Two-dimensional Haar Wavelet Research Articles

Convolutional neural networks and seismogram fingerprints as a tool for recognizing the nature of seismic events

In the presented article, two facts are convincingly demonstrated based on the results of numerous experiments. Firstly, fingerprints are informative enough representations of signal seismograms, despite their compression, to carry information about the nature of a seismic event. Secondly, the study showed that it is practically possible to design and train an artificial neural network capable of classifying events by origin based on their fingerprints with high accuracy. Fingerprints are a tenthousandfold compressed representation of the original seismogram obtained using the onedimensional wavelet transform and the twodimensional Haar wavelet transform. They carry information about all significant frequencytime phenomena contained in the primary seismogram. Convolutional neural networks were chosen as a class of neural network classifier based on the conducted review of publications on this topic. They have proven themselves to be excellent in recognizing objects and persons in raster images. And the fingerprints used in this study are binary images measuring 64×64 pixels. The convolutional neural network prepared to work with them has one of the simplest architectures for this type of network and a very small number of adjustable parameters. By means of it, the classification accuracy of 95% was easily achieved. To prove that this result is not accidental, a strategy for modeling the architecture of convolutional neural networks using specially developed software, the Trova system, is demonstrated. This software allows the researcher to easily and conveniently operate seismograms, obtain binary fingerprints from them, correctly augment data, create, train and test neural networks. An important feature of the Trova system is the advanced functionality of the graphical representation of multidimensional modeling results.

NSNet: An N-Shaped Convolutional Neural Network with Multi-Scale Information for Image Denoising

Deep learning models with convolutional operators have received widespread attention for their good image denoising performance. However, since the convolutional operation prefers to extract local features, the extracted features may lose some global information, such as texture, structure, and color characteristics, when the object in the image is large. To address this issue, this paper proposes an N-shaped convolutional neural network with the ability to extract multi-scale features to capture more useful information and alleviate the problem of global information loss. The proposed network has two main parts: a multi-scale input layer and a multi-scale feature extraction layer. The former uses a two-dimensional Haar wavelet to create an image pyramid, which contains the corrupted image’s high- and low-frequency components at different scales. The latter uses a U-shaped convolutional network to extract features at different scales from this image pyramid. The method sets the mean-squared error as the loss function and uses the residual learning strategy to learn the image noise directly. Compared with some existing image denoising methods, the proposed method shows good performance in gray and color image denoising, especially in textures and contours.

Two-dimensional Haar Wavelet Method for Numerical Solution of Delay Partial Differential Equations

In this paper, a two-dimensional Haar wavelet collocation method is applied to obtain the numerical solution of delay and neutral delay partial differential equations. Both linear and nonlinear problems can be solved using this method. Some benchmark test problems are given to verify the efficiency and accuracy of the aforesaid method. The results are compared with the exact solution and performance of the two-dimensional Haar collocation technique is measured by calculating the maximum absolute and root mean square errors for different numbers of grid points. The results are also compared with finite difference technique and one-dimensional Haar wavelet technique. The numerical results show that the two-dimensional Haar method is simply applicable, accurate and efficient.

A computational approach for solving fractional Volterra integral equations based on two-dimensional Haar wavelet method

In this paper, an operational matrix (OM) method based on two-dimensional Haar wavelets (2D-HWs) is proposed for solving generalized 2D fractional Volterra integral equations (2D-FVIEs). The proposed method converts these equations into an algebraic equations system. Presenting theorems, we prove the existence and uniqueness of the mentioned equations and estimate the error bound for function approximation. Our method validity and applicability are demonstrated through some examples.

