Haar Basis Research Articles

Analysis of finite Haar wavelet transform and its implementation

In this paper, we give frame‐based finite Haar wavelet transform of ‐scale and ‐level (FHWT( ). The formulation of FHWT is based on the finite nonuniform wavelet transform (FNUWT), and the reconstruction is done using ‐frame reconstruction property. We formulate the FNUWT based on the technique of nonuniform sampling and study the reconstruction property of FNUWT in terms of the ‐frame reconstruction property. FHWT is a particular model of FNUWT based on critical sampling. Using the principles of ‐frame, we prove that FHWT( is a unitary operator. Also, our formulation enables us to define the finite Haar wavelet basis (FHWB) explicitly. In our study, we establish another variant of the fast Haar wavelet transform with varying rate of sampling, and with fast computations. Finally, we formulate the algorithms of finite Haar wavelet transform and its inverse through block implementations of ‐frame. It is observed that the number of computations involved in FHWT( ) is far less than the number of computations required in the fast Fourier transform.

Numerical Approach Based on the Haar Wavelet Collocation Method for Solving a Coupled System with the Caputo–Fabrizio Fractional Derivative

In the present paper, we consider an effective computational method to analyze a coupled dynamical system with Caputo–Fabrizio fractional derivative. The method is based on expanding the approximate solution into a symmetry Haar wavelet basis. The Haar wavelet coefficients are obtained by using the collocation points to solve an algebraic system of equations in mathematical physics. The error analysis of this method is characterized by a good convergence rate. Finally, some numerical examples are presented to prove the accuracy and effectiveness of this method.

Image fusion research based on the Haar-like multi-scale analysis

In view of the serious color and definition distortion in the process of the traditional image fusion, this study proposes a Haar-like multi-scale analysis model, in which Haar wavelet has been modified and used for the medical image fusion to obtain even better results. Firstly, when the improved Haar wavelet basis function is translated, inner product and down-sampled with each band of the original image, the band is decomposed into four sub-images containing one low-frequency subdomain and three high-frequency subdomains. Secondly, the different fusion rules are applied in the low-frequency domain and the high-frequency domains to get the low-frequency sub-image and the high-frequency sub-images in each band. The four new sub-frequency domains are inverse-decomposed to reconstruct each new band. The study configures and synthesizes these new bands to produce a fusion image. Lastly, the two groups of the medical images are used for experimental simulation. The Experimental results are analyzed and compared with those of other fusion methods. It can be found the fusion method proposed in the study obtain the superior effects in the spatial definition and the color depth feature, especially in color criteria such as OP, SpD, CR and SSIM, comparing with the other methods.

TS-TWC: A time series representation learning framework based on Time-Wavelet contrasting

Nowadays, labeling time series datasets requires specialized knowledge and is time-consuming. Moreover, transfer learning faces a gap between the source and target domains. Therefore, we propose a Time Series representation learning framework based on Time-Wavelet Contrasting (TS-TWC) in this paper. It is pre-trained on unlabeled samples and fine-tuned on a small amount of labeled data. The features in the wavelet domain are used as a complement and are called wavelet series. First, the time series and its wavelet series are augmented by an attention-based augmentation structure. Then, a Time-Wavelet contrasting module contrasts the time series with its augmented data, as well as the time series views with their corresponding wavelet series views. In addition, a triple-view contrasting module uses the Daubechies and Haar wavelet bases to increase the number of views for contrastive learning. The aim is to reduce the pre-training batch size and improve the learning effect. In the fine-tuning and inference stages, this module also includes a tri-view fusion structure to assist in learning or extracting discriminative representations. Finally, our model is tested on five pairs of datasets using transfer learning. Experiments show that the proposed framework is effective in learning transferable representations in pre-training and obtaining discriminative representations in fine-tuning. It outperforms the state-of-the-art models on most of the metrics.

Introducing higher-order Haar wavelet method for solving three-dimensional partial differential equations

In this paper, we develop a collocation method for solving three-dimensional partial differential equations using Haar wavelet and Kronecker tensor product. The approach is based on a series of Haar wavelet basis functions to approximate sixth-order mixed derivatives. The proposed method is mathematically fast, less error and straightforward for the numerical solution of many types of three-dimensional Poisson, biharmonic and Helmholtz equations. Some numerical examples verify the accuracy and efficiency of the proposed method. Finally, we conclude that numerical results computed by our proposed method are more accurate than numerical results obtained in the existing methods in the literature. We find that the CPU time consumed by the suggested approach is lesser than the CPU time of existing methods. Thus, the process is fast, efficient and has a low numerical error.

