Structure Tensor Research Articles

Schouten-Codazzi Gravity

Abstract We propose a new phenomenological second order gravity theory to be denoted as ''Schouten-Codazzi' Gravity'' (SCG), as it is based on Schouten and Codazzi tensors. The theory is related, but is clearly distinct from Cotton Gravity. By assuming as source the energy momentum of General Relativity, we form a second order system with its geometric sector given by the sum of the Schouten tensor and a generic second order symmetric tensor complying with the following properties: (i) it must satisfy the Codazzi differential condition and (ii) it must be concomitant with the invariant characterization based on the algebraic structure of curvature tensors for specific spacetimes or classes of spacetimes. We derive and briefly discuss the properties of SCG solutions for static spherical symmetry (vacuum and perfect fluid), FLRW models and spherical dust fluids. While we do recognize that SCG is ``work in progress'' in an incipient stage that still requires significant theoretical development, we believe that the theory provides valuable guidelines in the search for alternatives to General Relativity

On a new class of non-dynamical ABCD algebras for classical and quantum integrable systems

We consider classical and quantum non-dynamical quadratic abcd Lax algebras with classical and quantum gl(n)⊗gl(n)-valued abcd-tensors satisfying a set of quadratic non-dynamical Yang-Baxter-type equations generalizing those of Fredel and Maillet [1]. We establish a relation of some of these equations with the so-called “semi-dynamical” Yang-Baxter equations of [2]. We show that the linearization of the corresponding quadratic structures lead to linear tensor structures with the classical gl(n)⊗gl(n)-valued r-matrices satisfying usual “permuted” classical Yang-Baxter equations [1,3–5]. We consider example of our construction associated with the deformed Zn-graded r-matrix of [10–12] and explicitly construct the corresponding abcd-tensors — both classical and quantum. Small n examples are also considered in some details.

Conformal bootstrap equations from the embedding space operator product expansion

We describe how to implement the conformal bootstrap program in the context of the embedding space OPE formalism introduced in previous work. To take maximal advantage of the known properties of the scalar conformal blocks for symmetric-traceless exchange, we construct tensorial generalizations of the three-point and four-point scalar conformal blocks that have many nice properties. Further, we present a special basis of tensor structures for three-point correlation functions endowed with the remarkable simplifying property that it does not mix under permutations of the external quasi-primary operators. We find that in this approach, we can write the M-point conformal bootstrap equations explicitly in terms of the standard position space cross-ratios without the need to project back to position space, thus effectively deriving all conformal bootstrap equations directly from the embedding space. Finally, we lay out an algorithm for generating the conformal bootstrap equations in this formalism. Collectively, the tensorial generalizations, the new basis of tensor structures, as well as the procedure for deriving the conformal bootstrap equations lead to four-point bootstrap equations for quasi-primary operators in arbitrary Lorentz representations expressed as linear combinations of the standard scalar conformal blocks for spin-ℓ exchange, with finite ℓ-independent terms. Moreover, the OPE coefficients in these equations conveniently feature trivial symmetry properties. The only inputs necessary are the relevant projection operators and tensor structures, which are all fixed by group theory. To illustrate the procedure, we present one nontrivial example involving scalars S and vectors V, namely ⟨SSSV⟩.

Two-loop five-point integrals: light, heavy and large-spin correlators

We evaluated all two-loop conformal integrals appearing in five-point correlation functions of protected operators of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 Super Yang-Mills in several kinematical regimes. Starting from the correlation function of the lightest operators of the theory, we were able to extract structure constants of up to two spinning operators for small and large values of spin and the tensor structure label (polarization). We conjectured an universal all-loop behaviour for the large-spin, small-polarization structure constants and comment on the subtleties of analytically continuing it from finite to large spin. We also consider correlation functions of heavier operators that get factorized in the more fundamental object called decagon. We fixed this object at two loops in general kinematics and studied its physical properties under OPE and null limits.

Cryo-X-Ray Phase Contrast Imaging Enables Combined 3D Structural Quantification and Nucleic Acid Analysis of Myocardial Biopsies.

Snap-frozen biopsies serve as a valuable clinical resource of archival material for disease research, as they enable a comprehensive array of downstream analyses to be performed, including extraction and sequencing of nucleic acids. Obtaining three-dimensional (3D) structural information before multi-omics is more challenging but can potentially allow for better characterization of tissues and targeting of clinically relevant cells. Conventional histological techniques are limited in this regard due to their destructive nature and the reconstruction artifacts produced by sectioning, dehydration, and chemical processing. These limitations are particularly notable in soft tissues such as the heart. In this study, the feasibility of using synchrotron-based cryo-X-ray phase contrast imaging (cryo-X-PCI) of snap-frozen myocardial biopsies is assessed and 3D structure tensor analysis of aggregated myocytes, followed by nucleic acid (DNA and RNA) extraction and analysis. It is shown that optimal sample preparation is the key driver for successful structural and nucleic acid preservation which is unaffected by the process of cryo-X-PCI. It is proposed that cryo-X-PCI has clinical value for 3D tissue analysis of cardiac and potentially non-cardiac soft tissue biopsies before nucleic acid investigation.

