Multiresolution Property Research Articles

Equivariant neural operators for gradient-consistent topology optimization

Abstract Most traditional methods for solving partial differential equations (PDEs) require the costly solving of large linear systems. Neural operators (NOs) offer remarkable speed-ups over classical numerical PDE solvers. Here, we conduct the first exploration and comparison of NOs for three-dimensional topology optimization. Specifically, we propose replacing the PDE solver within the popular Solid Isotropic Material with Penalization (SIMP) algorithm, which is its main computational bottleneck. For this, the NO not only needs to solve the PDE with sufficient accuracy but also has the additional challenge of providing accurate gradients which are necessary for SIMP’s density updates. To realize this, we do three things: (i) We introduce a novel loss term to promote gradient-consistency. (ii) We guarantee equivariance in our NOs to increase the physical correctness of predictions. (iii) We introduce a novel NO architecture called U-Net Fourier neural operator (U-Net FNO), which combines the multi-resolution properties of U-Nets with the Fourier neural operator (FNO)’s focus on local features in frequency space. In our experiments we demonstrate that the inclusion of the novel gradient loss term is necessary to obtain good results. Furthermore, enforcing group equivariance greatly improves the quality of predictions, especially on small training datasets. Finally, we show that in our experiments the U-Net FNO outperforms both a standard U-Net, as well as other FNO methods.

Singularly perturbative behaviour of nonlinear advection–diffusion-reaction processes

The purpose of this paper is to use a wavelet technique to generate accurate responses for models characterized by the singularly perturbed generalized Burgers-Huxley equation (SPGBHE) while taking multi-resolution features into account. The SPGBHE’s behaviours have been captured correctly depending on the dominance of advection and diffusion processes. It should be noted that the required response was attained through integration and by marching on time. The wavelet method is seen to be very capable of solving a singularly perturbed nonlinear process without linearization by utilizing multi-resolution features. Haar wavelet method results are compared with corresponding results in the literature and are found in agreement in determining the numerical behaviour of singularly perturbed advection–diffusion processes. The most outstanding aspects of this research are to utilize the multi-resolution properties of wavelets by applying them to a singularly perturbed nonlinear partial differential equation and that no linearization is needed for this purpose.

Wavelets collocation method for singularly perturbed differential–difference equations arising in control system

In this paper, we present a wavelet collocation method for efficiently solving singularly perturbed differential–difference equations (SPDDEs) and one-parameter singularly perturbed differential equations (SPDEs) taking into account the singular perturbations inherent in control systems. These equations represent a class of mathematical models that exhibit a combination of differential and difference equations, making their analysis and solution challenging. The terms that include negative and positive shifts were approximated using Taylor series expansion. The main aim of this technique is to convert the problems by using operational matrices of integration of Haar wavelets into a system of algebraic equations that can be solved using Newton’s method. The adaptability and multi-resolution properties of wavelet functions offer the ability to capture system behavior across various scales, effectively handling singular perturbations present in the equations. Numerical experiments were conducted to showcase the effectiveness and accuracy of the wavelet collocation method, demonstrating its potential as a reliable tool for analyzing and solving SPDDEs in control system.

A wavelet-based real-time fire detection algorithm with multi-modeling framework

This paper presents a wavelet-based real-time automated fire detection algorithm that takes into consideration the multi-resolution property of the wavelet transforms. Unlike conventional fire detection algorithms, which fail to capture temporal dependency within the fire sensor signals, the proposed wavelet-based features characterize temporal dynamics of chemical sensor signals generated from various types of fire, such as flaming, heating and smoldering fires. We propose a new feature selection technique based on types of fire to select the best features that can effectively discriminate between normal and various fire conditions. Then, a real-time fire detection algorithm with a multi-modeling framework is developed to effectively utilize the selected features and construct multiple fire detectors that are sensitive in monitoring various kinds of fires without prior knowledge. In addition, we develop a novel multi-sensor fusion system that incorporates various chemical sensors and collects an accurate and reliable fire dataset from different real-life fire scenarios in order to validate the performance of the proposed and existing fire detection algorithms. The experimental results with real-life and public fire data show that the proposed algorithm outperforms others with early detection time with a reasonable false alarm rate regardless of the type of fire.

