Anisotropic Regularization Research Articles

The anisotropic regularity criteria for 3D Navier–Stokes equations involving one velocity component

The anisotropic regularity criterion for the Navier–Stokes equations in terms of the one component of the velocity gradient (∂iuj,1≤i,j≤3) is investigated. It is an improvement of results of Qian (2016) and is also a complement of the results of Guo et al. (2018) for i=j. Besides, the consistent regularity criterion without distinguishing i=j and i≠j is also obtained. Finally, the regularity result based on the fractional derivatives for one component of the velocity field Dhαu3 with α∈[0,1] is also established, which can be reduced to some previous results with α=0 and α=1.

Splitting method for elliptic equations with line sources

In this paper, we study the mathematical structure and numerical approximation of elliptic problems posed in a (3D) domain Ω when the right-hand side is a (1D) line source Λ. The analysis and approximation of such problems is known to be non-standard as the line source causes the solution to be singular. Our main result is a splitting theorem for the solution; we show that the solution admits a split into an explicit, low regularity term capturing the singularity, and a high-regularity correction term w being the solution of a suitable elliptic equation. The splitting theorem states the mathematical structure of the solution; in particular, we find that the solution has anisotropic regularity. More precisely, the solution fails to belong to H1 in the neighbourhood of Λ, but exhibits piecewise H2-regularity parallel to Λ. The splitting theorem can further be used to formulate a numerical method in which the solution is approximated via its correction function w. This recasts the problem as a 3D elliptic problem with a 3D right-hand side belonging to L2, a problem for which the discretizations and solvers are readily available. Moreover, as w enjoys higher regularity than the full solution, this improves the approximation properties of the numerical method. We consider here the Galerkin finite element method, and show that the singularity subtraction then recovers optimal convergence rates on uniform meshes, i.e., without needing to refine the mesh around each line segment. The numerical method presented in this paper is therefore well-suited for applications involving a large number of line segments. We illustrate this by treating a dataset (consisting of ~3000 line segments) describing the vascular system of the brain.

A micromorphically regularized Cam-clay model for capturing size-dependent anisotropy of geomaterials

We introduce a regularized anisotropic modified Cam-clay (MCC) model which captures the size-dependent anisotropic elastoplastic responses for clay, mudstone, shales, and sedimentary rock. By homogenizing the multiscale anisotropic effects induced by clay particle aggregate, clusters, peds, micro-fabric, and mineral contact across length scales, we introduce two distinctive anisotropic mechanisms for the MCC model at the material point and mesoscale levels. We first employ a mapping that links the anisotropic stress state to a fictitious isotropic principal stress-space to introduce anisotropy at the material point scale. Then, the mesoscale anisotropy is introduced via an anisotropic regularization mechanism. This anisotropic regularization mechanism is triggered by introducing gradient-dependence of the internal variables through a penalty method such that the resultant gradient-enhanced plastic flow may exhibit anisotropic responses non-coaxial to the stress gradient of the yield function. The influence of the size-dependent anisotropy on the formation of the shear band and the macroscopic responses of the effective media are analyzed in 2D and 3D numerical examples.

A Flexible Space-Variant Anisotropic Regularization for Image Restoration with Automated Parameter Selection

We propose a new space-variant anisotropic regularization term for variational image restoration, based on the statistical assumption that the gradients of the target image distribute locally accor...

Damage detection in carbon fiber–reinforced polymer composite via electrical resistance tomography with Gaussian anisotropic regularization

Electrical resistance tomography is a method for sensing the spatial distribution of electrical conductivity. Therefore, this type of tomography is suitable for sensing damages, which affect electrical conductivity. The utilization of resistance tomography for the structural health monitoring of carbon fiber–reinforced polymer composites is questionable owing to its low spatial resolution and the strong anisotropy of carbon fiber–reinforced polymer composites. This article deals with the employment of resistance tomography with regularization based on a Gaussian anisotropic smoothing filter for the detection of cuts. The advantages of the filter are shown through the image reconstruction of rectangular composite specimens with three different laminate stacking sequences. The cuts are implemented by a milled groove. Visual comparison of the images shows a substantial improvement in the shape reconstruction ability. In addition to visual comparison, the image reconstructions are assessed in terms of the reconstruction error and cross-correlation.