A new three-dimensional strength criterion considering the transverse isotropy based on the α-Spatial Mobilized Plane

Natural geotechnical materials are affected by sedimentation and exhibit significant anisotropy. To study the transverse isotropy characteristics of soil, the influence of intermediate principal stress and loading direction must be considered. Currently, research on transverse isotropy primarily focuses on the modified stress space, which is cumbersome to apply in multi-yield surface constitutive models. To describe the three-dimensional mechanical properties of geomaterials in real stress space, the α-Spatial Mobilized Plane strength criterion is introduced. Then, combined with the structure tensor, the transverse isotropic three-dimensional strength criterion can account for the effect of the loading angle. Finally, the three-dimensional strengths of Fukakusa clay, unsaturated SP-SC soils, uncemented Monterey sands, Yamaguchi marble, San Francisco Bay mud, Toyoura sand, and Santa Monica Beach sand are predicted on the π-plane. The results show that the αmn-SMP criterion, in the context of transverse isotropy, can describe the three-dimensional mechanical properties reasonably, and it can provide an accurate strength criterion for geotechnical engineering practice.

A Low Transformed Tubal Rank Tensor Model Using a Spatial-Tubal Constraint for Sample Clustering with Cancer Multi-omics Data

Background: Since each dimension of a tensor can store different types of genomics data, compared to matrix methods, utilizing tensor structure can provide a deeper understanding of multi-dimensional data while also facilitating the discovery of more useful information related to cancer. However, in reality, there are issues such as insufficient utilization of prior knowledge in multi-omics data and limitations in the recovery of low-tubal-rank tensors. Therefore, the method proposed in this article was developed. Objective: In this paper, we proposed a low transformed tubal rank tensor model (LTTRT) using a spatial-tubal constraint to accurately partition different types of cancer samples and provide reliable theoretical support for the identification, diagnosis, and treatment of cancer. Method: In the LTTRT method, the transformed tensor nuclear norm based on the transformed tensor singular value decomposition is characterized by the low-rank tensor, which can explore the global low-rank property of the tensor, resolving the challenge of the tensor nuclear norm-based method not achieving the lowest tubal rank. Additionally, the introduction of weighted total variation regularization is conducive to extracting more information from sequencing data in both spatial and tubal dimensions, exploring cross-correlation features of multiple genomic data, and addressing the problem of overlooking prior knowledge from various perspectives. In addition, the L1-norm is used to improve sparsity. A symmetric Gauss‒Seidel-based alternating direction method of multipliers (sGS-ADMM) is used to update the LTTRT model iteratively. Results: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. result: The experiments of sample clustering on multiple integrated cancer multi-omics datasets show that the proposed LTTRT method is better than existing methods. Experimental results validate the effectiveness of LTTRT in accurately partitioning different types of cancer samples. Conclusion: The LTTRT method achieves precise segmentation of different types of cancer samples.

Two-gluon one-photon vertex in a magnetic field and its explicit one-loop approximation in the intermediate field strength regime

We find the general structure for the two-gluon one-photon vertex in the presence of a constant magnetic field. We show that, when accounting for the symmetries satisfied by the strong and electromagnetic interactions under parity, charge conjugation and gluon interchange, and for gluons and photons on mass-shell, there exist only three possible tensor structures that span the vertex. These correspond to external products of the polarization vectors for each of the particles in the vertex. We also explicitly compute the one-loop approximation to this vertex in the intermediate field strength regime, which is the most appropriate one to describe possible effects of the presence of a magnetic field to enhance photon emission during preequilibrium in peripheral relativistic heavy-ion collisions. We show that the most favored direction for the photon to propagate is in the plane transverse to the field, which is consistent with a positive contribution to ν2 and may help to understand the larger than expected elliptic flow coefficient measured in this kind of reaction. Published by the American Physical Society 2024

Enhancing seismic feature orientations: A novel approach using directional derivatives and Hilbert transform of gradient structure tensor

