Haar Basis Research Articles

Quantum Image Watermarking Based on Wavelet and Geometric Transformation

Like the digital watermark, which has been highlighted in previous studies, the quantum watermark aims to protect the copyright of any image and to validate its ownership using visible or invisible logos embedded in the cover image. In this paper, we propose a method to include an image logo in a cover image based on quantum fields, where a certain amount of texture is encapsulated to encode the logo image before it is included in the cover image. The method also involves transforming wavelets such as Haar base transformation and geometric transformation. These combination methods achieve a high degree of security and robustness for watermarking technology. The digital results obtained from the experiment show that the values of Peak Signal Noise Ratio (PSNR) and Mean Squared Error (MSE) that apply to one of the set cover images were 68.74 and 0.093, respectively.

Two Weight Commutators on Spaces of Homogeneous Type and Applications

In this paper, we establish the two weight commutator theorem of Calderon–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $$A_2$$ weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderon–Zygmund operators: Cauchy integral operator on $${\mathbb {R}}$$ , Cauchy–Szego projection operator on Heisenberg groups, Szego projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

Wavelet-based numerical and semianalytical methods of computational local structural analysis

Numerical or semianalytical solution of problems of structural mechanics of high dimensionality is computationally costly process in many cases. However, structural engineer does not normally face the task of obtaining a solution of the problem with high accuracy at all points of the considered domain occupied by the structure. As a rule, all subdomains that are potentially dangerous in terms of structural strength are well known in advance. Operational and variational formulations of boundary problems of structural mechanics with the use of method of extended domain are presented. After corresponding (finite element or finite difference) discretization and passage to governing equations considering problems are transformed to a multilevel space with the use of multilevel wavelet transform (discrete Haar basis is used). Special algorithms of averaging are presented.

Solving of nonlinear Fredholm integro-differential equation in a complex plane with rationalized Haar wavelet bases

Everyone knows about the complicated solution of the nonlinear Fredholm integro-differential equation in general. Hence, often, authors attempt to obtain the approximate solution. In this paper, a numerical method for the solutions of the nonlinear Fredholm integro-differential equation (NFIDE) of the second kind in the complex plane is presented. In fact, by using the properties of Rationalized Haar (RH) wavelet, we try to give the solution of the problem. So far, as we know, no study has yet been attempted for solving the NFIDE in the complex plane. For this purpose, we introduce the continuous integral operator and real valued function. The Banach fixed point theorem guarantees that, under certain assumptions, the integral operator has a unique solution. Furthermore, we give an upper bound for the error analysis. An algorithm is presented to compute and illustrate the solutions for some numerical examples.

Solving Two-Dimensional Nonlinear Fredholm Integral Equations Using Rationalized Haar Functions in the Complex Plane

As far as we are aware, no research has been published about two-dimensional integral equations in the complex plane by using Haar bases or any other kinds of wavelets. We introduce a numerical method to solve two-dimensional Fredholm integral equations, using Haar wavelet bases. To attain this purpose, first, an operator and then an orthogonal projection should be defined. Regarding the characteristics of Haar wavelet, we solve an integral equation without using common mathematical methods. We prove the convergence and an upper bound that mentioned in the method by employing the Banach fixed point theorem. Moreover, the rate of convergence our method is $$O(n(2q)^n)$$ . We present several examples of different kinds of functions and solve them by this method in this study.

A Framework for Efficient Information Aggregation in Smart Grid

The two-way communication of information between agents in the smart grid, while making way for better monitoring and control, comes at the cost of elevated communication traffic. Compressive sensing is a technique that exploits sparsity of power consumption data (in the Haar basis) and achieves sub-Nyquist compression. Household power consumption data, however, have varying sparseness due to, for example, multistate appliances. Compressing this data with a fixed ratio can lead to nonoptimal results (less compression or large reconstruction error). In this regard, a dynamic compression scheme that estimates a signal's sparsity and decides the amount of compression is desirable. We demonstrate that this approach, when applied with existing estimators of sparsity, has its limitations in overemphasizing one objective compared to the other. We propose a new measure derived from coefficient of variation and demonstrate that it achieves a better tradeoff between reconstruction performance and compression ratio. In addition, we employ a dynamic spatial compression scheme to account for spatial correlation between data of neighboring nodes and present a framework that incorporates dynamic temporal and spatial compression. We present the results on three publicly available datasets at different sampling rates and outline key findings of the study.