Wavelet based algorithm for numerical study of (1+2)-dimensional time fractional diffusion problems

An effective and robust scheme is developed for solutions of two-dimensional time fractional heat flow problems. The proposed scheme is based on two-dimensional Haar wavelets coupled with finite differences. The time fractional derivative is approximated by an L_{1}-formula while spatial part is approximated by two-dimensional Haar wavelets. The proposed methodology first converts the problem to a discrete form and then with collocation approach to a system of linear equations which is easily solvable. To check the efficiency of the scheme, two error norms, E_{infty } an E_{mathrm{rms}}, have been computed. The stability of the scheme has been discussed which is an important part of the manuscript. It is also observed that the spectral radius of the amplification matrix satisfies a stability condition. From computation it is clear that computed results are comparable with the exact solution.

Digital Image Processing with Two-Dimensional Haar Wavelets

Digital Image Processing with Two-Dimensional Haar Wavelets

Applying Haar Wavelets in Tasks of Digital Processing of Two-Dimensional Signals

In this article, a system application of local basis functions discussed, defined on compact media. formulas for estimating the accuracy of calculating the spectral energy by the method of two-dimensional Haar wavelets and main concepts of Haar fast transformation, spectral energy of Haar coefficients will be discussed in this article. effectiveness will be shown in order to solve problems of sampling of compact signals. Signals cannot only be functions of time, defined at finite intervals, but also functions of arguments of a different physical nature, for example, distances along surfaces.

Theoretical Error Analysis of Solution for Two-Dimensional Stochastic Volterra Integral Equations by Haar Wavelet

The finding an efficient way to the approximate solutions of the stochastic integral equations is an essential requirement. In this paper we discuss the convergence analysis of the two-dimensional Haar wavelet functions (2D-HWFs) method for solve 2D linear stochastic Volterra integral equation. The illustrative examples are included to demonstrate the validity and applicability of this numerical method.

Separating clods and stones from potato tubers based on color and shape

The separation of clods and stones from the harvested potato tuber has always been a prevalent problem in the world. However, the precision of sorting was restricted by the potato tubers covered with mud on the surface. This paper studied methods of separating clods and stones from potato tubers based on shape and color. An image acquisition system consisted of a light source, a camera, a computer was built for this experiment. The color features were extracted from the components of RGB and HSV images by the two-dimensional Haar Wavelet Transform and put into SVM (support vector machine) to classify the object after principal component analysis. The shape features which contained the original contour and corrected contour described by the mathematical statistical methods was extracted and used for separation by SVM. The experimental result showed that it was effective to separate clods and stones from potato tubers based on the extracted color and shape features, respectively. The combination of color and shape features could increase the accuracy rate of classification, especially for potato tubers and clods. The overall accuracy rate was 97.8% in 2016 and 98.1% in 2017. It was evident that the color features dominate in the classification model. Shape features based on the correcting image showed positive effect in classification. It turned out that the combination of shape and color features can obviously improve classification performance.

Measurement of entanglement entropy in the two-dimensional Potts model using wavelet analysis.

A method is introduced to measure the entanglement entropy using a wavelet analysis. Using this method, the two-dimensional Haar wavelet transform of a configuration of Fortuin-Kasteleyn (FK) clusters is performed. The configuration represents a direct snapshot of spin-spin correlations since spin degrees of freedom are traced out in FK representation. A snapshot of FK clusters loses image information at each coarse-graining process by the wavelet transform. It is shown that the loss of image information measures the entanglement entropy in the Potts model.

Application of RNAMlet to surface defect identification of steels

As three main production lines of steels, continuous casting slabs, hot rolled steel plates and cold rolled steel strips have different surface appearances and are produced at different speeds of their production lines. Therefore, the algorithms for the surface defect identifications of the three steel products have different requirements for real-time and anti-interference. The existing algorithms cannot be adaptively applied to surface defect identification of the three steel products. A new method of adaptive multi-scale geometric analysis named RNAMlet was proposed. The idea of RNAMlet came from the non-symmetry anti-packing pattern representation model (NAM). The image is decomposed into a set of rectangular blocks asymmetrically according to gray value changes of image pixels. Then two-dimensional Haar wavelet transform is applied to all blocks. If the image background is complex, the number of blocks is large, and more details of the image are utilized. If the image background is simple, the number of blocks is small, and less computation time is needed. RNAMlet was tested with image samples of the three steel products, and compared with three classical methods of multi-scale geometric analysis, including Contourlet, Shearlet and Tetrolet. For the image samples with complicated backgrounds, such as continuous casting slabs and hot rolled steel plates, the defect identification rate obtained by RNAMlet was 1% higher than other three methods. For the image samples with simple backgrounds, such as cold rolled steel strips, the computation time of RNAMlet was one-tenth of the other three MGA methods, while the defect identification rates obtained by RNAMlet were higher than the other three methods.

