Nonlocal Constraints Research Articles

Lagrange multiplier approach to unilateral indentation problems: Well-posedness and application to linearized viscoelasticity with non-invertible constitutive response

The Boussinesq problem describing indentation of a rigid punch of arbitrary shape into a deformable solid body is studied within the context of a linear viscoelastic model. Due to the presence of a non-local integral constraint prescribing the total contact force, the unilateral indentation problem is formulated in the general form as a quasi-variational inequality with unknown indentation depth, and the Lagrange multiplier approach is applied to establish its well-posedness. The linear viscoelastic model that is considered assumes that the linearized strain is expressed by a material response function of the stress involving a Volterra convolution operator, thus the constitutive relation is not invertible. Since viscoelastic indentation problems may not be solvable in general, under the assumption of monotonically non-increasing contact area, the solution for linear viscoelasticity is constructed using the convolution for an increment of solutions from linearized elasticity. For the axisymmetric indentation of the viscoelastic half-space by a cone, based on the Papkovich–Neuber representation and Fourier–Bessel transform, a closed form analytical solution is constructed, which describes indentation testing within the holding-unloading phase.

On a two-dimensional fractional thermoelastic system with nonlocal constraints describing a fractional Kirchhoff plate

We show herein the existence and uniqueness of solutions for coupled fractional order partial differential equations modeling a thermoelastic fractional Kirchhoff plate model associated with initial, Dirichlet, and nonlocal boundary conditions involving fractional Caputo derivative. Some efficient results of existence and uniqueness are obtained by employing the energy inequality method.

Influence of a slow moving vehicle on traffic: Well-posedness and approximation for a mildly nonlocal model

In this paper, we propose a macroscopic model that describes the influence of a slow moving large vehicle on road traffic. The model consists of a scalar conservation law with a non-local constraint on the flux. The constraint level depends on the trajectory of the slower vehicle which is given by an ODE depending on the downstream traffic density. After proving well-posedness, we first build a finite volume scheme and prove its convergence, and then investigate numerically this model by performing a series of tests. In particular, the link with the limit local problem of [M.L. Delle Monache and P. Goatin, J. Differ. Equ. 257(11), 4015-4029 (2014)] is explored numerically.

Attainability property for a probabilistic target in wasserstein spaces

In this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called \emph{multiagent system} made of several possible interacting agents.

Learning an SAR Image Despeckling Model Via Weighted Sparse Representation

Synthetic aperture radar (SAR) images are inherently degraded by the speckle noise due to the coherent imaging, which may affect the performance of subsequent image analysis task. To address this problem, a weighted sparse representation-based method is proposed in this article for SAR image despeckling. The homomorphic transformation is first adopted to convert multiplicative noise into additive one. Second, similar patches are grouped together to learn the adaptive dictionaries and sparse coefficients based on nonlocal self-similarity constraint. Moreover, weighted regularizations are adopted for coefficients to boost the performance. Finally, despeckling images are obtained via exponential transformation. Experimental results on synthetic and real-world SAR images demonstrate that our proposed method outperforms several state-of-the-art methods in terms of both quantitative measurements and visual quality.

Learning Interactions of Local and Non-Local Phonotactic Constraints from Positive Input

Learning Interactions of Local and Non-Local Phonotactic Constraints from Positive Input

Video Super-Resolution Reconstruction Based on Deep Learning and Spatio-Temporal Feature Self-Similarity

To address the problems in the existing video super-resolution methods, such as noise, over smooth and visual artifacts, which are caused by the reliance on limited external training or mismatch of internal similarity patch instances, this study proposes a novel video super-resolution reconstruction algorithm based on deep learning and spatio-temporal feature similarity (DLSS-VSR). The video super-resolution reconstruction mechanism with the joint internal and external constraints is established utilizing the complementary advantages of both external deep correlation mapping learning and internal spatio-temporal nonlocal self-similarity prior constraint. A deep learning model based on deep convolutional neural network is constructed to learn the nonlinear correlation mapping between low-resolution and high-resolution video frame patches. A novel spatio-temporal feature similarity calculation method is proposed, which considers both internal video spatio-temporal self-similarity and external clean nonlocal similarity. For the internal spatio-temporal feature self-similarity, we improve the accuracy and robustness of similarity matching by proposing a similarity measure strategy based on spatio-temporal moment feature similarity and structural similarity. The external nonlocal similarity prior constraint is learned by the patch group-based Gaussian mixture model. The time efficiency for spatio-temporal similarity matching is further improved based on saliency detection and region correlation judgment strategy, which achieves a better tradeoff between super-resolution accuracy and speed. Experimental results demonstrate that the DLSS-VSR algorithm achieves competitive super-resolution quality compared to other state-of-the-art algorithms in both subjective and objective evaluations.