Fast Frequency Domain Screen-Shooting Watermarking Algorithm Based on ORB Feature Points

With high performances of image capturing tools, image information can be easily obtained by screenshots that make image copyright protection a challenging task. The existing screen-shooting watermarking algorithms suffer from a huge running time, in addition to their low robustness against different screenshot attacks, such as different distances and capturing angles of the screenshots. In this paper, a fast and robust high-capacity flexible watermarking algorithm for screenshot images is proposed. Firstly, Oriented FAST and Rotated BRIEF (ORB) feature points are extracted from the input image. Secondly, the feature points are then sorted in a descending order according to their response values. Then, the first five non-overlapping feature points are selected for the embedding by using Hamming window-based filtering method. Furthermore, we exploit the multi-resolution property of Discrete Wavelet Transform (DWT) and energy compaction property of Singular Value Decomposition (SVD) to embed the watermark. Therefore, the classical DWT combined with Singular Value Decomposition (SVD) are adopted to improve the robustness and capacity of the proposed watermarking algorithm. At the extraction side, the sum of the response values for the three RGB channels of the color-ripped image is calculated to improve the feature point localization accuracy. Experimental results show that the proposed screen-shooting watermarking algorithm improves running speed while ensuring the robustness. Furthermore, it has less time complexity and high robustness compared with the state-of-the-art watermarking algorithms against different screenshot attacks.

Extraction of Group Velocity Dispersion Curves of Surface Waves Using Continuous Wavelet Transform

Surface wave tomography is performed by first estimating surface wave dispersion curves and then inverting them. The objective of this paper is the efficient and reliable estimation of the group velocity dispersion curves of surface waves. In this paper, group velocity dispersion curves of surface waves recorded at a single station with known location of seismic source or between a pair of stations are estimated using continuous wavelet transform (CWT). The advantage of CWT is its multi-resolution property and flexible choice of analyzing function. In the proposed method, the number of CWT filters used are almost half of the total number of filters used in the conventional frequency-time analysis (FTAN) method. The CWT coefficients of the seismogram are the functions of time and scale (analogues to wave period). The arrival time to reach the peak of the envelope function of CWT coefficients is estimated to calculate group velocity of the surface waves. Group velocity dispersion curve of the surface waves is the plot of change of group velocity with the period. Synthetic test data and ambient noise recordings of MesoAmerica Seismic Experiment (MASE) stations are acquired to investigate the performance of the proposed method. It is observed that the proposed method effectively retrieves the weaker surface waves and results the broader band group-velocity dispersion curve when compared to the conventional FTAN method. The proposed method is also found to be computationally efficient.

Improving the Efficiency of Variationally Enhanced Sampling with Wavelet-Based Bias Potentials.

Collective variable-based enhanced sampling methods are routinely used on systems with metastable states, where high free energy barriers impede the proper sampling of the free energy landscapes when using conventional molecular dynamics simulations. One such method is variationally enhanced sampling (VES), which is based on a variational principle where a bias potential in the space of some chosen slow degrees of freedom, or collective variables, is constructed by minimizing a convex functional. In practice, the bias potential is taken as a linear expansion in some basis function set. So far, primarily basis functions delocalized in the collective variable space, like plane waves, Chebyshev, or Legendre polynomials, have been used. However, there has not been an extensive study of how the convergence behavior is affected by the choice of the basis functions. In particular, it remains an open question if localized basis functions might perform better. In this work, we implement, tune, and validate Daubechies wavelets as basis functions for VES. The wavelets construct orthogonal and localized bases that exhibit an attractive multiresolution property. We evaluate the performance of wavelet and other basis functions on various systems, going from model potentials to the calcium carbonate association process in water. We observe that wavelets exhibit excellent performance and much more robust convergence behavior than all other basis functions, as well as better performance than metadynamics. In particular, using wavelet bases yields far smaller fluctuations of the bias potential within individual runs and smaller differences between independent runs. Based on our overall results, we can recommend wavelets as basis functions for VES.