Seismic dip calculation serves as a widely employed technique in the realms of seismic interpretation and reservoir characterization, strategically employed to highlight faults and attributes within the seismic volume. Among the various methodologies utilized for estimating structural dip and azimuth, the Gradient Structure Tensor (GST) stands out. This approach involves leveraging the dominant eigenvector of the positive definite GST matrix to ascertain the inline and crossline dip of seismic data.In the initial phase of our innovative proposal, we employed the spectral balancing technique to enhance the fidelity of seismic data. Subsequently, leveraging this groundwork, we introduced an Analytical Directional Gradient Structure Tensor technique, a distinctive adaptation of GST. This novel approach involves the calculation of directive derivatives in both perpendicular and parallel directions to seismic features. By incorporating directive derivatives, our method excels in capturing subtle stratigraphic nuances, particularly in the dipping direction of interest. To validate the accuracy and effectiveness of our approach, we present compelling evidence through the examination of synthetic and real-field seismic volume outcomes. This underscores the robustness and reliability of our proposed method in enhancing the precision of seismic dip calculations and providing valuable insights into subsurface geological features.

A 3D finite deformation constitutive model for anisotropic shape memory polymer composites integrating viscoelasticity and phase transition concept

The phase transition theory of shape memory polymers (SMPs) often involves a phenomenological assumption that the reference configuration of the newly transformed phase deviates from that of the initial phase. This distinction serves as a crucial mechanism in the manifestation of the shape memory effect. However, elucidating the precise definition of the reference configuration of the transformed phase poses a significant challenge in the formulation of the constitutive model. To tackle this challenge, a three-dimensional (3D) finite deformation constitutive model incorporating effective phase evolution for SMPs has been developed. This model merges insights from the classical viscoelastic framework with the phase transition theory. The anisotropic thermo-viscoelastic constitutive model is further developed by introducing hyperelastic fibers, which integrate the anisotropy of the fibers into a continuous thermodynamic framework through structure tensors. Implemented within the ABAQUS software via a user material (UMAT) subroutine, the proposed model has been meticulously validated against experimental data, showcasing its prowess in simulating stress-strain responses and shape memory characteristics of SMPs and their composites (SMPCs). This innovative model stands as an invaluable instrument for the design and of sophisticated SMP and SMPC structures.

5-D spatial–temporal information-based infrared small target detection in complex environments

Recently, infrared (IR) small target detection problem has attracted increasing attention. Tensor component analysis-based techniques have been widely utilized, while they are faced with challenges such as tensor structures, background and target estimation, and real-time performance. In this paper, we propose a 5-D spatial–temporal factor-based completion model (5D-STFC) for IR small target detection. Specifically, a 5-D whitened spatial–temporal patch-tensor is constructed. Then, we devise a spatial–temporal factor-based low-rank background estimation norm and a Moreau envelope-derived sparsity estimation norm based on joint spatial–temporal knowledge. Furthermore, we establish a comprehensive completion model for component analysis. To efficiently solve this model, we design a multi-block alternating direction method of multipliers (multi-block ADMM)-based optimization scheme. Extensive experiments conducted on five real IR sequences demonstrate the superiority of 5D-STFC over nine state-of-the-art competitive methods. It can be concluded that 5D-STFC is excellent and practical in target detectability, background suppressibility, overall performance, and real-time performance.

A covariant formulation for finite strain modelling of orthotropic elasticity and orthotropic plasticity with plasticity-induced evolution of orthotropy: Application to natural fibres

We introduce a rate-independent model for orthotropic elastic and orthotropic plastic material behaviour in a hyper-elasto-plastic framework at finite strains. The model is based on the postulate of covariance and does not rely on a multiplicative decomposition of the deformation gradient. Furthermore, a plastic-deformation-induced evolution of orthotropy is considered, similar to the notion of plastic spin. We propose that the orthotropic strain energy function and the orthotropic yield criterion are guided by identical structural tensors that evolve with plasticity. The modelled material behaviour is significant for natural fibres such as flax, hemp, or pulp fibres. Our formulation has three findings. Firstly, the covariant formulation of plasticity provides rate equations for the plastic variables and the structural tensors suitable for reproducing stress–strain diagrams of natural fibres. Secondly, the introduction of plastic-deformation-induced evolution of orthotropy in the proposed covariant setting results in a non-associative plasticity algorithm. Thirdly, the covariant setting allows the incorporation of suitable constitutive equations for the structural tensors to evolve orthotropy. The latter successfully models the stiffness increase in stress–strain diagrams of cyclic tensile tests of natural fibres.

