Multiresolution Scheme Research Articles

A multiresolution local-timestepping scheme for particle–laden multiphase flow simulations using a level-set and point-particle approach

We present a novel storage-and-access approach for Lagrangian particles in a multiresolution local-timestepping framework for multiphase flow simulations. The proposed method applies a block-particle mapping strategy for efficient access of all particles on a specific refinement level while traversing through the multiresolution tree. This allows for extending the local timestepping algorithm with its refinement level-dependent timestep sizes to the evolution of particles, which results in significant speed-up in comparison to standard timestepping. For multi-scale multiphase flow simulations, the particle model is combined with a level-set based multiphase model on a Cartesian grid. To maintain robustness of fluid-state interpolation for particles near the level set-based fluid–fluid interface, WENO-based interpolation is applied which includes both real- and ghost-fluid cells. This enforces the sharp interface property also for interpolated fluid states, and suppresses spurious oscillations in the event of discontinuities.We validate the particle model with a one-dimensional simulation of a single particle in an air–helium shock tube for one-way coupling, and with two-dimensional simulations of a particle injected in a quiescent domain for the feedback force. Simulations of two-dimensional aerodynamic fragmentation in shear-induced entrainment and Rayleigh–Taylor piercing regimes use Lagrangian particles as sub-grid scale representation of small droplets post-breakup. Finally, three-dimensional massively-parallel simulations of single- and triple-bubble collapse near a wall coated with free-floating particles are presented. The particle cleaning radius of the single-bubble setup agrees reasonably well with experimental reference data. These simulations consider 106 particles and an Eulerian grid with effectively 1012 finite-volume cells at compression rates of more than 90% for the particulate phase. This underlines the advantageous effect of embedding the particles in the multiresolution tree with its spatial and temporal adaptivity, which is necessary for performing such large scale simulations efficiently.

Absolutely convergent fixed-point fast sweeping WENO methods for steady state of hyperbolic conservation laws

Fixed-point iterative sweeping methods were developed in the literature to efficiently solve steady state solutions of Hamilton-Jacobi equations and hyperbolic conservation laws. Similar as other fast sweeping schemes, the key components of this class of methods are the Gauss-Seidel iterations and alternating sweeping strategy to achieve fast convergence rate. A family of characteristics of the corresponding hyperbolic partial differential equations (PDEs) are covered in a certain direction simultaneously in each sweeping order. Furthermore, good properties of fixed-point iterative sweeping methods which are different from other fast sweeping methods include that they have explicit forms and do not involve inverse operation of nonlinear local systems, and they can be applied to general hyperbolic equations using any monotone numerical fluxes and high order approximations easily. In a recent article (Wu et al. 2016) [31], a fifth order fixed-point sweeping WENO scheme was designed for solving steady state of hyperbolic conservation laws, and it was shown that the scheme converges to steady state solution much faster than the regular total variation diminishing (TVD) Runge-Kutta time-marching approach by stability improvement of high order schemes with a forward Euler time-marching. An open problem is that for some benchmark numerical examples, the iteration residue of the fixed-point sweeping WENO scheme hangs at a truncation error level, or even higher error levels, instead of settling down to machine zero. This issue makes it difficult to determine the convergence criterion for the iteration and challenging to apply the method to complex problems. To solve this issue, in this paper we apply the multi-resolution WENO scheme developed in Zhu and Shu (2018) [45] to the fifth order fixed-point sweeping WENO scheme and obtain an absolutely convergent fixed-point fast sweeping method for steady state of hyperbolic conservation laws, i.e., the residue of the fast sweeping iterations converges to machine zero / round off errors for all benchmark problems tested. Extensive numerical experiments, including solving difficult problems such as the shock reflection, supersonic flow past plates, and supersonic, transonic and subsonic flows past an airfoil, etc., are performed to show the accuracy, computational efficiency, and absolute convergence of the presented fifth order sweeping scheme.

