Multiresolution Setting Research Articles

Multi-resolution partial differential equations preserved learning framework for spatiotemporal dynamics

Traditional data-driven deep learning models often struggle with high training costs, error accumulation, and poor generalizability in complex physical processes. Physics-informed deep learning (PiDL) addresses these challenges by incorporating physical principles into the model. Most PiDL approaches regularize training by embedding governing equations into the loss function, yet this depends heavily on extensive hyperparameter tuning to weigh each loss term. To this end, we propose to leverage physics prior knowledge by “baking” the discretized governing equations into the neural network architecture via the connection between the partial differential equations (PDE) operators and network structures, resulting in a PDE-preserved neural network (PPNN). This method, embedding discretized PDEs through convolutional residual networks in a multi-resolution setting, largely improves the generalizability and long-term prediction accuracy, outperforming conventional black-box models. The effectiveness and merit of the proposed methods have been demonstrated across various spatiotemporal dynamical systems governed by spatiotemporal PDEs, including reaction-diffusion, Burgers’, and Navier-Stokes equations.

A novel approach for time–frequency localization of scaling functions and design of three-band biorthogonal linear phase wavelet filter banks

Design of time–frequency localized filters and functions is a classical subject in the field of signal processing. Gabor's uncertainty principle states that a function cannot be localized in time and frequency domain simultaneously and there exists a nonzero lower bound of 0.25 on the product of its time variance and frequency variance called time–frequency product (TFP). Using arithmetic mean (AM)– geometric mean (GM) inequality, product of variances and sum of variances can be related and it can be shown that sum of variances has lower bound of one. In this paper, we compute the frequency variance of the filter from its discrete Fourier transform (DFT) and propose an equivalent summation based discrete-time uncertainty principle which has the lower bound of one. We evaluate the performance of the proposed discrete-time time–frequency uncertainty measure in multiresolution setting and show that the proposed DFT based concentration measure generate sequences which are even more localized in time and frequency domain than that obtained from the Slepian, Ishii and Furukawa's concentration measures. The proposed design approach provides the flexibility in which the TFP can be made arbitrarily close to the lowest possible lower bound of 0.25 by increasing the length of the filter. In the other proposed approach, the sum of the time variance and frequency variance is used to formulate a positive definite matrix to measure the time–frequency joint localization of a bandlimited function from its samples. We design the time–frequency localized bandlimited low pass scaling and band pass wavelet functions using the eigenvectors of the formulated positive definite matrix. The samples of the time–frequency localized bandlimited function are obtained from the eigenvector of the positive definite matrix corresponding to its minimum eigenvalue. The TFP of the designed bandlimited scaling and wavelet functions are close to the lowest possible lower bound of 0.25 and 2.25 respectively. We propose a design method for time–frequency localized three-band biorthogonal linear phase (BOLP) wavelet perfect reconstruction filter bank (PRFB) wherein the free parameters can be optimized for time–frequency localization of the synthesis basis functions for the specified frequency variance of the analysis scaling function. The performance of the designed filter bank is evaluated in classification of seizure and seizure-free electroencephalogram (EEG) signals. It is found that the proposed filter bank outperforms other existing methods for the classification of seizure and seizure-free EEG signals.

An adaptive multiresolution semi‐intrusive scheme for UQ in compressible fluid problems

SummaryThis paper deals with the introduction of a multiresolution strategy into the semi‐intrusive scheme, recently introduced by the authors, aiming to propagate uncertainties in unsteady compressible fluid applications. The mathematical framework of the multiresolution setting is presented for the cell‐average case, and the coupling with the semi‐intrusive scheme is described from both the theoretical and algorithmic point‐of‐view. Some reference test cases are performed to demonstrate the convergence properties and the efficiency of the overall scheme: the linear advection problem for both smooth and discontinuous initial conditions, the inviscid Burgers equation, and an uncertain shock tube problem obtained by modifying the well‐known Sod shock problem. For all the cases, the convergence curves are computed with respect to semi‐analytical (exact) solutions. In the case of the shock tube problem, an original technique to obtain a reference highly‐accurate numerical stochastic solution has also been developed. Copyright © 2015 John Wiley & Sons, Ltd.

A multiresolution framework for local similarity based image denoising

In this paper, we present a generic framework for denoising of images corrupted with additive white Gaussian noise based on the idea of regional similarity. The proposed framework employs a similarity function using the distance between pixels in a multidimensional feature space, whereby multiple feature maps describing various local regional characteristics can be utilized, giving higher weight to pixels having similar regional characteristics. An extension of the proposed framework into a multiresolution setting using wavelets and scale space is presented. It is shown that the resulting multiresolution multilateral (MRM) filtering algorithm not only eliminates the coarse-grain noise but can also faithfully reconstruct anisotropic features, particularly in the presence of high levels of noise.