A correction to “Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis [Applied Mathematics and Computation 265 (2015) 304–312

In the recent paper, “Rationalized Haar wavelet bases to approximate solution of nonlinear Fredholm integral equations with error analysis [Applied Mathematics and Computation 265 (2015) 304–312]”, The authors have approximated the solutions of nonlinear Fredholm integral equations (NFIEs) of the second type by using the successive approximations method. In any stage, the rationalized Haar wavelets (RHWs) and the corresponding operational matrices were applied to approximate the integral operator. In Theorem 4.2, page 307 of the reference [4], the authors introduced an upper bound for the error and explicitly stated that the rate of convergence of ui to u is O(qi), in which i is the number of iterations, q is the contraction constant, u is the exact solution, and ui is the approximate solution in ith iteration. This statement is not true and we prove carefully that the rate of convergence will be O(iqi).

THEORETIC NUMBER BASIC METHODS OF INFORMATION TECHNOLOGIES FOR DIAGNOSTATION OF TECHNOLOGICAL OBJECTS OF THE OIL & GAS INDUSTRY

The existing methods of formation and coding of technological data at the grass-roots levels of distributed computerized systems for diagnosing technological objects in the oil and gas industry are analyzed. Methods of processing signals are analyzed. The mathematical model of multibase spectral cosine signal transformation, with the possibility of using different theorems of numerical bases, is considered. Correlation analysis of signals can be performed on the basis of different automobile and inter-correlation functions, but the highest speed and the smallest hardware complexity are characterized by functions in the theory of the Haar numerical basis. Where it is evident that in practice the widest use has been found in the form of information converters based on an analog digital converter of various types. The research of analog digital converters has been carried out on hardware complexity as well as on time complexity. The research was carried out on single-channel analogue digital converters with source codes, the theorem of the numerical basis of Haar, Krestinson, Rademacher and unitary basis. The paper deals with the development of digital correlators and transducers where among many structures of analog digital converters the most commonly used with source codes in different theorems of numerical bases and methods of formation of source codes are identified. The structure of the analog digital converter is proposed for the use of comparators with paraphase outputs and the implementation of a logical element that excludes OR on logical elements of I-NE with paraphase inputs and are connected by inverted outputs. Thus, the proposed improvement of the structure allows to increase its speed and reduce the time complexity of transformations by 2.5 -3 times. This reduces hardware complexity by about 30%. The method and structures of the transformation of the input analog signal by the multi-base ADA of Haar-Krestenson using the comparators with paraphase outputs in discrete digital samples for diagnostics of technological objects in the oil and gas industry and further calculation in computer technology are developed

ПРИМЕНЕНИЕ ТЕОРИИ ВЕЙВЛЕТОВ ПРИ СЖАТИИ И ФИЛЬТРАЦИИ ГЕОИНФОРМАЦИИ

The purpose of the article is to develop a detailed and accessible technology for the application of wavelets in the processing of geo-information, the subject of research is wavelet-based filtering and compression of geo-information. The research methodology is based on the modern theory of wavelets in the light of linear algebra. Research methods involve study and generalization, abstraction, formalization, mathematical modeling using computer programs compiled by the authors.After the introduction and formulation of the problem, the basic positions of linear algebra are presented, on which the content of the article is based when constructing orthonormal bases in one- and two-dimensional cases. First, the application of the general theory to the decomposition of the vector of initial data in the Haar and Shannon bases is given. Further, on the basis of the Haar basis, orthonormal bases of wavelet transforms and filtering information are constructed. The procedure for creating wavelet filters by a sequence of convolutions, the use of MSA analysis for constructing an orthonormal basis of the wavelet transform is considered. Implemented the practical possibility of wavelet filtering based on specific programs for modeling geo-information data fields and images, data compression and filtering. The result of the work is the methods of constructing orthonormal bases by various methods of wavelet transform, based on which algorithms and corresponding computer programs for geoinformation compression are compiled using the example of terrain and photographic images. The efficiency of geoinformation compression and noise filtering using wavelets was investigated. A method has been developed for determining the value of a filter depending on the accuracy of the initial geo-information, illustrated by the example of calculating the filter value for compressing information about the heights of the terrain. The same technique is recommended for image filtering.

Tail-greedy bottom-up data decompositions and fast multiple change-point detection

This article proposes a “tail-greedy”, bottom-up transform for one-dimensional data, which results in a nonlinear but conditionally orthonormal, multiscale decomposition of the data with respect to an adaptively chosen unbalanced Haar wavelet basis. The “tail-greediness” of the decomposition algorithm, whereby multiple greedy steps are taken in a single pass through the data, both enables fast computation and makes the algorithm applicable in the problem of consistent estimation of the number and locations of multiple change-points in data. The resulting agglomerative change-point detection method avoids the disadvantages of the classical divisive binary segmentation, and offers very good practical performance. It is implemented in the R package breakfast, available from CRAN.