Hyperfine Decoupling of ESR Spectra Using Wavelet Transform.

The objective of spectral analysis is to resolve and extract relevant features from experimental data in an optimal fashion. In continuous-wave (cw) electron spin resonance (ESR) spectroscopy, both values of a paramagnetic center and hyperfine splitting caused by its interaction with neighboring magnetic nuclei in a molecule provide important structural and electronic information. However, in the presence of - and/or -anisotropy and/or large number of resonance lines, spectral analysis becomes highly challenging. Either high-resolution experimental techniques are employed to resolve the spectra in those cases or a range of suitable ESR frequencies are used in combination with simulations to identify the corresponding and values. In this work, we present a wavelet transform technique in resolving both simulated and experimental cW-ESR spectra by separating the hyperfine and super-hyperfine components. We exploit the multiresolution property of wavelet transforms that allow the separation of distinct features of a spectrum based on simultaneous analysis of spectrum and its varying frequency. We retain the wavelet components that stored the hyperfine and/or super-hyperfine features, while eliminating the wavelet components representing the remaining spectrum. We tested the method on simulated cases of metal-ligand adducts at L-, S-, and X-band frequencies, and showed that extracted values, hyperfine and super-hyperfine coupling constants from simulated spectra, were in excellent agreement with the values of those parameters used in the simulations. For the experimental case of a copper(II) complex with distorted octahedral geometry, the method was able to extract and hyperfine coupling constant values, and revealed features that were buried in the overlapped spectra.

Technique Detects Well Interference Using Long-Term Pressure Data

This article, written by JPT Technology Editor Chris Carpenter, contains highlights of paper SPE 206354, “Well-Interference Detection From Long-Term Pressure Data Using Machine Learning and Multiresolution Analysis,” by Dante Orta Aleman, SPE, and Roland Horne, SPE, Stanford University. The paper has not been peer reviewed. Methods such as interference tests and tracers have been developed to obtain knowledge of reservoir heterogeneity and connectivity from dynamic data. However, detecting well connectivity using interference tests requires long periods with stable reservoir pressure and constant flow-rate conditions. Conversely, the long duration and high frequency of well-production data have high value for detecting connectivity if noise, abrupt changes in flow rate, and missing data are considered. In this work, a methodology to detect interference from long-term pressure and flow-rate data was developed using multiresolution analysis in combination with machine-learning algorithms. Introduction Multiple methodologies have been proposed to detect well connectivity using production data. Most such methods apply a statistical or physical model to relate production and injection rates. In previous works, pressure data rarely has been used and data were fitted to a predefined mathematical model with strong physical assumptions. The approach of the current work is to infer connectivity by estimating the pressure response of a well to a second “interfering” well. Because the pressure response at a given well is the convolution of all the connected wells’ flow rates with the reservoir response, the methodology attempts to deconvolve the flow rates and assign a pressure response to each. Previous work showed that, by using wavelet decomposition and deep learning, capture of the pressure response of a single well using only flow-rate data as input is possible. The complete paper expands on the idea of using the multiresolution properties of wavelets to simplify the flow-rate/pressure response of more than one well and to capture that response by using a machine-learning model with a flexible function representation.

