Nonlocal Regularization Research Articles

Numerical assessment of the time-dependent behaviour of a tunnel with a yielding support system for a potential repository in claystone

For the disposal of heat-generating radioactive waste and spent fuel in claystone formations, support structures for underground excavations are essential for the operational safety of geological disposal facilities (GDF). Claystone formations of moderate strength, at considerable depths, are characterised by their squeezing, creeping, and, sometimes, swelling behaviour that results in continuous tunnel convergence over time. Under these conditions, a rigid support system can be subjected to very high loads, requiring a considerable thickness and/or high-performance concretes. Therefore, yielding support systems are being currently investigated as a promising alternative for GDF. This work presents a numerical study of the behaviour of a tunnel in a claystone formation with a yielding support system. The use of a compressible mortar between the rock and the lining is assumed, as a means of reducing loads and mitigating the effects of creep deformations. A key aspect of the analyses is that the host rock is characterised by a constitutive model that includes a number of features that are relevant for the satisfactory description of the hydro-mechanical behaviour of stiff clayey materials, such as mechanical anisotropy, creep, strain softening, and its ability to simulate localised deformations through a nonlocal regularisation. An elastoplastic constitutive model was also developed to represent the behaviour of the compressible mortar. Results provide relevant insights into the performance of the adopted yielding support system, particularly regarding the effect of time-dependent deformations and the additional relaxation of the rock on the fractured zone near the excavation.

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Low-rank tensor completion via nonlocal self-similarity regularization and orthogonal transformed tensor Schatten-p norm

Evaluation of Three Weight Functions for Nonlocal Regularization of Sand Models

Evaluation of Three Weight Functions for Nonlocal Regularization of Sand Models

The use of a nonlocal critical state model in modelling triaxial and plane strain tests on overconsolidated clays

When modelling the phenomenon of strain localisation in strain-softening soils with the finite element (FE) method, nonlocal approaches have been commonly employed to avoid mesh dependency and numerical instability. This paper first presents the FE formulation of a critical state model for highly overconsolidated clays incorporating a nonlocal method. The performance of the nonlocal strain regularisation is subsequently assessed through a series of coupled hydro-mechanical (HM) analyses of undrained and drained triaxial compression tests on London clay. The mechanism behind the evolution of strain localisation in triaxial tests is investigated and a comparison with equivalent plane strain analyses is discussed. Finally, a comprehensive sensitivity study is presented, investigating the influence of the two nonlocal parameters, in the adopted nonlocal algorithm, on the predicted stress–strain responses. A key outcome is the derived linear relationship between the two parameters, which enables a unique stress–strain response to be achieved in either axisymmetric or plane strain analyses with multiple combinations of the two parameters. Such a modelling capability is essential in applications of the proposed nonlocal strain regularisation in large scale boundary value problems in which restrictions on element size exist.

Multispectral and hyperspectral images fusion based on subspace representation and nonlocal low-rank regularization

ABSTRACT Multispectral image (MSI) and hyperspectral image (HSI) fusion is a popular method for HSI super-resolution reconstruction (HSI-SR). MSI-HSI fusion problem is ill-posed and demands several image priors or regularization terms to solve accurately, which is a challenging issue. In this paper, we propose an MSI-HSI fusion model via subspace representation and nonlocal low-rank regularization (SRNLRR). SRNLRR model incorporates the global spectral correlations and spatial nonlocal similarities of HSI to improve the fusion results, where the priors complement each other. First, we use the mode-n tensor-matrix product to project latent high spatial resolution HSI (HR-HSI) into spectral subspace, which can capture spectral low-rankness and reduce computational complexity. Then, based on low-rank representation (LRR) and nonlocal processing strategy, we design a spatial nonlocal LRR regularization (spa-NLRR) and a spectral global LRR regularization (spe-GLRR). These two regularizations analyze the spatial nonlocal similarities and spectral global correlations from intermediate-level vision. Finally, we use the residual regularization program to obtain more image information and input it into the fusion model. We use alternate minimization (AM) methods to optimize the SRNLRR model and employ the alternating direction method (ADM) on spatial/spectral LRR learning. Comparison experiments between the SRNLRR method and six advanced methods on three HSI datasets indicate the superiority of our method.