Multi-Scale Analysis of Terahertz Time-Domain Spectroscopy for Inversion of Thermal Growth Oxide Thickness in Thermal Barrier Coatings

To address the inverse problem of thermal growth oxide (TGO) thickness in thermal barrier coatings (TBCs), a novel multi-scale analysis (MSA) method based on terahertz time-domain spectroscopy (THz-TDS) is introduced. The proposed method involves a MSA technique based on four wavelet basis functions (db4, sym3, haar, coif3). Informative feature parameters characterizing the TGO thickness were extracted by performing continuous wavelet transform (CWT) and max-pooling operations on representative wavelet coefficients. Subsequently, multi-linear regression and machine learning regression models were employed to predict and assess the wavelet feature parameters. Experimental results revealed a discernible trend in the wavelet feature parameters obtained through CWT and max-pooling in the MSA, wherein the visual representation of TGO thickness initially increases and then gradually decreases. Significant variations in these feature parameters with changes in both thickness and scale enabled the effective inversion of TGO thickness. Building upon this, multi-linear regression and machine learning regression prediction were performed using multi-scale data based on four wavelet basis functions. Partial-scale data were selected for multi-linear regression, while full-scale data were selected for machine learning regression. Both methods demonstrated high accuracy prediction performance. In particular, the haar wavelet basis function exhibited excellent predictive performance, as evidenced by regression coefficients of 0.9763 and 0.9840, further confirming the validity of MSA. Hence, this study effectively presents a feasible method for the inversion problem of TGO thickness, and the analysis confirms the promising application potential of terahertz time-domain spectroscopy’s multi-scale analysis in the field of TBCs evaluation. These findings provide valuable insights for further reference.

Wavelet Denoising Image Processing Based on MATLAB

This article uses MATLAB to simulate the wavelet denoising algorithm. The threshold denoising was simulated using the function method, triple standard deviation method, fixed threshold algorithm, Haar, sym5, Coif3, and db3 wavelet bases. The simulation results show that using the fixed threshold method, the Haar wavelet has the best denoising effect.

Optimal Control for Parabolic Uncertain System Based on Wavelet Transformation

In this paper, we study a new type of optimal control problem subject to a parabolic uncertain partial differential equation where the expected value criterion is adopted in the objective function. The basic idea of Haar wavelet transformation is to transform the proposed problem into an approximate uncertain optimal control problem with arbitrary accuracy because the dimension of Haar basis tends to infinity. The relative convergence theorem is proved. An application to an optimal control problem with an uncertain heat equation is dealt with to illustrate the efficiency of the proposed method.

Error analysis and numerical solution of generalized Benjamin–Bona–Mahony–Burgers equation using 3-scale Haar wavelets

In this paper, we propose an extended numerical algorithm for the numerical solution of the Benjamin–Bona–Mahony–Burgers equation. This algorithm involves the application of wavelet theory. First, we use the Quasilinearization technique of linearization and apply the 3-scale Haar wavelet approach for truncation error. This algorithm is constructed from two wavelet functions that make it robust and highly accurate. A multi-resolution is used to generate the Haar basis function. We consider three cases of a mathematical problem for the accuracy of the presented algorithm. The obtained results show good agreement with analytical solutions and have better accuracy.

A Brief Survey of the Graph Wavelet Frame

In recent years, the research of wavelet frames on the graph has become a hot topic in harmonic analysis. In this paper, we mainly introduce the relevant knowledge of the wavelet frames on the graph, including relevant concepts, construction methods, and related theory. Meanwhile, because the construction of graph tight framelets is closely related to the classical wavelet framelets on ℝ, we give a new construction of tight frames on ℝ. Based on the pseudosplines of type II, we derive an MRA tight wavelet frame with three generators ψ1, ψ2, and ψ3 using the oblique extension principle (OEP), which generate a tight wavelet frame in L2(ℝ). We analyze that three wavelet functions have the highest possible order of vanishing moments, which matches the order of the approximation order of the framelet system provided by the refinable function. Moreover, we introduce the construction of the Haar basis for a chain and analyze the global orthogonal bases on a graph . Based on the sequence of framelet generators in L2(ℝ) and the Haar basis for a coarse‐grained chain, the decimated tight framelets on graphs can be constructed. Finally, we analyze the detailed construction process of the wavelet frame on a graph.

