Construction Of Wavelets Research Articles

Non-separable multidimensional multiresolution wavelets: A Douglas-Rachford approach

After re-casting the wavelet construction problem as a feasibility problem with constraints arising from the requirements of compact support, smoothness and orthogonality, the Douglas–Rachford algorithm is employed in the search for multi-dimensional, non-separable, compactly supported, smooth, orthogonal, multiresolution wavelets in the case of translations along the integer lattice and isotropic dyadic dilations. An algorithm for the numerical construction of such wavelets is described. By applying the algorithm, new one-dimensional wavelets are produced as well as genuinely non-separable two-dimensional wavelets.

On some $q$-Bessel type continuous wavelet transform

In this paper we continue as in \cite{Rezguietal} to exploit the modified variants of Bessel function in the framework of $q$-theory to construct wavelet operators. A generalized $q$-Bessel type function has been introduced leading to an associated mother wavelet which in turns has induced a continuous wavelet transform. Finally, Plancherel/Parceval type relations have been proved. Such variants of wavelets permit in some sense to approximate solutions of ODEs and PDEs by transforming them to recurrent sequences.

Geometrization of the Huygens–Fresnel principle: Applications to Fraunhofer diffraction

The classical wave theory can trace its historical origins to the seminal works of Christian Huygens, Thomas Young, and Augustin Fresnel. To explain some of light’s observed properties, such as rectilinear propagation, reflection, and refraction, Huygens proposed a simple geometrical construction of secondary spherical wavelets with centers of disturbance located on a primary wavefront. More than a century later, Young formulated the law of interference to both predict the formation of fringes in his now famous double slit experiment and also to estimate the wavelengths associated with different colors. A decade after that, Fresnel combined Huygens’ construction with Young’s interference law to qualitatively and quantitatively describe diffraction, which is the bending of light upon encountering an obstacle or an aperture. This grand synthesis, called the Huygens–Fresnel principle, acts as a powerful pictorial aid and conceptual tool that can describe a wide variety of complicated optical phenomena. However, the applications of the principle and its later developments, such as the Kirchhoff–Fresnel integral, are strewn with several simplifying assumptions and approximations that are aimed at minimizing the mathematical challenges involved. Consequently, two distinct formalisms are necessary to account for diffraction effects when the source of light or observation screen is placed nearby and far away from the aperture or obstacle. Recently, a hyperbola framework for analyzing wave interference at a multi-slit barrier was shown to successfully circumvent all conventionally imposed ad hoc conditions. The method commences directly from the Huygens–Fresnel principle and the ensuing predictions pertaining to the distribution of fringe characteristics, namely, positions, widths, and intensities on a detection screen can, therefore, justifiably claim accuracy in both the near field (Fresnel regime) and the far field (Fraunhofer regime). In this paper, the analysis that was previously carried out for the special case of slits of negligible widths is further extended to encompass slits of finite widths as well.

Construction of non-stationary biorthogonal Wavelets based on cardinal Chebyshev B-spline

In this paper, we propose a novel method for designing non-stationary biorthogonal wavelets that leverages Bezout equations to compute wavelet filter coefficients. Our approach integrates cardinal Chebyshev B-splines of varying degrees (second, third, and fourth) as bases for constructing these wavelets, providing explicit formulations for both low-pass and high-pass filters at all levels. By employing this method, we streamline the design process, making it more efficient and accessible while accommodating diverse processing requirements. The utilization of Bezout equations offers a systematic framework for deriving wavelet filter coefficients, enhancing the reproducibility and reliability of the design process. This systematic approach ensures that the resulting wavelets possess desired properties such as vanishing moments, smoothness, and optimal localization. Moreover, by incorporating cardinal Chebyshev B-splines, we can effectively capture intricate signal and image features, addressing challenges encountered in traditional wavelet design methods. Our method not only simplifies the design process but also provides a comprehensive solution by explicitly formulating filters and dual filters across all levels. This comprehensive approach ensures that the resulting wavelets meet the specific demands of applications in signal and image processing, including compression, denoising, and pattern recognition. These advancements hold significant implications for various domains, including information technology, healthcare, and engineering, where efficient and reliable processing techniques are crucial. Moving forward, it is imperative to refine algorithms, assess parameter robustness, and conduct thorough empirical validations across diverse datasets and application scenarios. This will ensure the efficacy and scalability of our proposed method and pave the way for further advancements in non-stationary wavelet design and its applications. By continuously improving and validating our approach, we can foster innovation and enhance practical outcomes in a wide range of fields.