Model test and numerical analysis of shield tunnel face instability in sandy soil with strain-softening properties

During excavation of a shield tunnel in a sandy stratum, variations in the state of the sand affect the stratum stability of the shield tunnel face. When dense sand is loaded, its strength decreases gradually owing to dilatancy and it exhibits strain-softening behavior. Model tests and finite element numerical simulations were performed to clarify the failure characteristics of the tunnel face induced by strain-softening. In the model tests, a dense sand stratum was constructed; the movement of the supporting plate of the excavation face was controlled to simulate the progressive failure of the shield tunnel face. Based on a material state-dependent plasticity model, a numerical analysis model of the tunnel face was established to capture the strain-softening behavior of dense sand by including a nonlocal regularization technique. Comparing the model test and numerical simulation results, the evolution of the material state of the sand with progressive failure of the tunnel face was analyzed, and the failure characteristics caused by the strain-softening of sand were revealed.

Tensor robust PCA with nonconvex and nonlocal regularization

Tensor robust principal component analysis (TRPCA) is a classical way for low-rank tensor recovery, which minimizes the convex surrogate of tensor rank by shrinking each tensor singular value equally. However, for real-world visual data, large singular values represent more significant information than small singular values. In this paper, we propose a nonconvex TRPCA (N-TRPCA) model based on the tensor adjustable logarithmic norm. Unlike TRPCA, our N-TRPCA can adaptively shrink small singular values more and shrink large singular values less. In addition, TRPCA assumes that the whole data tensor is of low rank. This assumption is hardly satisfied in practice for natural visual data, restricting the capability of TRPCA to recover the edges and texture details from noisy images and videos. To this end, we integrate nonlocal self-similarity into N-TRPCA, and further develop a nonconvex and nonlocal TRPCA (NN-TRPCA) model. Specifically, similar nonlocal patches are grouped as a tensor and then each group tensor is recovered by our N-TRPCA. Since the patches in one group are highly correlated, all group tensors have strong low-rank property, leading to an improvement of recovery performance. Experimental results demonstrate that the proposed NN-TRPCA outperforms existing TRPCA methods in visual data recovery. The demo code is available at https://github.com/qguo2010/NN-TRPCA.

Prospectively accelerated dynamic speech magnetic resonance imaging at 3 T using a self-navigated spiral-based manifold regularized scheme.

This work develops and evaluates a self-navigated variable density spiral (VDS)-based manifold regularization scheme to prospectively improve dynamic speech magnetic resonance imaging (MRI) at 3 T. Short readout duration spirals (1.3-ms long) were used to minimize sensitivity to off-resonance. A custom 16-channel speech coil was used for improved parallel imaging of vocal tract structures. The manifold model leveraged similarities between frames sharing similar vocal tract postures without explicit motion binning. The self-navigating capability of VDS was leveraged to learn the Laplacian structure of the manifold. Reconstruction was posed as a sensitivity-encoding-based nonlocal soft-weighted temporal regularization scheme. Our approach was compared with view-sharing, low-rank, temporal finite difference, extra dimension-based sparsity reconstruction constraints. Undersampling experiments were conducted on five volunteers performing repetitive and arbitrary speaking tasks at different speaking rates. Quantitative evaluation in terms of mean square error over moving edges was performed in a retrospective undersampling experiment on one volunteer. For prospective undersampling, blinded image quality evaluation in the categories of alias artifacts, spatial blurring, and temporal blurring was performed by three experts in voice research. Region of interest analysis at articulator boundaries was performed in both experiments to assess articulatory motion. Improved performance with manifold reconstruction constraints was observed over existing constraints. With prospective undersampling, a spatial resolution of 2.4 × 2.4 mm2/pixel and a temporal resolution of 17.4 ms/frame for single-slice imaging, and 52.2 ms/frame for concurrent three-slice imaging, were achieved. We demonstrated implicit motion binning by analyzing the mechanics of the Laplacian matrix. Manifold regularization demonstrated superior image quality scores in reducing spatial and temporal blurring compared with all other reconstruction constraints. While it exhibited faint (nonsignificant) alias artifacts that were similar to temporal finite difference, it provided statistically significant improvements compared with the other constraints. In conclusion, the self-navigated manifold regularized scheme enabled robust high spatiotemporal resolution dynamic speech MRI at 3 T.