Anisotropic regularity principle in sequence spaces and applications

We refine a recent technique introduced by Pellegrino, Santos, Serrano and Teixeira and prove a quite general anisotropic regularity principle in sequence spaces. As applications, we generalize previous results of several authors regarding Hardy–Littlewood inequalities for multilinear forms.

Uniformly moving non-singular dislocations with elliptical core shape in anisotropic media

To allow for “relativistic”-like core contraction effects, an anisotropic regularization of steadily moving straight dislocations of arbitrary orientation is introduced, with two scale parameters [Formula: see text] and [Formula: see text] along the direction of motion and transverse to it, respectively. The dislocation core shape is an ellipse. When [Formula: see text], the model reduces to the Peierls–Eshelby dislocation, the fields of which are non-differentiable on the slip plane. For finite [Formula: see text] and [Formula: see text], fields are everywhere differentiable. Applying the author’s so-called “causal” Stroh formalism to the model, explicit expressions for the regularized fields in anisotropic elasticity are derived for any velocity. For faster-than-wave velocities, Mach-cone angles are found insensitive to the ratio [Formula: see text], as must be. However, the larger [Formula: see text], the weaker the intensity of the cone branches. An expression is given for the radiative dissipative force opposed to motion. From this expression, it is inferred that the concept of a “radiation-free” intersonic velocity can, when not applicable, be replaced by that of a “least-radiation” velocity.

Lagrangian solutions to the Vlasov-Poisson system with a point charge

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [ 17 ], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [ 8 ] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.

Anomaly Detection for Hyperspectral Images Based on Anisotropic Spatial-Spectral Total Variation and Sparse Constraint

A novel anomaly detection method for hyperspectral images (HSIs) is proposed based on anisotropic spatial-spectral total variation and sparse constraint. HSIs are assumed to be not only smooth in spectral dimension but also piecewise smooth in spatial dimension. The proposed method adopts the anisotropic spatial-spectral total variation model which combines 2D spatial total variation and 1D spectral variation to explore the spatial-spectral smooth property of HSIs. Meanwhile, the sparse property of anomalies is exploited for its low probability in the image. To utilize both the spatial and spectral information of HSIs, we preserve the original cubic form of HSIs and divide the HSIs into three 3D arrays, each representing the background, the anomaly, and the noise respectively. By using anisotropic spatial-spectral total variation regularization on the background component and sparse constraint on the anomaly component, this anomaly detection problem has therefore been formulated as a constraint optimization problem whose solution has been derived by alternately using Split Bregman Method and Go Decomposition (GoDec) Method. Experimental results on hyperspectral datasets illustrate that our proposed method has a better detection performance than state-of-the-art hyperspectral anomaly detection methods.

Consistently Sampled Correlation Filters with Space Anisotropic Regularization for Visual Tracking.

Most existing correlation filter-based tracking algorithms, which use fixed patches and cyclic shifts as training and detection measures, assume that the training samples are reliable and ignore the inconsistencies between training samples and detection samples. We propose to construct and study a consistently sampled correlation filter with space anisotropic regularization (CSSAR) to solve these two problems simultaneously. Our approach constructs a spatiotemporally consistent sample strategy to alleviate the redundancies in training samples caused by the cyclical shifts, eliminate the inconsistencies between training samples and detection samples, and introduce space anisotropic regularization to constrain the correlation filter for alleviating drift caused by occlusion. Moreover, an optimization strategy based on the Gauss-Seidel method was developed for obtaining robust and efficient online learning. Both qualitative and quantitative evaluations demonstrate that our tracker outperforms state-of-the-art trackers in object tracking benchmarks (OTBs).

