Haar Wavelet 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.

ОБОСНОВАНИЕ АНАЛИТИЧЕСКОГО АППАРАТА ДЛЯ АВТОМАТИЗАЦИИ ПРОЦЕДУР ОБНАРУЖЕНИЯ ЛЕСНЫХ ПОЖАРОВ

The results of the development of a scientific and methodological apparatus for the formation of feature vectors are presented in the interests of automating procedures for detecting forest fires based on the results of processing video image frames. Analytical algorithms for generating feature vectors and solving the problem of detecting their differences in automatic mode are presented. The possibility of synthesizing feature vectors based on image decomposition in the Haar wavelet basis is considered. The optimal Haar wavelet parameters have been determined to ensure maximum contrast of feature vectors. The results of the experiment are presented.

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.

Wavelet transform-based characterization of printing ink penetration depth and image phase dissimilarity

Abstract In this paper, continuous wavelet transform and discrete wavelet transform are used to detect transient anomalies entrained in normal information and to demonstrate their components. Multi-scale analysis of wavelet transform, Haar wavelet basis and multi-scale edge detection algorithms are utilized to determine the modal extreme points and identify the edge points for faster and more accurate extraction of edge features of the image. In order to further validate the applicability and feasibility of wavelet transform for printing images and to determine the quality inspection criteria based on ink penetration depth and image phase anisotropy, MATLAB software is utilized to perform simulation tests. The results show that the wavelet transform can remove the noise generated by uneven illumination and printing background during the printing process and can detect the edges of the printing image with an error accuracy of ±0.063mm and meet the error correction accuracy of <0.4mm as required by the printing standard. The experiments verify the feasibility of the wavelet transform, which can characterize the depth of penetration of the printing ink and the image anisotropy and provides a theoretical basis for improving the quality of printing. The experiment confirms that wavelet transform can be used to measure printing ink penetration depth and image anisotropy, giving a theoretical basis for improving printing quality.

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.

Neural Free‐Viewpoint Relighting for Glossy Indirect Illumination

AbstractPrecomputed Radiance Transfer (PRT) remains an attractive solution for real‐time rendering of complex light transport effects such as glossy global illumination. After precomputation, we can relight the scene with new environment maps while changing viewpoint in real‐time. However, practical PRT methods are usually limited to low‐frequency spherical harmonic lighting. All‐frequency techniques using wavelets are promising but have so far had little practical impact. The curse of dimensionality and much higher data requirements have typically limited them to relighting with fixed view or only direct lighting with triple product integrals. In this paper, we demonstrate a hybrid neural‐wavelet PRT solution to high‐frequency indirect illumination, including glossy reflection, for relighting with changing view. Specifically, we seek to represent the light transport function in the Haar wavelet basis. For global illumination, we learn the wavelet transport using a small multi‐layer perceptron (MLP) applied to a feature field as a function of spatial location and wavelet index, with reflected direction and material parameters being other MLP inputs. We optimize/learn the feature field (compactly represented by a tensor decomposition) and MLP parameters from multiple images of the scene under different lighting and viewing conditions. We demonstrate real‐time (512 x 512 at 24 FPS, 800 x 600 at 13 FPS) precomputed rendering of challenging scenes involving view‐dependent reflections and even caustics.

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.

Spherical-Orthogonal-Symmetric Haar Wavelet for Driver's Visual Detection

This work proposes a spherical-orthogonal-symmetric Haar wavelet to decompose and reconstruct spherical iris signals to obtain stronger geometric features of iris surface. It compares its feature extraction abilities of spherical harmonics, semi-orthogonal and nearly orthogonal spherical Haar wavelet. The developed spherical-orthogonal-symmetric Haar wavelet with a convolutional neural network is also proposed for drivers’ iris recognition. It can effectively capture the local fine features of iris spherical surface, and has stronger ability of iris recognition than semi-orthogonal or nearly orthogonal spherical Haar wavelet bases.

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.

Application of wavelet analysis for processing tomograms of narrow cracks

The previously proposed method of increasing the visibility of cracks on tomograms obtained from X-ray sensing of sections of metal elements of building structures has been developed. A multiscale decomposition of tomogram lines is performed using the Haar wavelet basis. The detailed decomposition coefficients corresponding to the level of the assumed crack width are multiplied by a constant, the value of which is determined based on the noise level and the difference between the amplitude of the crack image and the surrounding background. In contrast to the previous work, where these parameters were assumed to be set, here we suggest the methods for its evaluating from the measurements. In particular, a simple formula is obtained that relates the variance of uncorrelated noise in the projection data to that of noise in the tomogram. The method is implemented as a computational algorithm based on which a computer program is developed. Numerical modeling was performed. The errors of the first and second kind were used toevaluate the effectiveness of the method determining whether a pixel belongs to a crack image. Binary classification was used for their calculation. Dependences of errors on the number of projections and the noise level on them were obtained. The results of the simulations showed that the use of the proposed method can reduce errors by 1.7-2.5 times.

The Nyquist sampling rate for spiraling curves

We consider the problem of reconstructing a compactly supported function from samples of its Fourier transform taken along a spiral. We determine the Nyquist sampling rate in terms of the density of the spiral and show that, below this rate, spirals suffer from an approximate form of aliasing. This sets a limit to the amount of undersampling that compressible signals admit when sampled along spirals. More precisely, we derive a lower bound on the condition number for the reconstruction of functions of bounded variation, and for functions that are sparse in the Haar wavelet basis.

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.

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 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).

Haar Wavelet Technique for Solving Fractional Differential Equations with An Application

In this article, the approxmated solutions of ordinary differential equations of fractional order using Haar wavelet and B-spline bases are introduced. The algorithm of collection method is updated using two basis. Several initial value problems has been solved to show the applicability and efficacy of the Haar wavelet and B-spline basis. An application of Lane-Eman equation has been introduced and studied. The approximated results have clearly shown the advantage and the efficiency of the modified method in terms of accuracy and computational time.