Bilevel optimal parameter learning for a high-order nonlocal multiframe super-resolution problem

This work elaborated an improved method to multiframe super-resolution (SR), which involves a nonlocal first-order regularization combined with a nonlocal p-Laplacian term. The nonlocal TV term excels at edge preserving, whilst the nonlocal p-Laplacian is commonly used to perfectly reconstruct image textures. Firstly, we discuss the existence and uniqueness of a solution to our new model in a well posed framework. Then, we derive a modified Primal-dual iteration to compute the super-resolved solution. Furthermore, we introduce a new bilevel optimization approach to learn two regularization parameters. The included tests validate that the introduced optimization procedure performs favorably compared to numerous SR approaches in terms of efficiency and accuracy.

Accelerated Bayesian Reciprocal LASSO

Bayesian reciprocal LASSO (BRL) is a recently proposed nonlocal regularization method for Bayesian linear regression models. This paper develops a modified version of BRL, accommodating faster posterior sampling than published methods, by bypassing the use of auxiliary latent variables. We present a slice-within-Gibbs algorithm based on the elliptical slice sampler that matches the predictive accuracy of previous BRL implementations. Simulation studies and real data analyses show that the new method (XBRL) outperforms its Bayesian cousin (BRL) in out-of-sample prediction across a wide range of scenarios while offering the advantage of faster posterior computation. We have implemented the XBRL algorithm as part of the R package BayesRecipe available at: https://github.com/himelmallick/BayesRecipe.

Numerical modelling of shield tunnel face failure through a critical state sand plasticity model with nonlocal regularization

During shield tunneling in sand strata, variations in the state of sand can lead to localized instability and strength loss, which brings about numerical difficulties. This paper numerically investigates how nucleation and propagation of plastic deformation zones, including those leading to localized banding, possibly promote failure of shield tunnel faces excavated in sand. To this end, a material state-dependent model is used to capture the mechanical responses of sand. Nonlocal enhancement is introduced, through the volumetric strain increment that drives the change of void ratio and related material hardening/softening behavior, to regularize ill-posed boundary value problems caused by the activation of strain localization. Simulations of tunnel face failure indicate that lower initial void ratios result in more concentrated plastic deformation in the vicinity region near the face and a strain-softening trend in global deformation responses. Conversely, initial looser states drive the deformation zone to propagate spatially to the ground surface, and the resultant global deformation responses exhibit strain-hardening. The second-order work is used to explain the link between material instability around the tunnel faces and initial density. It was found that material instability usually occupies a fraction of plastic deformation regions, and shrinks as sand initial density increases.

Hierarchical graph augmented stacked autoencoders for multi-view representation learning

With recent success of deep neural networks, stacked autoencoder networks have received a lot of attention for robust unsupervised representation learning. However, recent autoencoder methods cannot make full use of multi-view information and thus fail to further improve many real-world applications by exploring the geometric structures of multi-view data. To address the above-mentioned issue, we introduce hierarchical graph augmented stacked autoencoders (HGSAE) for unsupervised multi-view representation learning. Specifically, a hierarchical graph structure is first adapted to stacked autoencoders to learn view-specific representations, aiming to preserve the geometric information of multi-view data through local and non-local graph regularizations. A general or common representation can then be learned by reconstructing each single view using fully connected neural networks. By doing this, the proposed method not only preserves the geometric information in multi-view data but also automatically balances the complementarity/consistency among different views. Extensive experiments on six popular unsupervised representation learning datasets demonstrate the effectiveness of our method when compared with recent state-of-the-art autoencoder methods.