Adaptive sequential design for regression on multi-resolution bases

This work investigates the problem of construction of designs for estimation and discrimination between competing linear models. In our framework, the unknown signal is observed with the addition of a noise and only a few evaluations of the noisy signal are available. The model selection is performed in a multi-resolution setting. In this setting, the locations of discrete sequential D and A designs are precisely constraint in a small number of explicit points. Hence, an efficient stochastic algorithm can be constructed that alternately improves the design and the model. Several numerical experiments illustrate the efficiency of our method for regression. One can also use this algorithm as a preliminary step to build response surfaces for sensitivity analysis.

Non-rigid Groupwise Registration using B-Spline Deformation Model

In this work, we extend a previously demonstrated entropy based groupwise registration method to include a non-rigid deformation model based on B-splines. We describe an open source implementation of the groupwise registration algorithm using the Insight Toolkit ITK. We provide the source code, parameters, input and output data that we used for validation.We describe an efficient implementation of the algorithm by using a stochastic optimization scheme embedded in a multi-resolution setting. The objective function is optimized using gradient descent algorithm combined with line search for the step size. The derivative of the objective function is evaluated efficiently by computing Jacobian of B-spline deformation field locally.We demonstrate the algorithm in application to different imaging modalities including proton density, FA, T1 and T2 MR images. We validate the algorithm on synthetic datasets varying from 2 to 30 images by recovering randomly applied affine and B-spline transforms.

Multiscale Galerkin method using interpolation wavelets for two‐dimensional elliptic problems in general domains

AbstractOne major hurdle in developing an efficient wavelet‐based numerical method is the difficulty in the treatment of general boundaries bounding two‐ or three‐dimensional domains. The objective of this investigation is to develop an adaptive multiscale wavelet‐based numerical method which can handle general boundary conditions along curved boundaries. The multiscale analysis is achieved in a multi‐resolution setting by employing hat interpolation wavelets in the frame of a fictitious domain method. No penalty term or the Lagrange multiplier need to be used in the present formulation. The validity of the proposed method and the effectiveness of the multiscale adaptive scheme are demonstrated by numerical examples dealing with the Dirichlet and Neumann boundary‐value problems in quadrilateral and quarter circular domains. Copyright © 2003 John Wiley & Sons, Ltd.

Adaptive multiscale wavelet-Galerkin analysis for plane elasticity problems and its applications to multiscale topology design optimization

The multiscale Galerkin formulation of two-dimensional elasticity problems is presented. For easy interpolation and boundary handlings as well as efficient adaptive analysis, two-dimensional interpolation wavelets are used as the multiscale trial functions in the Galerkin formulation. After the validity of the present multiscale adaptive method is verified with some benchmark problems, the present wavelet-based method is applied to the multiscale topology optimization that progresses design resolution levels dyadically from low to high levels. By this application, we show the potential of the multiscale method and the possibility of developing a fully integrated analysis and topology design optimization in the multiscale multiresolution setting.

Design of optimal binary filters under joint multiresolution–envelope constraint

This paper examines binary filter design by jointly applying multiresolution design and envelope constraint. In multiresolution design, the value of the designed filter for a configuration is defined at a resolution sufficiently low to have observed the configuration. Preference is given to higher resolutions, but the number of training observations is taken into account. In envelope design, the designed filter is constrained to lie in the envelope between two humanly designed filters. For small samples, the envelope-designed filter has the benefit of being in accord with expert knowledge, whereas for large samples statistical training provides more accurate filter design. To obtain the advantages of both approaches, they can be applied in combination. This can be done in more than a single way. This paper explores joint multiresolution–envelope design, extends the basic propositions for envelope design to the multiresolution setting, considers design consistency, and provides experimental support for the joint approach.

Multiresolution Based on Weighted Averages of the Hat Function II: Nonlinear Reconstruction Techniques

In this paper we describe and analyze a class of nonlinear \mr schemes for the multiresolution setting which corresponds to discretization by local averages with respect to the hat function. These schemes are based on the essentially-non-oscillatory (ENO) interpolatory procedure described in [A. Harten, B. Engquist, S. Osher, and S. Chakravarthy, J. Comput. Phys., 71 (1987), pp. 231--302]. We show that by allowing the approximation to fit the local nature of the data, one can improve the compression capabilities of the multiresolution algorithms. The question of stability for nonlinear (data-dependent) reconstruction techniques is also addressed.