Localized Anisotropic Regularity Conditions for the Navier–Stokes Equations

We establish a sufficient regularity condition for local solutions of the Navier–Stokes equations. For a suitable weak solution (u, p) on a domain D we prove that if $$\partial _3 u$$ belongs to the space $$L_t^{s_0}L_x^{r_0}(D)$$ where $$2/s_0 + 3/r_0 \le 2 $$ and $$9/4 \le r_0\le 5/2$$ , then the solution is Holder continuous in D.

The anisotropic integrability regularity criterion to 3D magnetohydrodynamics equations

In this article, we establish sufficient conditions for the regularity of solutions of 3D MHD equations in the framework of the anisotropic Lebesgue spaces. In particular, we obtain the anisotropic regularity criterion via partial derivatives, and it is a generalization of the some previous results. Besides, the anisotropic integrability regularity criteria in terms of the magnetic field and the third component of the velocity field are also investigated. Copyright © 2017 John Wiley & Sons, Ltd.

Multi-frame real image restoration based on double loops with alternative maximum likelihood estimation

A multi-frame blind restoration algorithm based on alternative maximum likelihood estimation using double loops is proposed to restore object images. An anisotropic regularization term is incorporated into the integral likelihood function to avoid over-smoothing of details of the restored images and improve the stability of image restoration. In order to make full use of the data relationship in the observed multi-frame images which contain the same objects with different point spread functions (PSF) in a short sequence, an iterative strategy based on double loops is employed in estimating the unknown different PSFs of the observed images. Experiments on synthetic and real turbulence-degraded images illustrate that the proposed algorithm is effective in restoring real turbulence-degraded images by using very few frames.

Anisotropic Regularity Conditions for the Suitable Weak Solutions to the 3D Navier–Stokes Equations

We are concerned with the problem, originated from Seregin (159–200, 2007), Seregin (J. Math. Sci. 143: 2961–2968, 2007), Seregin (Russ. Math. Surv. 62:149–168, 2007), what are minimal sufficiently conditions for the regularity of suitable weak solutions to the 3D Navier–Stokes equations. We prove some interior regularity criteria, in terms of either one component of the velocity with sufficiently small local scaled norm and the rest part with bounded local scaled norm, or horizontal part of the vorticity with sufficiently small local scaled norm and the vertical part with bounded local scaled norm. It is also shown that only the smallness on the local scaled L 2 norm of horizontal gradient without any other condition on the vertical gradient can still ensure the regularity of suitable weak solutions. All these conclusions improve pervious results on the local scaled norm type regularity conditions.

Distortion correction of EPI data using multimodal nonrigid registration with an anisotropic regularization

In this paper, a novel strategy for correcting both geometric and image intensity distortions of echo-planar imaging (EPI) MRI data is presented. To deal with small local distortions caused by rapid changes of the magnetic field, an improved multimodal registration framework using normalized mutual information (NMI) in combination with a multi-scale technique is presented to estimate a dense displacement field. To ensure the robustness of this high dimensional ill-posed inverse problem, a novel anisotropic regularization functional is used.In order to quantify geometric distortions, a new quality measure, called standardized contour distance (SCD), is introduced. It uses the outer structure shape (OSS) information as basis for the evaluation.The new registration method was evaluated with one monomodal phantom data set and two multimodal human brain data sets (BrainSuite trainings data, SPM Subject data). By comparing with recent and efficient techniques of the state of the art, in the monomodal case, the new approach achieves results comparable to the sum of squared differences as data term. In the multimodal cases, our new registration strategy improves the mean of the SCD from 0.96±0.11 to 0.60±0.13 in case of the SPM Subject data and from 0.92±0.07 to 0.78±0.11 in case of the BrainSuite trainings data.

Three-dimensional anisotropic regularization for limited angle tomography

Abstract Limited angle tomography is a challenging task in medical imaging. Due to practical limitations during the image acquisition, the sinogram is recorded incompletely and thus the quality of the reconstruction is deteriorated by streak artifacts. These artifacts are characterized by fast changes of the local intensity gradients and increase the total variation (TV). Generally, an energy functional is optimized which leads to a minimized Total Variation Minimization (TVM). As an outcome, noise and artifacts are reduced while edges are preserved. Anyway, often the orientation of the streak artifacts is not considered at all. Therefore, anisotropic regularization is used to reduce noise and distortions under specific directions.