A novel coded aperture for [formula omitted]-ray imaging based on compressed sensing

High spatial resolution is of great significance for the γ-ray imaging technique, whose performance is mainly limited by the spatial resolution of position sensitive detectors (PSDs). In this work, the compressive coded aperture is introduced in γ-ray imaging as the basic pattern, and enlarged with the “mosaic” method to increase the Fully Coded Field of View (FCFV) of the PSDs. In order to demonstrate the performance of the compressive coded aperture, Monte-Carlo simulations of the encoding process have been performed for the mosaiced compressive coded aperture as well as the Modified Uniformly Redundant Arrays (MURA) coded aperture. The encoded image of the compressive coded aperture is reconstructed using the sparse reconstruction algorithm based on interior point method, with Discrete Cosine Transform (DCT) basis, Haar wavelet basis and Discrete Fourier Transform (DFT) basis respectively, and the encoded image of the MURA coded aperture is reconstructed respectively with the sparse method and the Maximum-Likelihood, Expectation-Maximization (MLEM) algorithm. In comparison with the MURA coded aperture reconstructed with the MLEM method, higher resolution images can be obtained from lower resolution detectors with both compressive coded aperture and MURA coded aperture using the sparse method. However, with the sparse method, the compressive coded aperture leads to higher quality images than the MURA coded aperture does.

Close Encounters of the Binary Kind: Signal Reconstruction Guarantees for Compressive Hadamard Sampling With Haar Wavelet Basis

We investigate the problems of 1-D and 2-D signal recovery from subsampled Hadamard measurements using Haar wavelet as a sparsity inducing prior. These problems are of interest in, e.g., computational imaging applications relying on optical multiplexing or single-pixel imaging. However, the realization of such modalities is often hindered by the coherence between the Hadamard and Haar bases. The variable and multilevel density sampling strategies solve this issue by adjusting the subsampling process to the local and multilevel coherence, respectively, between the two bases; hence enabling successful signal recovery. In this work, we compute an explicit sample-complexity bound for Hadamard-Haar systems as well as uniform and non-uniform recovery guarantees; a seemingly missing result in the related literature. We explore the faithfulness of the numerical simulations to the theoretical results and show in a practically relevant instance, e.g., single-pixel camera, that the target signal can be recovered from a few Hadamard measurements.

Extraction of Walsh Harmonics by Linear Combinations of Dyadic Shifts

We solve two problems of extracting any term from a signal in the form of a finite sum of the Fourier series with respect to the discrete Walsh functions (in the first case) and the Walsh functions (in the second case) by a linear combination of group shifts of the original signal. We propose a vector version of the discrete time- and frequencythinning wavelet Haar bases, which widely used in encoding and decoding algorithms for the discrete Haar transform. Bibliography: 10 titles.

An approach based on Haar wavelet for the approximation of fractional calculus with application to initial and boundary value problems

In this paper, we propose the numerical approximation of fractional initial and boundary value problems using Haar wavelets. In contrast to the Haar wavelet methods available in literature, where the fractional derivative of the function is approximated using the Haar basis, we approximate the function and its classical derivatives using Haar basis functions. Moreover, error bounds in the approximation of fractional integrals and the fractional derivatives are derived, which depend on the index J of the approximation space VJ and the fractional order α. A neural network problem modeled by a system of nonlinear fractional differential equations is also solved using the proposed method. The numerical results show that the proposed numerical approach is efficient.

Vectorization of persistence barcode with applications in pattern classification of porous structures

Persistence barcode is a topological summary for persistent homology to exhibit topological features with different persistence. Persistence rank function (PRF), derived from persistence barcode, organizes persistence Betti numbers in the form of an integer-valued function. To obtain topological patterns of objects such as point clouds represented by finite-dimensional vectors for machine learning classification tasks, the vectorizing representations of barcodes is generated via decomposing PRF on a system of Haar basis. Theoretically, the generated vectorizing representation is proved to have 1-Wasserstein stability. In practice, to reduce training time and achieve better results, a technique of dimensionality reduction through out-of-sample mapping in supervised manifold learning is used to generate a low-dimensional vector. Experiments demonstrate that the representation is effective for capturing the topological patterns of data sets. Moreover, the classification of porous structures has become an essential problem in the fields such as material science in recent decades. The proposed method is successfully applied to distinguish porous structures on a novel data set of porous models.

