Multiscale Coefficient Research Articles

Physics-informed neural networks for learning the homogenized coefficients of multiscale elliptic equations

Multiscale elliptic equations with scale separation are often approximated by the corresponding homogenized equations with slowly varying homogenized coefficients (the G-limit). The traditional homogenization techniques typically rely on the periodicity of the multiscale coefficients, thus finding the G-limits often requires sophisticated techniques in more general settings even when multiscale coefficient is known, if possible. Alternatively, we propose a simple approach to estimate the G-limits from (noise-free or noisy) multiscale solution data, either from the existing forward multiscale solvers or sensor measurements. By casting this problem into an inverse problem, our approach adopts physics-informed neural networks (PINNs) algorithm to estimate the G-limits from the multiscale solution data by leveraging a priori knowledge of the underlying homogenized equations. Unlike the existing approaches, our approach does not rely on the periodicity assumption or the known multiscale coefficient during the learning stage, allowing us to estimate homogenized coefficients in more general settings beyond the periodic setting. We demonstrate that the proposed approach can deliver reasonable and accurate approximations to the G-limits as well as homogenized solutions through several benchmark problems.

Infrared and visible image fusion based on QNSCT and Guided Filter

Image fusion is the process of fusing multiple images of the same scene to obtain a more informative image for human eye perception. In this paper, a new fusion framework based on Quaternion Non-Subsampled Contourlet Transform (QNSCT) and Guided Filter detail enhancement is designed to address the problems of inconspicuous infrared targets and poor background texture in Infrared and visible image fusion. The proposed method uses the quaternion wavelet transform for the first time instead of the traditional Non-Subsampled Pyramid Filter Bank structure in the Non-Subsampled Contourlet Transform (NSCT). The flexible multi-resolution of quaternion wavelet and the multi-directionality of NSCT are fully utilized to refine the multi-scale decomposition scheme. On the other hand, the coefficient matrix obtained from the proposed QNSCT algorithm is fused using a weight refinement algorithm based on the guided filter. The fusion scheme is divided into four steps. First, the Infrared and visible images are decomposed into multi-directional and multiscale coefficient matrices using QNSCT. The experimental results show that the proposed algorithm not only extracts important visual information from the source image, but also preserves the texture information in the scene better. Meanwhile, the scheme outperforms state-of-the-art methods in both subjective and objective evaluations.

Dynamic multiscale spatiotemporal models for multivariate Gaussian data

We propose a novel class of multiscale spatiotemporal models for multivariate Gaussian data. First, we decompose the multivariate data and the underlying latent process with a novel multivariate multiscale decomposition. This decomposition results in multiscale coefficient matrices with elements that are multiscale approximations of spatial directional derivatives. We then assume that the associated latent multiscale coefficient matrices evolve through time with matrix-variate state-space equations. We allow for different speeds of change through time for each latent multiscale coefficient matrix, which induces distinct spatiotemporal dynamics for the mean process in different regions at multiple spatial scales of resolution. This flexible model framework allows for both stationary and nonstationary latent multiscale spatiotemporal processes. Further, we develop a singular matrix-variate forward filter backward sampler for efficient posterior exploration. Importantly for practical purposes, our proposed multiscale spatiotemporal algorithm scales linearly with dataset size and is fully parallelizable. To illustrate the usefulness and flexibility of our dynamic multivariate multiscale framework, we present an application to the spatiotemporal NCEP/NCAR Reanalysis-I dataset on stratospheric temperatures over North America from 1951 to 2016. Our analysis indicates substantial long-term trends in stratospheric temperatures.