Towed Streamer-Based Simultaneous Source Separation by Contourlet Transform

Simultaneous towed-streamer marine acquisition has the advantages of reducing the total time requirements and costs of surveys. However, the seismic records obtained are blended seismic data, so the successful deblending of such data is the key to this method. In this paper, we propose an effective deblending method with a new thresholding operator based on the shaping regularization framework in the contourlet domain. The new thresholding operator consists of an adaptive Bayesian threshold and a new thresholding function. Because of its multiresolution, locality, and directionality properties, the contourlet transform can effectively capture geometrical structures, which are the main features in natural images. To make the traditional Bayesian threshold adaptive in the contourlet domain, we propose a scale adjustment factor, a direction adjustment factor, and an attenuation factor to modify the threshold, and we also adopt local adaptive elliptic windows to estimate the standard deviations of useful signals; eventually, we obtain an adaptive Bayesian threshold. Furthermore, the new thresholding function can overcome the shortcomings of the existing soft and hard thresholding functions. Experimental results demonstrate that our method can effectively separate blended data.

A Wavelet-Based Compressive Deep Learning Scheme for Inverse Scattering Problems

Recently, physics-assisted deep learning schemes (DLSs) have demonstrated state-of-the-art performance for solving inverse scattering problems (ISPs). However, most learning approaches typically require a high computational overhead and a big memory footprint, which prohibits further applications. In this work, a wavelet-based compressive scheme (WCS) is proposed in solving ISPs, where the multi-subspace information is explored by wavelet bases and branched between each encoder and decoder path. It is shown that the proposed WCS can be simply adapted to commonly used DLSs, such as the back-propagation scheme (BPS) and the dominant current scheme (DCS), to reduce the computational and storage load. Specifically, benefiting from compressive and multi-resolution properties of wavelet and with the help of the factorized convolution method, more than 99.7% trainable weights are reduced in both illustrated BP-WCS and DC-WCS, whereas the performance deterioration is limited around 1% in terms of traditional BPS and DCS. Extensive numerical and experimental tests are conducted for quantitative validations. Comparisons are also made among UNet, a well-known compressive method (Mobile-UNet), and the proposed method. It is expected that the suggested compression technique would find its applications on deep learning-based electromagnetic inverse problems under source-limited scenarios.

Wavelet thresholding techniques implementation in retinal images for suppressing noises

In medical image processing, noise plays a major role of reducing the details of the images and blurs the features which are important for the diagnosis of the disease. Retinal images are high resolution images. Gaussian noise, Salt & Pepper noise affects certain important features of them which are needed for proper diagnosis of the retinal diseases. So removing noises in these images is a kind of difficulty that all researcher experiences. The main objective is to reduce the effect of Gaussian noise and Salt & Pepper noise in retinal images. Recently, wavelet transform is being used in all areas of image processing and especially in medical image denoising, it is highly used because it has multi resolution property and sparsity. So here, wavelet thresholding techniques such as Visu shrink, SURE Shrink and Bayes shrink are implemented to remove Gaussian noise and Salt & Pepper noise. The performance of all the three methods are evaluated using PSNR (Peak Signal to Noise Ratio), MSE (Mean Square Error), NK (Normalized cross correlation) and Time elapsed. Also, the performances of all the thresholding techniques were compared using above mentioned metrics.

Lung sound signal denoising using discrete wavelet transform and artificial neural network

Computerized analysis of Lung Sound (LS) is a promising method for assessing pulmonary function. However, the LS signal is severely contaminated by background noise from various sources. Conventional denoising methods may not be practical due to the noisy nature of the LS as well as its spectral overlap with different noise sources. This paper proposes an adaptive technique based on Discrete Wavelet Transform and Artificial Neural Network (DWT-ANN) to filtrate LS signals in a noisy environment. This new method mixes the multi-resolution property of DWT with ANN as a nonlinear adaptive filter. In this research, separate models for signal denoising with different SNRs (0, 5, 10, and 15 dB) were designed. Then a single model was introduced as a combined model to eliminate any information about input signals SNR before the denoising process. The results showed that the combined model, in addition to having a close performance to the individual models, could perform the denoising process well in the range of −2 to 20 dB, which is outside the range that the model has been trained with. In addition, comparing the results of our proposed method with the DWT method, it was observed that the SNR of the denoised signal was significantly enhanced. At SNR = 0 dB, this improvement is 9.18 compared to only 3.90 using the DWT method.

