithm, sets up a well-defined multiresolution framework for the optical flow estimation. Moreover, a very simple closure mechanism, approaching locally the solution by high-order polynomials is provided by truncating the wavelet basis at fine scales. Accuracy and efficiency of the proposed method is evaluated on image sequences of turbulent fluid flows.

The shearlet representation has gained increasing recognition in recent years as aframework for the efficient representation of multidimensional data. This representationconsists of a countable collection of functions defined at various locations, scales andorientations, where the orientations are obtained through the use of shear matrices. Whileshear matrices offer the advantage of preserving the integer lattice and being moreappropriate than rotations for digital implementations, the drawback is that the action ofthe shear matrices is restricted to cone-shaped regions in the frequency domain. Hence, inthe standard construction, a Parseval frame of shearlets is obtained by combiningdifferent systems of cone-based shearlets which are projected onto certain subspaces ofL2(ℝD) with the consequence thatthe elements of the shearlet system corresponding to the boundary of the cone regions losetheir good spatial localization property. In this paper, we present a new constructionyielding smooth Parseval frame of shearlets forL2(ℝD). Specifically, allelements of the shearlet systems obtained from this construction are compactly supportedand C∞ in the frequency domain, hence ensuring that the systemhas also excellent spatial localization.

Directional multiscale representations such as shearlets and curvelets have gainedincreasing recognition in recent years as superior methods for the sparse representationof data. Thanks to their ability to sparsely encode images and other multidimensionaldata, transform-domain denoising algorithms based on these representations are among thebest performing methods currently available. As already observed in the literature, theperformance of many sparsity-based data processing methods can be further improved byusing appropriate combinations of dictionaries. In this paper, we consider the problem of3D data denoising and introduce a denoising algorithm which uses combined sparsedictionaries. Our numerical demonstrations show that the realization of the algorithmwhich combines 3D shearlets and local Fourier bases provides highly competitive results ascompared to other 3D sparsity-based denosing algorithms based on both single and combineddictionaries.

Early damage detection on structures plays a very important role for ensuring safety and reliability. This paper provides an efficient method based on wavelet transforms in order to detect and localize damage on structures subjected to moving loads such as beams and bridges. A numerical model based on the experimental test-rig utilized in this study is developed by using a finite element commercial software. Different types of damage on the bridge of the numerical model are simulated and transient analyses are performed by incorporating a load which moves constantly along the beam nodes. Continuous wavelet transform diagrams using the vertical acceleration responses show that damage can be identified and localized even with significant percentages of noise. Nevertheless, the method is improved by filtering the signals, removing the border effects, and calculating the total wavelet energy of the beam from the coefficients along the selected range of scales. Thus, the accumulation of wavelet energy could indicate the presence of damage. Finally, laboratory experiments are conducted to validate this work and a good agreement between numerical and experimental results is obtained.

In the framework of an explicitly correlated formulation of the electronic Schrödingerequation known as the transcorrelated method, this work addresses some fundamental issuesconcerning the feasibility of eigenfunction approximation by hyperbolic wavelet bases.Focusing on the two-electron case, the integrability of mixed weak derivatives ofeigenfunctions of the modified problem and the improvement compared to the standardformulation are discussed. Elements of a discretization of the eigenvalue problem based onorthogonal wavelets are described, and possible choices of tensor product bases arecompared especially from an algorithmic point of view. The use of separable approximationsof potential terms for applying operators efficiently is studied in detail, and estimatesfor the error due to this further approximation are given.