Error analysis and numerical solution of Burgers–Huxley equation using 3-scale Haar wavelets

In this article, we develop an efficient and accurate numerical scheme based on the Crank–Nicolson finite difference method and Haar wavelet analysis to evaluate the numerical solution of the Burgers–Huxley equation. The present method is extended form of Haar wavelet 2D scaling which shows that it is reliable for solving nonlinear partial differential equations. The numerical results are more accurate than other existing methods available in the literature and very close to the exact solution. The Haar basis function is generated from multi-resolution analysis and used to evaluate fast and accurate approximate solutions on the collocation points. The convergence of the proposed method is demonstrated by its error analysis. We compared numerical solutions with the exact solutions and solutions available in the literature. The proposed method is found to be straight forward, accurate with small computational cost and can be easily implemented in mathematical software MATLAB.

Fast Haar Transforms for Graph Neural Networks

Graph Neural Networks (GNNs) have become a topic of intense research recently due to their powerful capability in high-dimensional classification and regression tasks for graph-structured data. However, as GNNs typically define the graph convolution by the orthonormal basis for the graph Laplacian, they suffer from high computational cost when the graph size is large. This paper introduces a Haar basis, which is a sparse and localized orthonormal system for a coarse-grained chain on the graph. The graph convolution under Haar basis, called Haar convolution, can be defined accordingly for GNNs. The sparsity and locality of the Haar basis allow Fast Haar Transforms (FHTs) on the graph, by which one then achieves a fast evaluation of Haar convolution between graph data and filters. We conduct experiments on GNNs equipped with Haar convolution, which demonstrates state-of-the-art results on graph-based regression and node classification tasks.

Haar Wavelet Method for Solving High-Order Differential Equations with Multi-Point Boundary Conditions

In This paper, the developed Haar wavelet method for solving boundary value problems is described. As known, the orthogonal Haar basis functions are applied widely for solving initial value problems, but In this study, the method for solving systems of ODEs associated with multipoint boundary conditions is generalized in separated or non-separated forms. In this technique, a system of high-order boundary value problems of ordinary differential equations is reduced to a system of algebraic equations. The experimental results confirm the computational efficiency and simplicity of the proposed method. Also, the implementation of the method for solving the systems arising in the real world for phenomena in fluid mechanics and construction engineering approves the applicability of the approach for a variety of problems.

Compression and Recovery of 3D Broad-Leaved Tree Point Clouds Based on Compressed Sensing

The terrestrial laser scanner (TLS) has been widely used in forest inventories. However, with increasing precision of TLS, storing and transmitting tree point clouds become more challenging. In this paper, a novel compressed sensing (CS) scheme for broad-leaved tree point clouds is proposed by analyzing and comparing different sparse bases, observation matrices, and reconstruction algorithms. Our scheme starts by eliminating outliers and simplifying point clouds with statistical filtering and voxel filtering. The scheme then applies Haar sparse basis to thin the coordinate data based on the characteristics of the broad-leaved tree point clouds. An observation procedure down-samples the point clouds with the partial Fourier matrix. The regularized orthogonal matching pursuit algorithm (ROMP) finally reconstructs the original point clouds. The experimental results illustrate that the proposed scheme can preserve morphological attributes of the broad-leaved tree within a range of relative error: 0.0010%–3.3937%, and robustly extend to plot-level within a range of mean square error (MSE): 0.0063–0.2245.

Probabilistic Godunov-type hydrodynamic modelling under multiple uncertainties: robust wavelet-based formulations

Intrusive stochastic Galerkin methods propagate uncertainties in a single model run, eliminating repeated sampling required by conventional Monte Carlo methods. However, an intrusive formulation has yet to be developed for probabilistic hydrodynamic modelling incorporating robust wetting-and-drying and stable friction integration under joint uncertainties in topography, roughness, and inflow. Robustness measures are well-developed in deterministic models, but rely on local, nonlinear operations that can introduce additional stochastic errors that destabilise an intrusive model. This paper formulates an intrusive hydrodynamic model using a multidimensional tensor product of Haar wavelets to capture fine-scale variations in joint probability distributions and extend the validity of robustness measures from the underlying deterministic discretisation. Probabilistic numerical tests are designed to verify intrusive model robustness, and compare accuracy and efficiency against a conventional Monte Carlo approach and two other alternatives: a nonintrusive stochastic collocation formulation sharing the same tensor product wavelet basis, and an intrusive formulation that truncates the basis to gain efficiency under multiple uncertainties. Tests reveal that: (i) A full tensor product basis is required to preserve intrusive model robustness, while the nonintrusive counterpart achieves identically accurate results at a reduced computational cost; and, (ii) the Haar wavelet basis requires at least three levels of refinements per uncertainty dimension to reliably capture complex probability distributions. Accompanying model software and simulation data are openly available online.