Formulation, analysis and computation of an optimization-based local-to-nonlocal coupling method

We present an optimization-based coupling method for local and nonlocal continuum models. Our approach couches the coupling of the models into a control problem where the states are the solutions of the nonlocal and local equations, the objective is to minimize their mismatch on the overlap of the local and nonlocal problem domains, and the virtual controls are the nonlocal volume constraint and the local boundary condition. We present the method in the context of Local-to-Nonlocal diffusion coupling. Numerical examples illustrate the theoretical properties of the approach.

FusionNDVI: A Computational Fusion Approach for High-Resolution Normalized Difference Vegetation Index

Normalized difference vegetation index (NDVI), derived from the near-infrared and red bands of a multispectral (MS) image, has been widely used in remote sensing. To obtain a high-resolution (HR) NDVI, existing attempts typically first generate an HR-MS image using pansharpening and then calculate the HR NDVI accordingly. However, some inaccurate spatial information will be simultaneously introduced into NDVIs, influencing their spatial quality seriously. To overcome this challenge, we investigate a computational fusion approach from a novel perspective for HR NDVI in this study. Rather than pansharpening an HR-MS image, we define an HR vegetation index calculated based on an available HR panchromatic image and an estimated HR red band (VIPR) and fuse the low-resolution (LR) NDVI and HR VIPR directly to acquire an HR NDVI. In particular, we adopt a nonlocal gradient sparsity constraint to force a similar nonlocal spatial structure in the fused NDVI and VIPR, where the VIPR is dynamically updated by adding a constraint to reconstruct the HR red band. We further integrate a data fidelity term to constrain the relationship between the fused NDVI and its LR version, and an efficient strategy based on the alternative direction multiplier method is developed to solve the nonconvex optimization problem. The extensive experimental results demonstrate that the proposed method achieves superior fusion performance over the state of the art, exhibiting its wide application aspect in remote sensing.

Nonlocal Band-Weighted Iterative Spectral Mixture Model for Hyperspectral Imagery Denoising

Although efficient hyperspectral image (HSI) denoising relies on complete and accurate description and modeling the spatial-spectral signal in HSI, the current approaches do not fully account for key characteristics of HSI, i.e., the mixed spectra effect, the spatial nonstationarity effect, and noise variance heterogeneity effect. To address this issue, this article presents a linear spectral mixture model with nonlocal means constraint (LSMM-NLMC), with the following advantages. First, LSMM-NLMC can effectively learn the signal in mixed pixels in HSI by estimating clean endmembers and abundances for image restoration. Second, LSMM-NLMC can efficiently address nonstationary spatial correlation effect by imposing NLMC on the latent scene signal. Last, LSMM-NLMC provides accurate noise characterization by accounting for noise variance heterogeneity effect using a band-dependent noise model and a band-weighted Mahalanobis distance for similarity measurement. A novel optimization method based on the expectation-maximization (EM) algorithm and the purified means approach is used to efficiently solve the resulting maximum a posterior (MAP) problem. The experiments on both simulated and real HSI data sets demonstrate that the visual quality and denoising accuracy are significantly improved by the proposed LSMM-NLMC compared with previous methods.

Stationary solutions of advective Lotka–Volterra models with a weak Allee effect and large diffusion

This paper is devoted to the Neumann problem of a stationary Lotka–Volterra model with diffusion and advection. In the model it is assumed that one population growth rate is described by weak Allee effect. We first obtain some sufficient conditions ensuring the existence of nonconstant solutions by using the Leray–Schauder degree theory. And then we study a limiting system (with nonlocal constraint) which stems from the original model as diffusion and advection of one of the species tend to infinity. Finally, we classify the global bifurcation structure of nonconstant solutions of the simplified 1D case.