Strain regularisation using a non-local method in Coupled Eulerian-Lagrangian analyses

Conventional finite element analyses generally suffer from mesh dependency when considering softening effects. Non-local strain regularisation techniques have been developed to address this issue, which are generally complex to implement. In view of this, this paper introduces an more efficient procedure for implementation of a non-local method in Abaqus for Coupled Eulerian-Lagrangian (CEL) large deformation finite element undrained analyses. Results from a series of biaxial compression simulations demonstrate that the non-local method with a strain-softening Tresca model in CEL avoids mesh dependency. Owing to the stationary Eulerian element and the built-in mapping algorithm in CEL, high computational efficiency is achieved, adding no more than 12% to the computational cost. Guidance is provided on selection of internal length scales, element sizes and the search radius to ensure efficient non-local calculations, and it is shown that a softening scaling rule can also be used to allow use of practical mesh densities for some boundary value problems. Simulations of a number of classical geotechnical problems demonstrate the type of boundary value problems where the non-local method can effectively mitigate the mesh dependency, whilst also highlighting that the method fails to control the strain localisation that develops at soil-structure interfaces due to geometrical issues.

Deep nonlocal low-rank regularization for complex-domain pixel super-resolution.

Pixel super-resolution (PSR) has emerged as a promising technique to break the sampling limit for phase imaging systems. However, due to the inherent nonconvexity of phase retrieval problem and super-resolution process, PSR algorithms are sensitive to noise, leading to reconstruction quality inevitably deteriorating. Following the plug-and-play framework, we introduce the nonlocal low-rank (NLR) regularization for accurate and robust PSR, achieving a state-of-the-art performance. Inspired by the NLR prior, we further develop the complex-domain nonlocal low-rank network (CNLNet) regularization to perform nonlocal similarity matching and low-rank approximation in the deep feature domain rather than the spatial domain of conventional NLR. Through visual and quantitative comparisons, CNLNet-based reconstruction shows an average 1.4 dB PSNR improvement over conventional NLR, outperforming existing algorithms under various scenarios.

Long-time convergence of a nonlocal Burgers’ equation towards the local N-wave

We study the long-time behaviour of the unique weak solution of a nonlocal regularisation of the (inviscid) Burgers equation where the velocity is approximated by a one-sided convolution with an exponential kernel. The initial datum is assumed to be positive, bounded, and integrable. The asymptotic profile is given by the ‘N-wave’ entropy solution of the Burgers equation. The key ingredients of the proof are a suitable scaling argument and a nonlocal Oleinik-type estimate.

Quarter match non-local means algorithm for noise removal

The notion of improving plays out in many forms in our lives. We look for better quality, faster speed, and leisurelier connections. To achieve our desired goals, we must ask questions. How to make a process stronger? How to make a process more efficient? How to make a process more effective? Image denoising plays a vital role in many professions and understanding how noise can be present in images has led to multiple denoising techniques. These techniques include total variation regularization, non-local regularization, sparse representation, and low-rank minimization just to name a few. Many of these techniques exist because of the concept of improvement. First, we have a change (problem). This change invokes thoughts and questions. How these changes occur and how they are handled play an essential role in the realization or malfunction of that process. With this understanding, first, we look to fully understand the process to achieve success. As it relates to image denoising, the non-local means is incredibly effective in image reconstruction. In particular, the non-local means filter removes noise and sharpens edges without losing too many fine structures and details. Also, the non-local means algorithm is amazingly accurate. Consequently, the disadvantage that plagues the non-local means filtering algorithm is the computational burden and it is due to the non-local averaging. In this paper, we investigate innovative ways to reduce the computational burden and enhance the effectiveness of this filtering process. Research examining image analysis shows there is a battle between noise reduction and the preservation of actual features, which makes the reduction of noise a difficult task. For exploration, we propose a quarter-match non-local means denoising filtering algorithm. The filters help to classify a more concentrated region in the image and thereby enhance the computational efficiency of the existing non-local means denoising methods and produce an enriched comparison for overlying in the restoration process. To survey the constructs of this new algorithm, the authors use the original non-local means filtering algorithm, which is coined, “State of the Art” and other selective processes to test the effectiveness and efficiency of the new model. When comparing the original non-local means with the new quarter match filtering algorithm, on average, we can reduce the computational cost by half, while improving the quality of the image. To further test our new algorithm, medical resonance (MR) and synthetic aperture radar (SAR) images are used as specimens for real-world applications.