Multiresolution approximation using shifted splines

We consider the construction of least squares pyramids using shifted polynomial spline basis functions. We derive the pre and post-filters as a function of the degree n and the shift parameter /spl Delta/. We show that the underlying projection operator is entirely specified by two transfer functions acting on the even and odd signal samples, respectively. We introduce a measure of shift invariance and show that the most favorable configuration is obtained when the knots of the splines are centered with respect to the grid points (i.e., /spl Delta/=1/2 when n is odd and /spl Delta/=0 when n is even). The worst case corresponds to the standard multiresolution setting where the spline spaces are nested.

Multiresolution Based on Weighted Averages of the Hat Function I: Linear Reconstruction Techniques

In this paper we analyze a particular example of the general framework developed in [A. Harten, {\it SIAM J. Numer. Anal}., 33 (1996) pp. 1205--1256], the case in which the discretization operator is obtained by taking local averages with respect to the hat function. We consider a class of reconstruction procedures which are appropriate for this multiresolution setting and describe the associated prediction operators that allow us to climb up the ladder from coarse to finer levels of resolution. In Part I we use data-independent (linear) reconstruction techniques as our approximation tool. We show how to obtain multiresolution transforms in bounded domains and analyze their stability with respect to perturbations.

Plotting and Scheming with Wavelets

Wavelets are acquiring a visibility and popularity that may soon be on the scale first enjoyed by fractals a few years back. Like fractals, wavelets have attractive and novel features, both as mathematical entities and in numerous applications. They are often touted as worthwhile alternatives to classical Fourier analysis, which works best when applied to periodic data: wavelet methods make no such assumptions. However, the mathematics of wavelets can seem intractable to the novice. Indeed, most introductions to wavelets assume that the reader is already well versed in Fourier techniques. Our main goal is simple: to convince the reader that at their most basic level, wavelets are fun, easy, and ideal for livening up dull conversations. We demonstrate how elementary linear algebra makes accessible this exciting and relatively new area at the border of pure and applied mathematics. In Plotting, we explore several ways of visually representing data, with the help of Matlab software. In Scheming, we discuss a simple wavelet-based compression technique, whose generalizations are being used today in signal and image processing, as well as in computer graphics and animation. The basic technique uses only addition, subtraction, and division by two! Only later, in Wavelets, do we come clean and reveal what wavelets are, while unveiling the multiresolution setting implicit in any such analysis. In Averaging and Differencing with Matrices, which may be read independently of Wavelets, we provide a matrix formulation of the compression scheme. In Wavelets on the World Wide Web we mention a natural form of progressive image transmission that lends itself to use by the emerging generation of web browsers (such wavelet-enhanced software is already on the market). In Wavelet Details, we attempt to put everything in context, while hinting at the more sophisticated mathematics that must be mastered if one wishes to delve deeper into the subject. Finally, in Closing Remarks, we mention some other common applications of wavelets. Along the way we find ourselves trying out an adaptive plotting technique for ordinary functions of one variable that differs from those currently employed by many of today's popular computer algebra packages. While this technique, as described here, is limited in its usefulness, it can be modified to produce acceptable results. We were much inspired by Stollnitz, DeRose, and Salesin's fine wavelets primers ([15, 16]), which, along with [17], [18], [19], we recommend heartily to beginners wlho

Multiresolution Representation of Data: A General Framework

In this paper we present a general framework for a multiresolution representation of data which is obtained by the discretization of mappings. This framework, which can be viewed as a generalization of the theory of wavelets, includes discretization corresponding to unstructured grids in several space dimensions, and thus is general enough to enable us to embed most numerical problems in a multiresolution setting. Furthermore, this framework allows for nonlinear (data-dependent) multiresolution representation schemes and thus enables us to design adaptive data-compression algorithms. In this paper we also study the stability of linear schemes for a multiresolution representation and derive sufficient conditions for existence of a multiresolution basis.

A multiresolution representation scheme for multimedia databases

We present a compression scheme that is useful for interactive video applications such as browsing a multimedia database. The focus of our approach the development of a compression scheme (and a corresponding retrieval scheme) that is optimal for any data rate. To browse a multimedia database, such a compression scheme is essential. We use a multiresolution setting, but eliminate the need for wavelets. This results in much better compression. We show experimental results and explain in detail how to extend our approach to multidimensional data.

Matrix decomposition and data reduction

In this paper, we present a class of decomposition techniques for data represented as matrices. The main idea is to transform a matrix into a sequence of components in order to represent and analyze the matrix in a multi-resolution setting. Data reduction is obtained by maintaining only entries of each matrix component that give significant contributions to the reconstruction of the original matrix. Important features of the matrix decomposition strategies are illustrated by several examples using a selection of digital images.