A New Biorthogonal Spline Wavelet-Based K-Layer Network for Underwater Image Enhancement

Wavelet decomposition is pivotal for underwater image processing, known for its ability to analyse multi-scale image features in the frequency and spatial domains. In this paper, we propose a new biorthogonal cubic special spline wavelet (BCS-SW), based on the Cohen–Daubechies–Feauveau (CDF) wavelet construction method and the cubic special spline algorithm. BCS-SW has better properties in compact support, symmetry, and frequency domain characteristics. In addition, we propose a K-layer network (KLN) based on the BCS-SW for underwater image enhancement. The KLN performs a K-layer wavelet decomposition on underwater images to extract various frequency domain features at multiple frequencies, and each decomposition layer has a convolution layer corresponding to its spatial size. This design ensures that the KLN can understand the spatial and frequency domain features of the image at the same time, providing richer features for reconstructing the enhanced image. The experimental results show that the proposed BCS-SW and KLN algorithm has better image enhancement effect than some existing algorithms.

From the construction of wavelets based on derivatives of the Gaussian function to the synthesis of filters with a finite impulse response

From the construction of wavelets based on derivatives of the Gaussian function to the synthesis of filters with a finite impulse response

A New Method for 2D-Adapted Wavelet Construction: An Application in Mass-Type Anomalies Localization in Mammographic Images

This contribution presents a wavelet-based algorithm to detect patterns in images. A two-dimensional extension of the DST-II is introduced to construct adapted wavelets using the equation of the tensor product corresponding to the diagonal coefficients in the 2D discrete wavelet transform. A 1D filter was then estimated that meets finite energy conditions, vanished moments, orthogonality, and four new detection conditions. These allow, when performing the 2D transform, for the filter to detect the pattern by taking the diagonal coefficients with values of the normalized similarity measure, defined by Guido, as greater than 0.7, and α=0.1. The positions of these coefficients are used to estimate the position of the pattern in the original image. This strategy has been used successfully to detect artificial patterns and localize mass-like abnormalities in digital mammography images. In the case of the latter, high sensitivity and positive predictive value in detection were achieved but not high specificity or negative predictive value, contrary to what occurred in the 1D strategy. This means that the proposed detection algorithm presents a high number of false negatives, which can be explained by the complexity of detection in these types of images.

Discrete-time scale-shift approach for wavelet construction and analysis

We provide a sufficient condition to construct a wavelet family from a purely discrete-time signal. The construction is based on an exact mathematical definition of the scale-shift operator in discrete-time and strongly motivated by a strict equivalence between voice transform and wavelet coefficients of a discrete-time signal. Particular classes of signals which satisfy this condition are given and furthers analysis are also considered.

Construction of Wavelets Based on Exponential Atomic Functions hupa(x) and Harmonic Functions scupb(x) and gk(x)

Construction of Wavelets Based on Exponential Atomic Functions hupa(x) and Harmonic Functions scupb(x) and gk(x)

Construction of wavelets based on exponential atomic functions hup_a (x) and harmonic scup_ b (x) and g_k (x).

The possibility of constructing wavelets based on convolutions with a rectangular pulse of exponential and harmonic atomic functions is investigated. Some properties of exponential and harmonic atomic functions are considered. The values of the uncertainty constants of the constructed wavelets are calculated.

Quadratic Phase Multiresolution Analysis and the Construction of Orthonormal Wavelets in L2(ℝ)

The multi-resolution analysis (MRA) associated with quadratic phase Fourier transform (QPFT) serves as a tool to construct orthogonal bases of the L2(R). Consequently, it assumes a pivotal role in facilitating potential applications of QPFT. Inspired by the sampling theorem applicable to band-limited signals in the QPFT domain, this paper formulates the development of the MRA linked with QPFT. Subsequently, we develop a method for constructing orthogonal bases for L2(R), followed by some examples.

Dual wavelet frame packets with non-square iterative matrices

Traditional wavelet frame packets are based on square iterative matrices and frame scaling functions. Since the Fourier transform of frame scaling functions are discontinuous, the resulting wavelet frame packets do not possess time-frequency localization. In this study, we will develop the new construction of wavelet frame packets by using non-square iterative matrices: one is to construct wavelet frame packets directly from known non-GMRA wavelet frames to obtain finer time-frequency structure, the other is based on GMRA wavelet frames constructed from smooth generalization scaling functions, the corresponding quasi-projection formula and fast algorithm are also given.

The Farey map exploited in the construction of a Farey mother wavelet

The wavelet analysis of a function passes through its so-called wavelet transform. Such a transform is mathematically defined as a convolution product of the analyzed function with another analyzing function known as the mother wavelet by involving the scale and the translation parameters. This means that the mother wavelet construction is the starting and major point in the wavelet analysis. Besides, the choice of the mother wavelet remains a major problem in wavelet applications such as statistical series, time series, signal, and image processing. This needs more candidates of mother wavelets to be constructed. The main aim of the present paper is to construct a new mother wavelet by exploiting the well-known Farey map. We showed indeed that such a map may be a mother wavelet owing properties such as admissibility, moments, 2-scale relation, and reconstruction rule already necessary in the wavelet analysis of functions. By a suitable choice of translation-dilation parameters on the original Farey map, we succeeded to prove the main properties of a Farey wavelet analysis. The constructed mother looks to be suitable for many complicated applications such as hyperbolic PDEs.