Intellectual Property-Based Lossless Image Compression for Camera Systems [Hardware Matters

This article presents a novel framework for intellectual property (IP)- based lossless image compression used for camera systems. The proposed approach presents two subframeworks: a) two-dimensional (2-D) Haar wavelet transformation (HWT)- based forward pixel calculator IP for image compression and b) 2-D HWTbased inverse pixel calculator IP for image decompression. Each framework is capable of fully compressing and decompressing images through onestage computation during forward and inverse transformations. The proposed approach has been tested on images of critical data sets of medical applications [e.g., computed tomography (CT) images and satellite applications, including NASA images]. Results of compressed and decompressed images along with compression efficiency through the proposed framework for IP-based lossless image compression are also reported in this article.

Fast CU partition strategy for HEVC based on Haar wavelet

As the latest video coding standard, high-efficiency video coding (HEVC) achieves better performance and supports higher resolution compared with the predecessor standard, H.264/advanced video coding (AVC). Intra-coding is an important feature in HEVC standard, which reduces the spatial redundancy significantly, due to the flexible coding structure, and high density of angular prediction modes. However, the improvement on coding efficiency is obtained at the expense of the extraordinary computation complexity. This study presents a novel coding unit (CU) partitioning technique for HEVC. By using a fast texture complexity detection method, which is based on two-dimensional Haar wavelet transform, texture complexity for each CU can be extracted. According to the Haar wavelet coefficients obtained, an early CU splitting termination is proposed to decide whether a CU should be decomposed into four lower dimensions CUs or not. Experimental results demonstrate that the fast CU partition strategy achieves better trade-off between rate-distortion performance and complexity reduction than the previous algorithms. Compared with the reference software HM16.7, the proposed algorithm can lessen the encoding time up to 46.22% on average, with a negligible bit rate increase of 0.45%, and quality losses lower than 0.04 dB, respectively.

The Two-Dimensional Wavelet Transform De-noising and Combining with Side Scan Sonar Image

This paper puts forward an image de-noising method based on 2D wavelet transform with the application of the method in seabed identification data collection system. Two-dimensional haar wavelets in image processing presents a unified framework for wavelet image compression and combining with side scan sonar image. Seabed identification target have 7 target detection in side scan sonar imagery result. The vibration signals were analyzed to perform fault diagnosis. The obtained signal was time-domain signal. The experiment result shows that the application of 2D wavelet transform image de-noising algorithm can achieve good subjective and objective image quality and help to collect high quality data and analyze the images for the data center with optimum effects, the features from time-domain signal were extracted. 3 vectors were formed which are v1, v2, v3. In Haar wavelet retained energy is 93.8 %, so from the results, it has been concluded that Haar wavelet transform shows the best results in terms of Energy from De-noised Image processing with side scan sonar imagery.