Optimal design problems governed by the nonlocal p -Laplacian equation

<p style='text-indent:20px;'>In the present work, a nonlocal optimal design model has been considered as an approximation of the corresponding classical or local optimal design problem. The new model is driven by the nonlocal <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian equation, the design is the diffusion coefficient and the cost functional belongs to a broad class of nonlocal functional integrals. The purpose of this paper is to prove the existence of an optimal design for the new model. This work is complemented by showing that the limit of the nonlocal <inline-formula><tex-math id="M3">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian state equation converges towards the corresponding local problem. Also, as in the paper by F. Andrés and J. Muñoz [J. Math. Anal. Appl. 429:288– 310], the convergence of the nonlocal optimal design problem toward the local version is studied. This task is successfully performed in two different cases: when the cost to minimize is the compliance functional, and when an additional nonlocal constraint on the design is assumed.

Hyperspectral and Multispectral Image Fusion via Nonlocal Low-Rank Tensor Approximation and Sparse Representation

The fusion of hyperspectral (HS) and multispectral (MS) images designed to obtain high-resolution HS (HRHS) images is a very challenging work. A series of solutions has been proposed in recent years. However, the similarity in the structure of the HS image has not been fully used. In this article, we present a novel HS and MS image-fusion method based on nonlocal low-rank tensor approximation and sparse representation. Specifically, the HS image and the MS image are considered the spatially and spectrally degraded versions of the HRHS image, respectively. Then, the nonlocal low-rank constraint term is adopted in order to form the nonlocal similarity and the spatial-spectral correlation. Meanwhile, we add the sparse constraint term to describe the sparsity of abundance. Thus, the proposed fusion model is established and its optimization is solved by alternative direction method of multipliers (ADMM). The experimental results on three synthetic data sets and one real data set show the advantages of the proposed method over several state-of-the-art competitors.

Local and nonlocal constraints for compressed sensing video and multi-view image recovery

In compressed sensing (CS) signal recovery tasks, employing prior constraints to exploit the inherent properties in the signal is critical to the recovery performance. Recently, multiple constraints were employed to improve the reconstruction for video sequence and multi-view image CS reconstruction since they can exploit intra- and inter-image properties simultaneously. However, correlation and dependencies among constraints were not taken into account in existing methods. This drawback prevents the multi-constraint technique from improving the reconstruction performance further. In this paper, we propose to employ local and nonlocal constraints in the CS recovery model to reconstruct video sequences and multi-view images. In the proposed CS recovery model, a pixel-wise total variation constraint is utilized to exploit the spatial local piecewise smoothness in the image. Meanwhile, a patch-based low-rank constraint is applied for the nonlocal inter-image similarity property. These two types of constraints are proposed to be applied in different dimensions with different granularities so as to alleviate the dependency and correlation between the constraints and exploit the intra- and inter-image features in images sufficiently. Besides, an efficient augmented Lagrange multipliers and alternating direction method is presented to solve the proposed local and nonlocal constraint-based CS recovery model (LNLC-CS). Experiments on test video sequences and multi-view images at various sampling rates have demonstrated the great advantages of applying independent constraints in different granularities and dimensions. The proposed method can outperform the existing state-of-the-art CS recovery approaches a lot in both the objective and the subjective quality of the recovered image. Experiments also show the proposed method has made a remarkable improvement over other competing constraint-based methods in the convergence speed and reconstruction time.

Double-Constraint Inpainting Model of a Single-Depth Image.

In real applications, obtained depth images are incomplete; therefore, depth image inpainting is studied here. A novel model that is characterised by both a low-rank structure and nonlocal self-similarity is proposed. As a double constraint, the low-rank structure and nonlocal self-similarity can fully exploit the features of single-depth images to complete the inpainting task. First, according to the characteristics of pixel values, we divide the image into blocks, and similar block groups and three-dimensional arrangements are then formed. Then, the variable splitting technique is applied to effectively divide the inpainting problem into the sub-problems of the low-rank constraint and nonlocal self-similarity constraint. Finally, different strategies are used to solve different sub-problems, resulting in greater reliability. Experiments show that the proposed algorithm attains state-of-the-art performance.