Multiresolution Daubechies finite wavelet domain method for transient dynamic wave analysis in elastic solids

AbstractThe multiresolution capability provided by the family of Daubechies wavelets is exploited to develop a new computational approach, termed as multiresolution finite wavelet domain method for the fast and hierarchical prediction of transient dynamic problems with focus on wave propagation in one‐ and two‐dimensional structural configurations. In the developed method, both scaling and wavelet functions are employed as basis functions for the approximation of displacements in the governing dynamic equations. Because of the orthogonal and the multiresolution properties of Daubechies wavelets, a two‐scale system of mass uncoupled dynamic equations, representing the coarse and the fine spatial wave component solutions is derived. Numerical results concerning wave propagation in one‐ and two‐dimensional elastic structures demonstrate substantial computational gains of the method in comparison with other well‐established numerical methods. Moreover, additional benefits of the involved coarse and fine solutions enabling the analysis and characterization of the resultant wave response are revealed and discussed.

Speckle filtering of ultrasonic images using weighted nuclear norm minimization in wavelet domain

Medical images such as ultrasound images suffer from multiplicative speckle noise which reduces the contrast of ultrasound images and adversely affects the diagnosis process. Discrete Wavelet transform (DWT) is widely accepted in the area of image processing due to its time localization, multi-resolution and sparseness properties. DWT decomposes the noisy image into approximation coefficients and detail coefficients. The detail wavelet coefficients of clean image have sparsity property, whereas in the case of noisy image, sparsity of detail coefficients is reduced due to the presence of speckle noise. Thus, in the proposed method, low rank based Weighted Nuclear Norm Minimization (WNNM) is applied on detail coefficients to reveal the sparsity property of DWT. WNNM is applied on the group matrix of non-local similar patches of detail subbands of DWT to approximate the low-rank denoised version of the subbands. Moreover, less amount of edge and structure information is present in the approximation coefficients of DWT. Thus, in the proposed method, Non-local Means (NLM) filter with Square-Chord distance is also used to denoise speckle noise from approximation coefficients of DWT. Exhaustive experiments are conducted on various images such as real US images, simulated kidney image and synthetic image. It is observed that the mean improvement in terms of PSNR and CNR values of the proposed hybrid method are 1.50% and 3.81% respectively over WNNM based Despeckling using Low-Rank Approximation (DLRA) method. Qualitative and quantitative analyses show that the proposed hybrid method performed better than existing speckle reduction methods in terms of both speckle reduction and edge preservation.

Financion: A quantum approach to financial market modelling

In this paper we will show that, in the financial market, the concept of chaos is removed and replaced with that of complexity. Indeed, it has always shown the characteristics of a dynamical system with high complexity. In particular, we present a new theory to describe the financial phenomenologies: a theory that is able to combine the different founding aspects of the financial market. This study provides a useful answer to the fundamental aspects of the market: its multiscale or multi resolution properties; its quantum-relativistic properties; the usefulness of describing, at a fundamental level, the interactions between bullish o + and bearish o – financial operators; the usefulness of declining financial interactions between operators in terms of scattering, annihilation or creation of elementary liquidity constituents; the introduction of a specific field, named financial field, and its gauge boson, the so-called “financion” as a mediator of financial interactions; the possibility of describing the financial market as a dynamical system, capable of absorbing financions and that is quanta of interaction liquidity. Thanks to this new paradigm, we have been able to describe and define the market scale length, i.e. revenue, price, risk (λ, ρ, σ). Furthermore, we have analyzed Kahneman and Tversky’s Prospect Theory to describe the decision-making process under risk conditions, not considering the price as a function of time, p = p(t), but by carrying out the analysis of price dynamics as trajectories in the transformed energy-entropy space ES of the financial market , where there are more stable and reliable indications than common indicators.