Bayesian spatial and spatiotemporal models based on multiscale factorizations

AbstractWe review the literature on spatial and spatiotemporal models based on spatial multiscale factorizations. Specifically, we review models based on wavelets and Kolaczyk–Huang factorizations for Gaussian and Poisson data. These multiscale models decompose spatial and spatiotemporal datasets into many small components, called multiscale coefficients, at multiple levels of spatial resolution. Then analysis proceeds independently for each multiscale coefficient. After that, aggregation equations are used to coherently combine the analyses from the multiple multiscale coefficients to obtain a statistical analysis at the original resolution level. The computational cost of such analysis grows linearly with sample size. Furthermore, computations for these models are scalable, parallelizable, and fast. Therefore, these multiscale models are tremendously useful for the analysis of massive spatial and spatiotemporal datasets.This article is categorized under: Statistical Learning and Exploratory Methods of the Data Sciences > Modeling Methods Statistical Models > Bayesian Models Data: Types and Structure > Image and Spatial Data

Batch similarity comparison of line traces based on multi-strategy machine learning

Conventional methods can scarcely achieve the rapid comparison of line traces. In this study, trace detection signals based on a single-point laser were collected and smoothed. The multi-scale wavelet coefficient of the trace detection signals was obtained after noise reduction by the dual-tree complex wavelet algorithm to minimize the adverse effects of data size on successive comparison calculations. Batch similarity comparison of trace features was achieved using multiple comparison strategies, including an optimized dynamic time regulation algorithm and the recognition of the changing rate gradient. Based on the results of boosting fusion multi-strategy comparison, optimized comparisons were achieved by machine learning, and a model for the rapid comparison of trace features was established. Finally, the viability of the proposed algorithm was verified by experiments.

A novel fusion framework of infrared and visible images based on RLNSST and guided filter

In this paper, we designed a novel image fusion framework based on Redundant Lifting Non-Separable Wavelet-based Shearlet Transform (RLNSST) and guided filter. Specifically, the Non-subsampled Pyramid (NSP) algorithm is substituted with Redundant Lifting Non-Separable Wavelet (RLNSW) in the Non-sub-sampled shearlet transform (NSST) algorithm, which is one of the state-of-the-art multi-scale decomposition algorithms and widely used in image fusion applications. On the other hand, guided filter-based saliency detection algorithm is used to fuse the coefficient matrices obtained by the improved NSST, named as RLNSST in this paper. We first decompose the infrared and visible image into multi-direction and multi-scale coefficient matrices by RLNSST decomposition. Then, Laplacian filter and Gaussian filter is utilized to generate the saliency map. Next, a novel weighted average based on guided filtering to retain edge detail information is used as the fusion rule. Finally, inverse RLNSST transform is used to get the fused image. Experiments show that our method can obtain better performance than other recently developed techniques.

Bearing fault diagnosis via kernel matrix construction based support vector machine

A novel approach on kernel matrix construction for support vector machine (SVM) is proposed to detect rolling element bearing fault efficiently. First, multi-scale coefficient matrix is achieved by processing vibration sample signal with continuous wavelet transform (CWT). Next, singular value decomposition (SVD) is applied to calculate eigenvector from wavelet coefficient matrix as sample signal feature vector. Two kernel matrices i.e. training kernel and predicting kernel, are then constructed in a novel way, which can reveal intrinsic similarity among samples and make it feasible to solve nonlinear classification problems in a high dimensional feature space. To validate its diagnosis performance, kernel matrix construction based SVM (KMCSVM) classifier is compared with three SVM classifiers i.e. classification tree kernel based SVM (CTKSVM), linear kernel based SVM (L-SVM) and radial basis function based SVM (RBFSVM), to identify different locations and severities of bearing fault. The experimental results indicate that KMCSVM has better classification capability than other methods.

An iteratively adaptive multi-scale finite element method for elliptic PDEs with rough coefficients