Single Remote Sensing Image Super-Resolution with an Adaptive Joint Constraint Model.

Remote sensing images have been widely used in many applications. However, the resolution of the obtained remote sensing images may not meet the increasing demands for some applications. In general, the sparse representation-based super-resolution (SR) method is one of the most popular methods to solve this issue. However, traditional sparse representation SR methods do not fully exploit the complementary constraints of images. Therefore, they cannot accurately reconstruct the unknown HR images. To address this issue, we propose a novel adaptive joint constraint (AJC) based on sparse representation for the single remote sensing image SR. First, we construct a nonlocal constraint by using the nonlocal self-similarity. Second, we propose a local structure filter according to the local gradient of the image and then construct a local constraint. Next, the nonlocal and local constraints are introduced into the sparse representation-based SR framework. Finally, the parameters of the joint constraint model are selected adaptively according to the level of image noise. We utilize the alternate iteration algorithm to tackle the minimization problem in AJC. Experimental results show that the proposed method achieves good SR performance in preserving image details and significantly improves the objective evaluation indices.

Combining synthesis sparse with analysis sparse for single image super-resolution

Sparse coding based-methods show great effectiveness in single image super-resolution (SR). Existing methods generally use only synthesis sparse coding. However, the analysis sparse coding model, which is an alternative to the synthesis sparse coding model, is typically neglected In this paper, we propose a novel single image SR method by combining synthesis sparse coding with analysis sparse coding. In contrast to the existing sparse coding-based SR methods, we replace synthesis sparse coding with analysis sparse coding in the low-resolution (LR) feature representation phase. Thus, we use an analysis dictionary to obtain the LR coefficients and use a synthesis dictionary to reconstruct the high-resolution (HR) patches. Due to the introducing of analysis sparse representation, the l0 or l1 optimization is replaced by soft threshold shrinkage, which is a more time-saving method. To improve the convergence of the training model, we introduce a linear mapping function that reveals the relationship between the HR coefficients and LR coefficients. An alternating optimization strategy is adopted to solve the improved training model. Furthermore, global and nonlocal constraints are taken into account to improve the quality of the reconstructed images. Compared with some existing SR methods, the experimental results demonstrate that our proposed method can obtain better performance.

Medical image restoration method via multiple nonlocal prior constraints

Medical image restoration method via multiple nonlocal prior constraints

TNLRS: Target-Aware Non-local Low-Rank Modeling with Saliency Filtering Regularization for Infrared Small Target Detection.

Recently, infrared small target detection problem has attracted substantial attention. Many works based on local low-rank model have been proven to be very successful for enhancing the discriminability during detection. However, these methods construct patches by traversing local images and ignore the correlations among different patches. Although the calculation is simplified, some texture information of the target is ignored, and targets of arbitrary forms cannot be accurately identified. In this paper, a novel target-aware method based on a non-local low-rank model and saliency filter regularization is proposed, with which the newly proposed detection framework can be tailored as a non-convex optimization problem, therein enabling joint target saliency learning in a lower dimensional discriminative manifold. More specifically, non-local patch construction is applied for the proposed target-aware low-rank model. By combining similar patches, we reconstruct them together to achieve a better generalization of non-local spatial sparsity constraints. Furthermore, to encourage target saliency learning, our proposed saliency filtering regularization term based on entropy is restricted to lie between the background and foreground. The regularization of the saliency filtering locally preserves the contexts from the target and surrounding areas and avoids the deviated approximation of the low-rank matrix. Finally, a unified optimization framework is proposed and solved with the alternative direction multiplier method (ADMM). Experimental evaluations of real infrared images demonstrate that the proposed method is more robust under different complex scenes compared with some state-of-the-art methods.

Homotopy Analysis Method for a Fractional Order Equation with Dirichlet and Non-Local Integral Conditions

The main purpose of this paper is to obtain some numerical results via the homotopy analysis method for an initial-boundary value problem for a fractional order diffusion equation with a non-local constraint of integral type. Some examples are provided to illustrate the efficiency of the homotopy analysis method (HAM) in solving non-local time-fractional order initial-boundary value problems. We also give some improvements for the proof of the existence and uniqueness of the solution in a fractional Sobolev space.