2D wavelet transform data compression with error level guarantee for Z-map models

AbstractMost of workpiece shapes in NC milling simulations are in Z-map representations that require a very large amount of data to precisely hold a high resolution model. An irreversible compression algorithm for Z-map models using a two-dimensional Haar wavelet transform is proposed to resolve this tight memory situation for an ordinary PC. A shape model is first transformed by using Haar wavelet to build a wavelet synopsis tree while the maximum errors caused by virtually truncating high-frequency components are simultaneously calculated. The total amount of the shape data can be reduced by truncating particular sections of the wavelet components that satisfy the error threshold given by the user. Our algorithm guarantees that any error due to its irreversible compression processes is smaller than the specified level measured against the original model. A series of experiments were conducted using an Apple iMac with a 3.2 GHz CPU and 8 GB of memory. The experiments were performed with 16 sample shape models on 512×512 to 8192×8192 grids to evaluate the compression efficiency of the proposed method. Experimental results confirmed that our compression algorithm requires approximately 20–30 ms for 512×512 models and 7 s for 8192×8192 models under a maximum error level of 10× 10−6 m (a typical criteria for NC milling simulations). The compressed binaries outputted by the proposed method are generally 25–35% smaller than the baseline results by gzip, one of common reversible compression libraries, while these two methods require almost the same level of computational costs.Highlights Discarding diagonal components in WT significantly reduces data amount. The proposed method outperforms a reversible method by 25–35% in size reduction. Most of computational time is consumed by the reversible compression step. The proposed method compresses 5122 models in 20 ms, 81922 models in 7 s.

A Galerkin method with two-dimensional Haar basis functions for the computation of the Karhunen–Loève expansion

We study the numerical approximation of a homogeneous Fredholm integral equation of second kind associated with the Karhunen–Loeve expansion of Gaussian random fields. We develop, validate, and discuss an algorithm based on the Galerkin method with two-dimensional Haar wavelets as basis functions. The shape functions are constructed from the orthogonal decomposition of tensor-product spaces of one-dimensional Haar functions, and a recursive algorithm is employed to compute the matrix of the discrete eigenvalue system without the explicit calculation of integrals, allowing the implementation of a fast and efficient algorithm that provides considerable reduction in CPU time, when compared with classical Galerkin methods. Numerical experiments confirm the convergence rate of the method and assess the approximation error and the sparsity of the eigenvalue system when the wavelet expansion is truncated. We illustrate the numerical solution of a diffusion problem with random input data with the present method. In this problem, accuracy was retained after dropping the coefficients below a threshold value that was numerically determined. A similar method with scaling functions rather than wavelet functions does not need a discrete wavelet transform and leads to eigenvalue systems with better conditioning but lower sparsity.

A two-dimensional Haar wavelets method for solving systems of PDEs

In this paper, we modify the idea of Haar wavelets method to obtain semi-analytical solutions for the systems of three-dimensional nonlinear partial differential equations. Theoretical considerations are discussed. To demonstrate the efficiency of the method, a test problem is presented. The approximate solutions of the system of three-dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature. The numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions.

A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

A convergence analysis of the numerical solution of boundary-value problems by using two-dimensional Haar wavelets

A Secure Compression Scheme for Real-time Applications Using 2D-WT and Cellular Automata

In this work is presented a numerical implementation of a system that combines a compression scheme with an improved encryption procedure, which is applied to digital images. For the compression stage is considered the two-dimensional Haar wavelet transform, where an energy criterion is contemplated. On the other hand, the encryption scheme regarded is based on the synchronization of the cellular automaton rule 90, this system presents a good performance to encrypt images and it is resistant to cryptanalysis attacks such as the Chosen/Known-plaintext attack. The numerical conjunction of these procedures could be an ap- pealing option for real-time applications such as video communications, video-surveillance among others.

A two-dimensional Haar wavelet approach for the numerical simulations of time and space fractional Fokker–Planck equations in modelling of anomalous diffusion systems

In this paper, the numerical solution for the fractional order Fokker–Planck equation has been presented using two dimensional Haar wavelet collocation method. Two dimensional Haar wavelet method is applied to compute the numerical solution of nonlinear time- and space-fractional Fokker–Planck equation. The approximate solutions of the nonlinear time- and space-fractional Fokker–Planck equation are compared with the exact solutions as well as solutions available in open literature. The present scheme is very simple, effective and convenient for obtaining numerical solution of the time and space-fractional Fokker–Planck equation.