We propose an iteratively adaptive Multi-scale Finite Element Method (MsFEM) for elliptic PDEs with rough coefficients. The choice of the local boundary conditions for the multi-sale basis functions determines the accuracy of the MsFEM numerical solution, and one needs to incorporate the global information of the elliptic equation into the local boundary conditions of the multi-scale basis functions to recover the underlying fine-mesh solution of the equation. In our proposed iteratively adaptive method, we achieve this global-to-local information transfer through the combination of coarse-mesh solving using adaptive multi-scale basis functions and fine-mesh smoothing operations. In each iteration step, we first update the multi-scale basis functions based on the approximate numerical solutions of the previous iteration steps, and obtain the coarse-mesh approximate solution using a Galerkin projection. Then we apply several steps of smoothing operations to the coarse-mesh approximate solution on the underlying fine mesh to get the updated approximate numerical solution. The proposed algorithm can be viewed as a nonlinear two-level multi-grid method with the restriction and prolongation operators adapted to the approximate numerical solutions of the previous iteration steps. Convergence analysis of the proposed algorithm is carried out under the framework of two-level multi-grid method, and the harmonic coordinates are employed to establish the approximation property of the adaptive multi-scale basis functions. We demonstrate the efficiency of our proposed multi-scale methods through several numerical examples including a multi-scale coefficient problem, a high-contrast interface problem, and a convection-dominated diffusion problem.

Dynamic Multiscale Spatiotemporal Models for Poisson Data

ABSTRACTWe propose a new class of dynamic multiscale models for Poisson spatiotemporal processes. Specifically, we use a multiscale spatial Poisson factorization to decompose the Poisson process at each time point into spatiotemporal multiscale coefficients. We then connect these spatiotemporal multiscale coefficients through time with a novel Dirichlet evolution. Further, we propose a simulation-based full Bayesian posterior analysis. In particular, we develop filtering equations for updating of information forward in time and smoothing equations for integration of information backward in time, and use these equations to develop a forward filter backward sampler for the spatiotemporal multiscale coefficients. Because the multiscale coefficients are conditionally independent a posteriori, our full Bayesian posterior analysis is scalable, computationally efficient, and highly parallelizable. Moreover, the Dirichlet evolution of each spatiotemporal multiscale coefficient is parametrized by a discount factor that encodes the relevance of the temporal evolution of the spatiotemporal multiscale coefficient. Therefore, the analysis of discount factors provides a powerful way to identify regions with distinctive spatiotemporal dynamics. Finally, we illustrate the usefulness of our multiscale spatiotemporal Poisson methodology with two applications. The first application examines mortality ratios in the state of Missouri, and the second application considers tornado reports in the American Midwest.

Introduction and application of the multiscale coefficient of variation analysis.

Quantifying how patterns of behavior relate across multiple levels of measurement typically requires long time series for reliable parameter estimation. We describe a novel analysis that estimates patterns of variability across multiple scales of analysis suitable for time series of short duration. The multiscale coefficient of variation (MSCV) measures the distance between local coefficient of variation estimates within particular time windows and the overall coefficient of variation across all time samples. We first describe the MSCV analysis and provide an example analytical protocol with corresponding MATLAB implementation and code. Next, we present a simulation study testing the new analysis using time series generated by ARFIMA models that span white noise, short-term and long-term correlations. The MSCV analysis was observed to be sensitive to specific parameters of ARFIMA models varying in the type of temporal structure and time series length. We then apply the MSCV analysis to short time series of speech phrases and musical themes to show commonalities in multiscale structure. The simulation and application studies provide evidence that the MSCV analysis can discriminate between time series varying in multiscale structure and length.

The multiscale analysis between stock market time series

This paper is devoted to multiscale cross-correlation analysis on stock market time series, where multiscale DCCA cross-correlation coefficient as well as multiscale cross-sample entropy (MSCE) is applied. Multiscale DCCA cross-correlation coefficient is a realization of DCCA cross-correlation coefficient on multiple scales. The results of this method present a good scaling characterization. More significantly, this method is able to group stock markets by areas. Compared to multiscale DCCA cross-correlation coefficient, MSCE presents a more remarkable scaling characterization and the value of each log return of financial time series decreases with the increasing of scale factor. But the results of grouping is not as good as multiscale DCCA cross-correlation coefficient.