A motion correction approach for oral and maxillofacial cone-beam CT imaging

Patient movement affects image quality in oral and maxillofacial cone-beam computed tomography imaging. While many efforts are made to minimize the possibility of motion during a scan, relatively little attention has been given to motion correction after acquisition. We propose a novel method which can improve the image quality after an oral and maxillofacial scan. The proposed method is based on our previous work and is a retrospective motion estimation and motion compensation (ME/MC) approach that iteratively estimates and compensates for rigid pose change over time. During motion estimation, image update and motion update are performed alternately in a multi-resolution scheme to obtain the motion. We propose use of a feature-based motion update and patch-based image update in the iterative estimation process, to alleviate the effect of limited scan field of view on estimation. During motion compensation, a fine-resolution image reconstruction was performed with compensation for the estimated motion. The proposed ME/MC method was evaluated with simulations, phantom and patient studies. Two experts in dentomaxillofacial radiology assessed the diagnostic importance of the resulting motion artifact suppression. The quality of the reconstructed images was improved after motion compensation, and most of the image artifacts were eliminated. Quantitative analysis by comparison to a reference image and by calculation of a sharpness metric agreed with the qualitative observation. The results are promising, and further evaluation is required to assess the clinical value of the proposed method.

HNSF Log‐Demons: Diffeomorphic demons registration using hierarchical neighbourhood spectral features

Many biomedical applications require accurate non-rigid image registration that can cope with complex deformations. However, popular diffeomorphic Demons registration algorithms suffer from difficulties for complex and serious distortions since they only use image greyscale and gradient information. To address these difficulties, a new diffeomorphic Demons registration algorithm is proposed using hierarchical neighbourhood spectral features namely HNSF Log-Demons in this paper. In view of three important properties of hierarchical neighbourhood spectral features based on line graph such as rotation invariance, invariance of linear changes of brightness, and robustness to noise, the hierarchical neighbourhood spectral features of a reference image and a moving image is first extracted and these novel spectral features are incorporated into the energy function of the diffeomorphic registration framework to improve the capability of capturing complex distortions. Secondly, the Nystr o ̈ m approximation based on random singular value decomposition is employed to effectively enhance the computational efficiency of HNSF Log-Demons. Finally, the hybrid multi-resolution strategy based on wavelet decomposition in the registration process is utilised to further improve the registration accuracy and efficiency. Experimental results show that the proposed HNSF Log-Demons not only effectively ensures the generation of smooth and reversible deformation field, but also achieves better performance than state-of-the-art algorithms.

A new fifth-order alternative finite difference multi-resolution WENO scheme for solving compressible flow

A new fifth-order alternative finite difference multi-resolution weighted essentially non-oscillatory (WENO) scheme is designed to solve the hyperbolic conservation laws in this paper. With the application of the original finite difference multi-resolution WENO schemes (Zhu and Shu, 2018; Zhu and Shu, 2020), a series of unequal-sized central spatial stencils are adopted to perform the WENO procedures, but the difference is that the methodology adopted in this paper is directly based on the point values of the solution rather than on the flux values. The proposed new multi-resolution WENO scheme can inherit many advantages of the original high order schemes, such as the arbitrary choice of the linear weights, the maintenance of the essentially non-oscillatory property in the vicinity of strong discontinuities, the smaller L1 and L∞ truncation errors than that of the same order classical WENOJS schemes (Balsara et al., 2016; Jiang and Shu, 1996) in smooth regions, and better convergence properties of the residue when solving some steady-state problems. Compared with the original multi-resolution WENO schemes, the presented WENO scheme also has some advantages. Firstly, an arbitrary monotone flux can be used in this framework, while the original method is only suitable for smooth flux splitting technique. Secondly, it has smaller L1 and L∞ truncation errors than that of the original same order multi-resolution WENO scheme. And finally, it avoids some time-consuming intermediate processes in the original multi-resolution WENO reconstruction procedures, thus resulting in less calculation time under the same conditions. Some inviscid and viscous numerical examples are provided to verify the superior performance of the new fifth-order alternative finite difference multi-resolution WENO scheme.

On New Means with Interesting Practical Applications: Generalized Power Means

Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these properties and improve upon the results obtained with other means, in the sense that they give sharper theoretical constants that are closer to the results obtained in practical examples. This has an immediate correspondence in several applications, as can be observed in the section devoted to a particular example.

Efficient propagation of sparse edits on 360[formula omitted] panoramas

We present an efficient method to propagate sparse user edits indicated by strokes on 360∘ panoramas. Our algorithm first projects each equirectangular pixel to its corresponding position on a 3D unit sphere, so each pixel can be characterized by a feature vector consisting of its 3D coordinates and RGB color values. We formulate edit propagation as an optimization problem that aims to satisfy the user edit constraints while preserving the manifold structure of the image at the same time. To solve the problem using a linear system efficiently, we first construct the K-D tree structure in the feature space to cluster pixels. Then we optimize the manifold structure where both the number of nearest neighbors and their corresponding weights are determined by the feature distributions. We further apply a multiresolution strategy to speedup the edit propagation. Our method is the first to perform interactive edit propagation on 360∘ panoramas. Experiments show that our method is able to generate seam-free and visually pleasing results, and users can receive instant feedback during interactive editing.