Global small solutions of 2-D incompressible MHD system

In this paper, we consider the global wellposedness of 2-D incompressible magneto-hydrodynamical system with smooth initial data which is close to some non-trivial steady state. It is a coupled system between the Navier–Stokes equations and a free transport equation with a universal nonlinear coupling structure. The main difficulty of the proof lies in exploring the dissipative mechanism of the system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the free transport equation, we first reformulate our system (1.1) in the Lagrangian coordinates (2.19). Then we employ anisotropic Littlewood–Paley analysis to establish the key a prioriL1(R+;Lip(R2)) estimate for the Lagrangian velocity field Yt. With this estimate, we can prove the global wellposedness of (2.19) with smooth and small initial data by using the energy method. We emphasize that the algebraic structure of (2.19) is crucial for the proofs to work. The global wellposedness of the original system (1.1) then follows by a suitable change of variables.

A fast deconvolution-based approach for single-image super-resolution with GPU acceleration

In this paper, we propose fast deconvolution-based image super-resolution (SR) with graphics processing unit (GPU)-accelerated computation. Recently, the deconvolution-based single-image SR has been proven to be very effective in upsampling images with favorable results. Based on the GPU-accelerated computation, we aim to realize the fast SR reconstruction and achieve balanceable performance in terms of both image quality and computational cost. To achieve this, we provide a novel and efficient deconvolution method to enhance the reconstruction results. We combine the gradient consistency in images with the anisotropic regularization which has been used in motion deblurring. Thus, we produce a directly parallelizable solution which is suitable for running on GPU by minimizing redundancy in computing. Experimental results demonstrate that the proposed method achieves superior performance in comparison with the existing methods with respect to image quality and runtime.

Lagrangian flows for vector fields with anisotropic regularity

We prove quantitative estimates for flows of vector fields subject to anisotropic regularity conditions: some derivatives of some components are (singular integrals of) measures, while the remaining derivatives are (singular integrals of) integrable functions. This is motivated by the regularity of the vector field in the Vlasov–Poisson equation with measure density. The proof exploits an anisotropic variant of the argument in [14,20] and suitable estimates for the difference quotients in such anisotropic context. In contrast to regularization methods, this approach gives quantitative estimates in terms of the given regularity bounds. From such estimates it is possible to recover the well posedness for the ordinary differential equation and for Lagrangian solutions to the continuity and transport equations.

Global Small Solutions to Three-Dimensional Incompressible Magnetohydrodynamical System

In this paper, we consider the global wellposedness of three-dimensional incompressible magneto-hydrodynamical (MHD) system with small and smooth initial data. The main difficulty of the proof lies in establishing the global in time $L^1$ estimate for gradient of the velocity field due to the strong degeneracy and anisotropic spectral properties of the linearized system. To achieve this and to avoid the difficulty of propagating anisotropic regularity for the transport equation, we first write our system in the Lagrangian formulation. Then we employ anisotropic Littlewood--Paley analysis to establish the key $L^1$ in time estimates to the velocity and the gradient of the pressure in the Lagrangian coordinate. With those estimates, we prove the global wellposedness of the Lagrangian formulation with smooth and small initial data by using the energy method. Toward this, we will have to use the algebraic structure of the Lagrangian formulation in a rather crucial way. The global wellposedness of the original system then follows by a suitable change of variables together with a continuous argument. We should point out that compared with the linearized systems of two-dimensional MHD equations of F. Lin, L. Xu, and P. Zhang [Global Small Solutions to 2-D MHD System, arXiv:1302.5877] and that of the three-dimensional modified MHD equations of F. Lin and P. Zhang [Comm. Pure Appl. Math., 67 (2014), pp. 531--580] our linearized system here is much more degenerate and moreover, the formulation of the initial data for the Lagrangian formulation is more subtle than that of F. Lin, L. Xu, and P. Zhang.