About local numerical solution of boundary problems of elasticity three-dimensional theory

The distinctive paper is devoted to wavelet-based multilevel method of local numerical solution of boundary problems of elasticity three-dimensional theory. As it is known, effective qualitative multilevel analysis of local and structure global stress-strain states is normally required in various technical applications. Operational and variational formulations of the problem (particularly with the use of wavelet (Haar) basis) are presented. Computer-oriented algorithms of fast direct and inverse discrete Haar transforms are described. Due to special algorithms of averaging within corresponding multigrain approach, problem reduction is provided.

Wavelet-based two-grid numerical method of structural analysis with the use of discrete Haar basis

The distinctive paper is devoted to so-called two-grid method of structural analysis based on discrete Haarbasis (three-dimensional problems are under consideration). Approximations of the mesh functions in discrete Haar bases of zero andfirst levels are described (the mesh function is represented as the sum where one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Special projectors are constructed for the spaces of the original grid vector functions to the space of their approximationon the first-level grid and its complement (the detailing component) to the initial state. Basic scheme of the two-grid method is presented. This method allows boundary problems solution of structural mechanics with the matrix operators’ use of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation (defined on a given interval) is a system of linear algebraic equations which is constructed with the use of finite element method or finite difference method. Block Gauss method can be used for direct solution.

Real-time, image-based slice-to-volume registration for ultrasound-guided spinal intervention

Real-time fusion of magnetic resonance (MR) and ultrasound (US) images could facilitate safe and accurate needle placement in spinal interventions. We develop an entirely image-based registration method (independent of or complementary to surgical trackers) that includes an efficient US probe pose initialization algorithm. The registration enables the simultaneous display of 2D ultrasound image slices relative to 3D pre-procedure MR images for navigation.A dictionary-based 3D–2D pose initialization algorithm was developed in which likely probe positions are predefined in a dictionary with feature encoding by Haar wavelet filters. Feature vectors representing the 2D US image are computed by scaling and translating multiple Haar basis filters to capture scale, location, and relative intensity patterns of distinct anatomical features. Following pose initialization, fast 3D–2D registration was performed by optimizing normalized cross-correlation between intra- and pre-procedure images using Powell’s method. Experiments were performed using a lumbar puncture phantom and a fresh cadaver specimen presenting realistic image quality in spinal US imaging. Accuracy was quantified by comparing registration transforms to ground truth motion imparted by a computer-controlled motion system and calculating target registration error (TRE) in anatomical landmarks.Initialization using a 315-length feature vector yielded median translation accuracy of 2.7 mm (3.4 mm interquartile range, IQR) in the phantom and 2.1 mm (2.5 mm IQR) in the cadaver. By comparison, storing the entire image set in the dictionary and optimizing correlation yielded a comparable median accuracy of 2.1 mm (2.8 mm IQR) in the phantom and 2.9 mm (3.5 mm IQR) in the cadaver. However, the dictionary-based method reduced memory requirements by 47× compared to storing the entire image set. The overall 3D error after registration measured using 3D landmarks was 3.2 mm (1.8 mm IQR) mm in the phantom and 3.0 mm (2.3 mm IQR) mm in the cadaver. The system was implemented in a 3D Slicer interface to facilitate translation to clinical studies.Haar feature based initialization provided accuracy and robustness at a level that was sufficient for real-time registration using an entirely image-based method for ultrasound navigation. Such an approach could improve the accuracy and safety of spinal interventions in broad utilization, since it is entirely software-based and can operate free from the cost and workflow requirements of surgical trackers.

ABOUT WAVELET-BASED MULTIGRID NUMERICAL METHOD OF STRUCTURAL ANALYSIS WITH THE USE OF DISCRETE HAAR BASIS

he distinctive paper is devoted to so-called multigrid (particularly two-grid) method of structural analysis based on discrete Haar basis (one-dimensional, two-dimensional and three-dimensional problems are under consideration). Approximations of the mesh functions in discrete Haar bases of zero and first levels are described (the mesh function is represented as the sum in which one term is its approximation of the first level, and the second term is so-called complement (up to the initial state) on the grid of the first level). Special projectors are constructed for the spaces of vector functions of the original grid to the space of their approximation on the first-level grid and its complement (the refinement component) to the initial state. Basic scheme of the two-grid method is presented. This method allows solution of boundary problems of structural mechanics with the use of matrix operators of significantly smaller dimension. It should be noted that discrete analogue of the initial operator equation is a system of linear algebraic equations which is constructed with the use of finite element method or finite difference method. Block Gauss method can be used for direct solution.

Uncertainty product for Vilenkin groups

We study a localization of functions defined on Vilenkin groups. To measure the localization, we introduce two uncertainty products [Formula: see text] and [Formula: see text] that are similar to the Heisenberg uncertainty product. [Formula: see text] and [Formula: see text] differ from each other by the metric used for the Vilenkin group [Formula: see text]. We discuss analogs of a quantitative uncertainty principle. Representations for [Formula: see text] and [Formula: see text] in terms of Walsh and Haar basis are given.