Baseline-Free Structural Damage Identification for Beam-Like Structures Using Curvature Waveforms of Propagating Flexural Waves.

Curvatures in mode shapes and operating deflection shapes have been extensively studied for vibration-based structural damage identification in recent decades. Curvatures of mode shapes and operating deflection shapes have proved capable of localizing and manifesting local effects of damage on mode shapes and operating deflection shapes in forms of local anomalies. The damage can be inversely identified in the neighborhoods of the anomalies that exist in the curvatures. Meanwhile, propagating flexural waves have also been extensively studied for structural damage identification and proved to be effective, thanks to their high damage-sensitivity and long range of propagation. In this work, a baseline-free structural damage identification method is developed for beam-like structures using curvature waveforms of propagating flexural waves. A multi-resolution local-regression temporal-spatial curvature damage index (TSCDI) is defined in a pointwise manner. A two-dimensional auxiliary TSCDI and a one-dimensional auxiliary damage index are developed to further assist the identification. Two major advantages of the proposed method are: (1) curvature waveforms of propagating flexural waves have relatively high signal-to-noise ratios due to the use of a multi-resolution central finite difference scheme, so that the local effects of the damage can be manifested, and (2) the proposed method does not require quantitative knowledge of a pristine structure associated with a structure to be examined, such as its material properties, waveforms of propagating flexural waves and boundary conditions. Numerical and experimental investigations of the proposed method are conducted on damaged beam-like structures, and the effectiveness of the proposed method is verified by the results of the investigations.

Multi-resolution ISPH-SPH for accurate and efficient simulation of hydroelastic fluid-structure interactions in ocean engineering

A SPH (Smoothed Particle Hydrodynamics)-based multi-resolution fully Lagrangian meshfree hydroelastic FSI (Fluid-Structure Interaction) solver is developed for accurate and adaptive reproductions of ocean engineering problems. The presented hydroelastic FSI solver comprises of projection-based ISPH (Incompressible SPH) fluid model and SPH structure model, through consideration of continuity/Navier-Stokes equations for fluid phase as well as linear/angular momentum conservation equations for structure phase. The FSI solver includes a consistent fluid-structure coupling scheme along with a novel multi-resolution scheme incorporating a common influence length, a modified SPH density definition and a SPH-based formulated SPP (Space Potential Particle) scheme in order to achieve consistent particle-based discretizations, precise satisfaction of fluid-structure interface boundary conditions and accurate volume conservation at the interface. The present ISPH fluid model corresponds to a refined version of ISPH incorporating several previously developed refined schemes, and hence the proposed FSI solver is referred to as “Enhanced Multi-resolution ISPH-SPH”. Validations are performed qualitatively/quantitatively through reproductions of classical as well as ocean engineering benchmark tests. Comparisons are also made with an MPS-based FSI solver as well as an Explicit ISPH-based one. A preliminary extension of the proposed solver to three-dimensions is also presented.

Research on Panoramic Stitching Algorithm of Lateral Cranial Sequence Images in Dental Multifunctional Cone Beam Computed Tomography.

In the design of dental multifunctional Cone Beam Computed Tomography, the linear scanning strategy not only saves equipment cost, but also avoids the demand for patients to be repositioned when acquiring lateral cranial sequence images. In order to obtain panoramic images, we propose a local normalized cross-correlation stitching algorithm based on Gaussian Mixture Model. Firstly, the Block-Matching and 3D filtering algorithm is used to remove quantum and impulse noises according to the characteristics of X-ray images; Then, the segmentation of the irrelevant region and the extraction of the region of interest are performed by Gaussian Mixture Model; The locally normalized cross-relation is used to complete the registration with the multi-resolution strategy based on wavelet transform and Particle Swarm Optimization algorithm; Finally, image fusion is achieved by the weighted smoothing fusion algorithm. The experimental results show that the panoramic image obtained by this method has significant performance in both subjective vision and objective quality evaluation and can be applied to preoperative diagnosis of clinical dental deformity and postoperative effect evaluation.