A Novel Scrambling Digital Image Watermark Algorithm Based on Double Transform Domains

Digital watermark technology is a very good method for protecting copyright. In this paper, in terms of requisition of imperceptibility and robustness of watermarking, the multidirectional, multiscale, and band-pass coefficient features of Curvelet transform are introduced and a novel image watermark scheme based on Curvelet and human visual system is proposed. Digital watermark information is embedded into the first 16 directions with larger energy in the fourth layer. Experimental results indicate that the proposed watermark scheme is feasible and simple. Simultaneously, the embedded watermark images just have tiny difference with the original images and the extracted watermark is accurate. Moreover, it is imperceptible and robust against various methods of signals processing such as cropping, noise adding, and rotating and altering.

The Research on Multiscale Image Fusion Based on the Dualtree Complex Wavelet

In order to effectively ease and solve fusion effect and the contradiction of the algorithm complexity, this paper puts forward a fusion rule on rapid extraction of multi-scale fusion coefficient, this fusion rules first used in the source image multi-scale decomposition of the scale fusion is the extraction of coefficient based on the neighborhood the fusion of window way, the low frequency of the improved neighborhood entropy to extract matching measure (that is, between the input image similarity degree), high frequency with the cross scale neighborhood gradient to extract matching measure, and gives the fusion coefficient formula. Because of the wavelet transform has moved degeneration, this paper puts forward the application of double tree after wavelet transform to do image multi-scale decomposition.

Heterogeneous multiscale finite element method with novel numerical integration schemes

In this paper we introduce two novel numerical integration schemes, within the framework of the heterogeneous multiscale method (HMM) when finite element method is used as the macroscopic solver, to resolve the elliptic problem with multiscale coefficient. For non-self-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.

DYNAMICS OF AVALANCHE ACTIVITIES IN FINANCIAL MARKETS

We study the dynamical properties of avalanche activities in the Korean Treasury Bond (KTB) futures price and the S&P 500 stock index. We apply the detrended fluctuation analysis, multiscale sample entropy and wavelet coefficient correlation to them, which revealed the scale-free dynamics of the bursting time series, avalanche size, and laminar time. We found that the laminar time and the avalanche size are anti-correlated in a short scale but in a large scale strongly correlated in KTB503, and are strongly correlated over all scales in S&P 500.

Multiscale Poisson Data Smoothing

SummaryThe paper introduces a framework for non-linear multiscale decompositions of Poisson data that have piecewise smooth intensity curves. The key concept is conditioning on the sum of the observations that are involved in the computation of a given multiscale coefficient. Within this framework, most classical wavelet thresholding schemes for data with additive homoscedastic noise can be used. Any family of wavelet transforms (orthogonal, biorthogonal or second generation) can be incorporated in this framework. Our second contribution is to propose a Bayesian shrinkage approach with an original prior for coefficients of this decomposition. As such, the method combines the advantages of the Haar–Fisz transform with wavelet smoothing and (Bayesian) multiscale likelihood models, with additional benefits, such as extendability towards arbitrary wavelet families. Simulations show an important reduction in average squared error of the output, compared with the present techniques of Anscombe or Fisz variance stabilization or multiscale likelihood modelling.

A Method of Detecting Seismic Singularities Using Combined Wavelet with Fractal

A new method of utilizing combined wavelet with fractal to detect seismic singularities is proposed aiming at improving the detecting precision. Firstly the fractal property of seismic wave is comprehensively investigated based on its power spectrum analysis, and the conclusion is made that seismic wave is of self-affine fractal property only in several higher frequency bands. This not only provides the theoretic foundation for reasonably applying fractal to seismic wave analysis, but also reveals the shortage of employing single integrated fractal dimension to detect seismic singularities. Secondly an algorithm of frequency and time-ordered non-decimated wavelet packet transform is put forward to produce excellent platform for fractal analysis of seismic wave. On the basis of the above two aspects, a wavelet-fractal based method for detecting seismic singularities is developed, which consists of three successive steps: seismic wave is decomposed into multi-scale coefficient sequences, and then fractal-parameter space of seismic profile is constructed depending on the correlation dimension analysis of effective scale coefficient sequences, in the end, singular parameters are identified and they indicate the seismic singularities. An engineering example shows that the proposed method outperforms the existing wavelet and fractal concerned methods in rationality and practicability, so it provides a new approach for parameterized accurate seismic geophysical prospecting.