Generalized Haar wavelet operational matrix method for solving hyperbolic heat conduction in thin surface layers

It is remarkably known that one of the difficulties encountered in a numerical method for hyperbolic heat conduction equation is the numerical oscillation within the vicinity of jump discontinuities at the wave front. In this paper, a new method is proposed for solving non-Fourier heat conduction problem. It is a combination of finite difference and pseudospectral methods in which the time discretization is performed prior to spatial discretization. In this sense, a partial differential equation is reduced to an ordinary differential equation and solved implicitly with Haar wavelet basis. For the pseudospectral method, Haar wavelet expansion has been using considering its advantage of the absence of the Gibbs phenomenon at the jump continuities. We also derived generalized Haar operational matrix that extend usual domain (0,1] to (0,X]. The proposed method has been applied to one physical problem, namely thin surface layers. It is found that the proposed numerical results could suppress and eliminate the numerical oscillation in the vicinity jump and in good agreement with the analytic solution. In addition, our method is stable, convergent and easily coded. Numerical results demonstrate good performance of the method in term of accuracy and competitiveness compare to other numerical methods.

A Haar wavelet method for angularly discretising the Boltzmann transport equation

A novel, hierarchical Haar wavelet basis is introduced and used to discretise the angular dimension of the Boltzmann transport equation. This is used in conjunction with a finite element subgrid scale method. This combination is then validated using two steady-state radiation transport problems, namely a 2D dogleg-duct shielding problem and the 2D C5MOX OECD/NEA benchmark. It is shown that the scheme has many similarities to a traditional equal weighted discrete ordinates (Sn) angular discretisation, but the strong motivation for our hierarchical Haar wavelet method is the potential for adapting in angle in a simple fashion through elimination of redundant wavelets. Initial investigations of this adaptive approach are presented for a shielding and criticality eigenvalue example. It is shown that a 60% reduction in the number of angles needed on most spatial nodes - and rising up to 90% on nodes located in high streaming areas - can be attained without adversely affecting the accuracy of the solution.

A new multifractional process with random exponent

A first type of Multifractional Process with Random Exponent (MPRE) was constructed several years ago in (Ayache, Taqqu, 2005) by replacing in a wavelet series representation of Fractional Brownian Motion (FBM) the Hurst parameter by a random variable depending on the time variable. In the present article, we propose another approach for constructing another type of MPRE. It consists in substituting to the Hurst parameter, in a stochastic integral representation of the high-frequency part of FBM, a random variable depending on the integration variable. The MPRE obtained in this way offers, among other things, the advantages to have a representation through classical Ito integral and to be less difficult to simulate than the first type of MPRE, previously introduced in (Ayache, Taqqu, 2005). Yet, the study of Holder regularity of this new MPRE is a significantly more challenging problem than in the case of the previous one. Actually, it requires to develop a new methodology relying on an extensive use of the Haar basis.

Method for Structuring the Fourier Discrete Cosine Transform in the Modular Arithmetic of the Haar–Krestenson Number-Theoretic Basis

A theory and a solution are presented to the applied problem of structuring the Fourier discrete spectral cosine transform (FDSCT) in the modular arithmetic of the Haar–Krestenson number-theoretic basis. A high-performance algorithm for FDSCT was developed by adapting the orthogonal functions of the Fourier, Rademacher, Krestenson, and Haar bases to the asymptotic autocovariance of the signals being investigated. A method for structuring the FDSCT algorithm in the modular arithmetic of the residue number system of the Haar–Krestenson number-theoretic basis was implemented. The structure of an implementation of a special FDSCT processor and its microelectronic basic components are given.

Wavelet Transform Subspace-Based Optimization Method for Inverse Scattering

Inspired by the new fast Fourier transform subspace-based optimization method (NFFT-SOM), wavelet transform subspace-based optimization method (WT-SOM) that represents the minor part of induced current by wavelet bases is proposed in this paper to solve the inverse scattering problem. This paper provides a guideline for choosing the appropriate bases for the minor part of induced current in the inverse problem, and it is shown that Daubechies 20 wavelet bases and Fourier bases are better choices than Daubechies 4 wavelet bases and Haar wavelet bases. Compared to NFFT-SOM, it is shown that WT-SOM can be used to improve the resolution of a specific region of a scatterer. By using only a part of the detail coefficients, the convergence rate of the algorithm can be greatly accelerated compared to using all the detail coefficients. For a piecewise constant scatterer, additive total variation (TV) regularization is used together with WT-SOM. We prove that TV-regularized WT-SOM has exactly the same performance when the contrast is represented by no matter natural pixels bases, Fourier bases, or wavelet bases.