A high order multi-resolution WENO numerical scheme for solving viscous quantum hydrodynamic model for semiconductor devices

In this article, fifth order finite volume multi-resolution weighted essentially non-oscillatory (MR-WENO) scheme is developed for solving one-dimensional non linear viscous quantum hydrodynamical model for semiconductor devices with constant temperature. This non linear model consists of three equations, two equations for the current density and electron density, and third equation is the Poisson equation. The first two equations include the viscous and quantum correction terms. Further, numerical technique is used to obtain the solution due to high non linearity of considered model. The multi-resolution technique is used to reduce the computational cost of high resolution numerical scheme. In the procedure of derivation of the MR-WENO scheme unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical scheme. Further, the results obtained from considered numerical scheme are compared with those of kinetic flux vector splitting numerical scheme.

High-order Runge-Kutta discontinuous Galerkin methods with multi-resolution WENO limiters for solving steady-state problems

Since the classical WENO schemes [27] might suffer from slight post-shock oscillations (which are responsible for the numerical residual to hang at a truncation error level) and the new high-order multi-resolution WENO schemes [59] are successful to solve for steady-state problems, we apply these high-order finite volume multi-resolution WENO techniques to serve as limiters for high-order Runge-Kutta discontinuous Galerkin (RKDG) methods in simulating steady-state problems. Firstly, a new troubled cell indicator is designed to precisely detect the cells which would need further limiting procedures. Then the high-order multi-resolution WENO limiting procedures are adopted on a sequence of hierarchical L2 projection polynomials of the DG solution within the troubled cell itself. By doing so, these RKDG methods with multi-resolution WENO limiters could gradually degrade from the optimal high-order accuracy to the first-order accuracy near strong discontinuities, suppress the slight post-shock oscillations, and push the numerical residual to settle down to machine zero in steady-state simulations. These new multi-resolution WENO limiters are very simple to construct and can be easily implemented to arbitrary high-order accuracy for solving steady-state problems in multi-dimensions.

On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

Well-balanced finite volume multi-resolution schemes for solving the Ripa models

In this article, fifth order well-balanced finite volume multi-resolution weighted essentially non-oscillatory (FV MR-WENO) schemes are constructed for solving one-dimensional and two-dimensional Ripa models. The Ripa system generalizes the shallow water model by incorporating horizontal temperature gradients. The presence of temperature gradients and source terms in the Ripa models introduce difficulties in developing high order accurate numerical schemes which can preserve exactly the steady-state conditions. The proposed numerical methods are capable to exactly preserve the steady-state solutions and maintain non-oscillatory property near the shock transitions. Moreover, in the procedure of derivation of the FV MR-WENO schemes unequal central spatial stencils are used and linear weights can be chosen any positive numbers with only restriction that their total sum is one. Various numerical test problems are considered to check the validity and accuracy of the derived numerical schemes. Further, the results obtained from considered numerical schemes are compared with those of a high resolution central upwind scheme and available exact solutions of the Ripa model.

Radiological image retrieval technique using multi-resolution texture and shape features

Medical image analysis plays a very indispensable role in providing the best possible medical support to a patient. With the rapid advancements in modern medical systems, these digital images are growing exponentially and reside in discrete places. These images help a medical practitioner in understanding the problem and then the best suitable treatment. Radiological images are very often found to be the critical constituent of medical images. So, in health care, manual retrieval of visually similar images becomes a very tedious task. To address this issue, we have suggested a content-based medical image retrieval (CBMIR) system that effectively analyzes a Radiological image’s primitive visual features. Since radiological images are in gray-scale form, these images contain rich texture and shape features only. So, we have suggested a novel multi-resolution radiological image retrieval system that uses texture and shape features for content analysis. Here, we have employed a multi-resolution modified block difference of inverse probability (BDIP) and block-level variance of local variance (BVLC) for shape and texture features, respectively. Our proposed scheme uses a multi-resolution and variable window size feature extraction strategy to maintain the block-level co-relation and extract more salient visual features. Further, we have used the MURA x-ray image dataset, which has 40561 images captured from 12173 different patients to demonstrate the proposed scheme’s retrieval performance. We have also performed and compared image retrieval experiments on Brodatz and STex texture, Corel-1K, and GHIM-10K natural image datasets to demonstrate the robustness and improvement over other contemporaries.