Multiplicative noise removal using complementary total generalized variation and nonlocal low rank regularization

The problem of multiplicative image denoising is a typical ill-posed inverse problem which has been extensively studied in the past few decades and is still an open problem. Exploiting appropriate image priors is critical to solve ill-posed inverse image restoration problem. The existing total generalized variation (TGV) based approaches have achieved significant performance in removing the multiplicative noise. However, these TGV regularized approaches only consider the local smoothness prior of the potential clean image, which limits their ability for image restoration. Other than the local smoothness property, the nonlocal self-similarity prior of images can also be exploited to offer helpful remedy for better image reconstruction. In this paper, to simultaneously make use of both the local smoothness prior and nonlocal self-similarity prior, we integrate the TGV regularizer with the nonlocal low rank regularizer to propose a novel variational model for multiplicative image denoising. Specifically, in the objective function, the nonlocal low rank regularization is expressed by the celebrated weighted nuclear norm minimization (WNNM) which has recently been applied to image denoising and presents state-of-art result. Then, we apply the split Bregman algorithm to efficiently solve the resulting model. Numerical experiments demonstrate that the proposed method achieve superior performance over the competing methods, including the state-of-the-art SAR-BM3D method.

Unsupervised learning spectral functions with neural networks

Reconstructing spectral functions from Euclidean Green’s functions is an ill-posed inverse problem that is crucial for understanding the properties of many-body systems. In this proceeding, we propose an automatic differentiation (AD) framework utilizing neural network representations for spectral reconstruction from propagator observables. We construct spectral functions using neural networks and optimize the network parameters unsupervisedly based on the reconstruction error of the propagator. Compared to the maximum entropy method, the AD framework demonstrates better performance in situations with high noise levels. It is noteworthy that neural network representations provide non-local regularization, which has the potential to significantly improve the solution of inverse problems.

Nonlocal implicit gradient enhancements for strain localization informed by controllability criteria for plastic solids

Nonlocal implicit gradient enhancements are widely used to suppress mesh dependency in simulations involving strain localization. For example, nonlocality is often introduced through internal variables that account for possible material softening. This work, however, shows that this approach may become ineffective if the solid displays plastic non-normality (i.e., non-associated plastic flow). For this purpose, we consider an over-nonlocal formulation and mathematically inspect the conditions at which regularization is lost in the presence of plastic non-normality. Specifically, such loss of regularization is linked to the loss of uniqueness and/or existence of the incremental plastic response that is kinematically compatible with the development of a deformation band. By doing so, we find a lower limit for the admissibility of the parameters controlling the effectiveness of nonlocal implicit gradient regularization. Furthermore, we show that such a lower limit is regulated by a plastic modulus reflecting the loss of controllability of the constitutive response, and, hence, depends on the degree of plastic non-normality. We also derive a closed-form expression relating the thickness of the deformation band to both the controllability modulus and gradient regularization constants, which suggests that the thickness of the process zone may change in response to the prevailing plastic flow characteristics and evolve during active plastic deformation. The proposed nonlocal enhancement is applied to a non-associated elasto-plastic model for porous sedimentary rocks, which is capable of displaying both shear-dominated and compaction-dominated bands. Numerical simulations reveal that effective regularization can be enforced only when the over-nonlocal weighting coefficient is larger than the above-mentioned lower limit.

Comparison of local and nonlocal regularization approaches in Eulerian-based finite element analyses

Finite element (FE) results might suffer from significant mesh dependency, especially when modeling shear bands in strain-softening soils. On the other hand, the shear bands in the field are generally thin. The computational costs might dramatically increase when modeling such thin shear bands, especially for large-scale problems, such as progressive landslides. To overcome these issues, FE analyses are generally performed with larger shear band thicknesses by adopting ‘softening scaling’ rules based on local and nonlocal (averaged) strains at the shear bands. The present study compares the performances of local and nonlocal approaches, with a specific focus on softening scaling and large deformation. Analyses are performed using a Eulerian-based FE program, which can model large strains in the shear band without numerical issues related to mesh distortion. Implementing strain-softening behavior in local and nonlocal methods, two idealized cases are simulated: (i) biaxial compression test, (ii) slope failure due to upslope surface loading. FE simulations show that the macroscopic response (i.e., load–displacement behavior) can be modeled using both local and nonlocal regularization techniques. Shear band thickness increases with the progress of shearing. In local analysis, mesh orientation has a considerable effect on shear band thickness. The existence of neighboring shear bands could affect nonlocal strain calculation and the simulation results.