Simulation Evaluation Method for Fusion Characteristics of the Optical Camouflage Pattern

A comprehensive evaluation system for a camouflage design combining local effect evaluation and global sampling is developed. Different from previous models, this method can sample and evaluate target camouflage in a wide range of combat areas, thereby obtaining a comprehensive evaluation effect. In evaluating local effects, the Gaussian pyramid model is adopted to decompose the image on a multi-scale so that it can conform to the multi-resolution property of human eyes. The Universal Image Quality Index (UIQI) conforming to features of eye movements is then adopted to measure the similarities between multi-scale targeted and background brightness, color and textural features. In terms of the imitation camouflage pattern design algorithm, uniform sampling is used to obtain the evaluation distribution in the background; while for the deformation camouflage pattern, the sampling distribution is improved to make it conform to the movement rule of the target in the background. The evaluation results of the model for different designs were investigated. It is suggested by the experimental results that the model can compare and evaluate the indicators involved in the process of camouflage design, including integration, polychromatic adaptability and algorithm stability. This method can be applied in the evaluation and contrast of camouflage pattern design algorithms, in parameter optimisation of camouflage design and in scheme comparison in engineering practice, and can provide support of evaluation methodology for camouflage design theories.

Active Subdivision Surfaces for the Semiautomatic Segmentation of Biomedical Volumes.

We present a new family of active surfaces for the semiautomatic segmentation of volumetric objects in 3D biomedical images. We represent our deformable model by a subdivision surface encoded by a small set of control points and generated through a geometric refinement process. The subdivision operator confers important properties to the surface such as smoothness, reproduction of desirable shapes and interpolation of the control points. We deform the subdivision surface through the minimization of suitable gradient-based and region-based energy terms that we have designed for that purpose. In addition, we provide an easy way to combine these energies with convolutional neural networks. Our active subdivision surface satisfies the property of multiresolution, which allows us to adopt a coarse-to-fine optimization strategy. This speeds up the computations and decreases its dependence on initialization compared to singleresolution active surfaces. Performance evaluations on both synthetic and real biomedical data show that our active subdivision surface is robust in the presence of noise and outperforms current state-of-the-art methods. In addition, we provide a software that gives full control over the active subdivision surface via an intuitive manipulation of the control points.

Temporal Multiresolution Graph Learning

Estimating time-varying graphs, i.e., a set of graphs in which one graph represents the relationship among nodes in a certain time slot, from observed data is a crucial problem in signal processing, machine learning, and data mining. Although many existing methods only estimate graphs with a single temporal resolution, the actual graphs often demonstrate different relationships in different temporal resolutions. In this study, we propose an approach for time-varying graph learning by leveraging a multiresolution property. The proposed method assumes that time-varying graphs can be decomposed by a linear combination of graphs localized at different temporal resolutions. We formulate a convex optimization problem for temporal multiresolution graph learning. In experiments using synthetic and real data, the proposed method demonstrates the promising objective performances for synthetic data, and obtains reasonable temporal multiresolution graphs from real data.

Texture Analysis using wavelet Transform

In this research application of wavelet based multiscale image analysis methods for texture analysis has been highlighted. These methods are based on multiresolution properties of the two-dimensional wavelet transform, which is used to extract the features needed to discriminate and differentiate various textures more accurately then existing methods, we also took into account the texture model, the noise distribution, and the inter-dependence of the texture features which further help in discriminating factor. Multiresolution approach is nothing but a modified wavelet transform called the tree-structured wavelet transform or wavelet packets for texture analysis and classification. This approach is motivated by the observation that a large class of natural textures can be modeled as quasi-periodic signals whose dominant frequencies are located in the middle frequency channels. With the transform, we are able to zoom into any desired frequency channels for further decomposition and thus we could extract more texture features as compared to other methods.