Accelerated 3D image reconstruction with a morphological pyramid and noise-power convergence criterion

Model-based iterative reconstruction (MBIR) for cone-beam CT (CBCT) offers better noise-resolution tradeoff and image quality than analytical methods for acquisition protocols with low x-ray dose or limited data, but with increased computational burden that poses a drawback to routine application in clinical scenarios. This work develops a comprehensive framework for acceleration of MBIR in the form of penalized weighted least squares optimized with ordered subsets separable quadratic surrogates. The optimization was scheduled on a set of stages forming a morphological pyramid varying in voxel size. Transition between stages was controlled with a convergence criterion based on the deviation between the mid-band noise power spectrum (NPS) measured on a homogeneous region of the evolving reconstruction and that expected for the converged image, computed with an analytical model that used projection data and the reconstruction parameters. A stochastic backprojector was developed by introducing a random perturbation to the sampling position of each voxel for each ray in the reconstruction within a voxel-based backprojector, breaking the deterministic pattern of sampling artifacts when combined with an unmatched Siddon forward projector. This fast, forward and backprojector pair were included into a multi-resolution reconstruction strategy to provide support for objects partially outside of the field of view. Acceleration from ordered subsets was combined with momentum accumulation stabilized with an adaptive technique that automatically resets the accumulated momentum when it diverges noticeably from the current iteration update. The framework was evaluated with CBCT data of a realistic abdomen phantom acquired on an imaging x-ray bench and with clinical CBCT data from an angiography robotic C-arm (Artis Zeego, Siemens Healthineers, Forchheim, Germany) acquired during a liver embolization procedure. Image fidelity was assessed with the structural similarity index (SSIM) computed with a converged reconstruction. The accelerated framework provided accurate reconstructions in 60 s (SSIM = 0.97) and as little as 27 s (SSIM = 0.94) for soft-tissue evaluation. The use of simple forward and backprojectors resulted in 9.3× acceleration. Accumulation of momentum provided extra ∼3.5× acceleration with stable convergence for 6–30 subsets. The NPS-driven morphological pyramid resulted in initial faster convergence, achieving similar SSIM with 1.5× lower runtime than the single-stage optimization. Acceleration of MBIR to provide reconstruction time compatible with clinical applications is feasible via architectures that integrate algorithmic acceleration with approaches to provide stable convergence, and optimization schedules that maximize convergence speed.

An Adaptive Multiresolution Ultra-weak Discontinuous Galerkin Method for Nonlinear Schrödinger Equations

This paper develops a high-order adaptive scheme for solving nonlinear Schrödinger equations. The solutions to such equations often exhibit solitary wave and local structures, which make adaptivity essential in improving the simulation efficiency. Our scheme uses the ultra-weak discontinuous Galerkin (DG) formulation and belongs to the framework of adaptive multiresolution schemes. Various numerical experiments are presented to demonstrate the excellent capability of capturing the soliton waves and the blow-up phenomenon.

A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws

A Class of Robust Low Dissipation Nested Multi-Resolution WENO Schemes for Solving Hyperbolic Conservation Laws

High Order Finite Difference Multi-resolution WENO Method for Nonlinear Degenerate Parabolic Equations

In this paper, we propose a new finite difference weighted essentially non-oscillatory (WENO) scheme for nonlinear degenerate parabolic equations which may contain non-smooth solutions. An alternative formulation is designed to approximate the second derivatives in a conservative form. In this formulation, the odd order derivatives at half points are used to construct the numerical flux, instead of the usual practice of reconstruction. Moreover, the multi-resolution WENO scheme is designed to circumvent the negative ideal weights and mapped nonlinear weights that appear when applying the standard WENO idea. We will describe the scheme formulation and present numerical tests for one- and two-dimensional, demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

Coronary vessel segmentation using multiresolution and multiscale deep learning

We present a coronary vessel segmentation method for X-Ray coronary angiography images using multiresolution and multiscale deep learning. Our segmentation method constructs a set of multiresolution images from an input image via bilinear interpolation, which can handle coronary vessels with uneven distribution of contrast. We incorporate Multiresolution and Multiscale Convolution Filtering into an U-Net Network, which can help to improve accuracy of segmentation results by dealing with various thickness of coronary vessels in different positions. We investigate two types of experiments of multiresolution strategy with U-Net and multiscale strategy with U-Net, respectively. Our method has been evaluated and compared both qualitatively with networks such as single U-Net, Attention U-Net, R2U-Net and R2AttU-Net, and quantitatively with 20 state-of-the-art visual segmentation methods using a benchmark X-Ray coronary angiography database. The experiments demonstrate that our segmentation method outperforms methods using each of these networks alone and these 20 methods significantly in terms of Dice Coefficient metric, which is considered as a major evaluation criteria of segmentation results.