Sparse Grid Approximation in Weighted Wiener Spaces

We study approximation properties of multivariate periodic functions from weighted Wiener spaces by sparse grid methods constructed with the help of quasi-interpolation operators. The class of such operators includes classical interpolation and sampling operators, Kantorovich-type operators, scaling expansions associated with wavelet constructions, and others. We obtain the rate of convergence of the corresponding sparse grid methods in weighted Wiener norms as well as analogues of the Littlewood–Paley-type characterizations in terms of families of quasi-interpolation operators.

General framework of a two-dimensional complex wavelet for fringe projection profilometry.

Single-shot high-accuracy reconstruction of three-dimensional (3D) objects plays an essential role in many fields. Wavelet-based fringe projection profilometry (WFPP) is a popular single-shot imaging method due to its multiple-resolution analysis and non-contact property. However, the construction of mother wavelets for high-accuracy WFPP is still challenging. In this paper, a general framework of two-dimensional (2D) compact support complex wavelet employing the dual-tree concept combined with the one-dimensional (1D) asymmetric wavelet function is proposed. The constructed wavelet has an asymmetric frequency envelope in the radial direction and concentrated energy in a single quadrant of the frequency domain. A method that the constructed wavelet applied in dual-angle rotation-based 2D WFPP is presented for higher accuracy. The theoretical analysis of the proposed method is presented, and a comparison of frequency characteristics between the constructed wavelet and other popular wavelets is conducted. Computer simulations and experiments have demonstrated that the proposed method provides a better reconstruction of 3D shapes in single-shot WFPP.

Non-stationary wavelet for ECG signal classification

Wavelet analysis has shown to be an interesting tool for representing ECG signals for classification. In this paper, we present a new ECG signal representation based on the notion of non-stationary wavelets. The main difference with the construction of standard wavelets is that the multiresolution spaces are generated by scale-dependent functions in order to achieve increased flexibility and sparseness. In order to customize the non-stationary wavelet to the given ECG classification task, we resort to the fireworks optimization algorithm, thus making the proposed method general and not constrained by the choice of a particular classifier. The proposed method is validated on AAMI classes of the well-known MIT data set. Results compared to standard stationary wavelets show a significant boost in accuracy.

Neural Network Approximation of Refinable Functions

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

Approximation properties of wavelets on p-adic fields

Refinable functions play an important role in the construction and properties of wavelets. Basically, most of the wavelets are generated from refinable functions. In this paper, a study on the approximation properties of refinable functions on [Formula: see text]-adic fields is carried out with necessary theoretical background. Various equivalent forms of approximation order and the connection between the approximation order and the Strang–Fix condition are derived. Finally a characterization for the approximation order of a refinable function is given in terms of order of the sum rules associated with the refinement mask.

Empirical Voronoi wavelets

Recently, the construction of 2D empirical wavelets based on partitioning the Fourier domain with the watershed transform has been proposed. If such approach can build partitions of completely arbitrary shapes, for some applications, it is desirable to keep a certain level of regularity in the geometry of the obtained partitions. In this paper, we propose to build such partition using Voronoi diagrams. This solution allows us to keep a high level of adaptability while guaranteeing a minimum level of geometric regularity in the detected partition.

Samplets: Construction and scattered data compression

We introduce the concept of samplets by transferring the construction of Tausch-White wavelets to scattered data. This way, we obtain a multiresolution analysis tailored to discrete data which directly enables data compression, feature detection and adaptivity. The cost for constructing the samplet basis and for the fast samplet transform, respectively, is O(N), where N is the number of data points. Samplets with vanishing moments can be used to compress kernel matrices, arising, for instance, in kernel based learning and scattered data approximation. The result are sparse matrices with only O(Nlog⁡N) remaining entries. We provide estimates for the compression error and present an algorithm that computes the compressed kernel matrix with computational cost O(Nlog⁡N). The accuracy of the approximation is controlled by the number of vanishing moments. Besides the cost efficient storage of kernel matrices, the sparse representation enables the application of sparse direct solvers for the numerical solution of corresponding linear systems. In addition to a comprehensive introduction to samplets and their properties, we present numerical studies to benchmark the approach. Our results demonstrate that samplets mark a considerable step in the direction of making large scattered data sets accessible for multiresolution analysis.