Toward Jordan decompositions for tensors

We expand on an idea of Vinberg to take a tensor space and the natural Lie algebra that acts on it and embed their direct sum into an auxiliary algebra. Viewed as endomorphisms of this algebra, we associate adjoint operators to tensors. We show that the group actions on the tensor space and on the adjoint operators are consistent, which means that the invariants of the adjoint operator of a tensor, such as the Jordan decomposition, are invariants of the tensor. We show that there is an essentially unique algebra structure that preserves the tensor structure and has a meaningful Jordan decomposition. We utilize aspects of these adjoint operators to study orbit separation and classification in examples relevant to tensor decomposition and quantum information.

Image dejittering on the perspective of spatially-varying mixed noise removal

Jittery image is visually abnormal in jags of edge and loss of coherence. The problem of image dejittering is challenging to resolve due to the ubiquitous blur and/or noise in jittery data. In this paper, we devote to the pixel-jitter (possibly blurry) image recovery on the perspective of spatially-varying mixed noise removal. By viewing jittery image as the corruption of ideal image with outliers and spatially-varying Gaussian noise, we proposed a two-phase (including filtering and diffusing phases) image dejittering approach. In the filtering phase, outliers posed by jitters around edges are inspected by median filters. In the diffusing phase, structure tensor based anisotropic diffusion is exploited to reduce the perturbations in piecewise smooth image regions. Upon the spectral decomposition of structure tensor, the variational model in diffusing phase can be solved by some state-of-the-art optimization methods. Numerical simulations on synthetic and real jittery data demonstrate the compelling performance of the proposed approach. The Matlab source codes of the proposed approach are available at the repositories of https://github.com/WenxingZhang.

Low-rank tensor completion based on tensor train rank with partially overlapped sub-blocks and total variation

Recently, the low-rank tensor completion method based on tensor train (TT) rank has achieved promising performance. Ket augmentation (KA) is commonly used in TT rank-based methods to improve the performance by converting low-dimensional tensors to higher-dimensional tensors. However, block artifacts are caused since KA also destroys the original structure and image continuity of original low-dimensional tensors. To tackle this issue, a low-rank tensor completion method based on TT rank with tensor augmentation by partially overlapped sub-blocks (TAPOS) and total variation (TV) is proposed in this paper. The proposed TAPOS preserves the image continuity of the original tensor and enhances the low-rankness of the generated higher-dimensional tensors, and a weighted de-augmentation method is used to assign different weights to the elements of sub-blocks and further reduce the block artifacts. To further alleviate the block artifacts and improve reconstruction accuracy, TV is introduced in the TAPOS-based model to add the piecewise smooth prior. The parallel matrix decomposition method is introduced to estimate the TT rank to reduce the computational cost. Numerical experiments show that the proposed method outperforms the existing state-of-the-art methods.

Micron-resolution fiber mapping in histology independent of sample preparation.

Detailed knowledge of the brain's nerve fiber network is crucial for understanding its function in health and disease. However, mapping fibers with high resolution remains prohibitive in most histological sections because state-of-the-art techniques are incompatible with their preparation. Here, we present a micron-resolution light-scattering-based technique that reveals intricate fiber networks independent of sample preparation for extended fields of view. We uncover fiber structures in both label-free and stained, paraffin-embedded and deparaffinized, newly-prepared and archived, animal and human brain tissues - including whole-brain sections from the BigBrain atlas. We identify altered microstructures in demyelination and hippocampal neurodegeneration, and show key advantages over diffusion magnetic resonance imaging, polarization microscopy, and structure tensor analysis. We also reveal structures in non-brain tissues - including muscle, bone, and blood vessels. Our cost-effective, versatile technique enables studies of intricate fiber networks in any type of histological tissue section, offering a new dimension to neuroscientific and biomedical research.

Tensor double arc-tangent norm minimization for multi-view clustering

Tensor structures are widely utilized in clustering tasks due to their ability to represent high-order relationships among multi-view data. The rank and decomposition of tensors are essential characteristics that have recently received considerable attention. Most approaches approximate tensor rank using the tensor nuclear norm (TNN) and non-convex rank functions. We introduce a multi-view clustering method based on an enhanced self-expression with a tensor double arc-tangent norm. Initially leveraging tensor singular value decomposition, this method impartially approximates the true rank of the tensor, avoiding rank overestimation. The enhanced self-expression further utilizes the latent connections among the original data. Then, we propose an improved alternating direction method of multipliers (ADMM) for rapid optimization of the model. Experiments conducted on six datasets confirm the effectiveness of the proposed model.

On neutral Tannakian subcategories of loop quiver representations

We explored the categories of (twisted) representations of a loop quiver. These representation categories have two choices of tensor structures: Kronecker tensor and Simpson tensor. By studying the rigidity properties, we have provided several examples of (semi-) Tannakian categories using the category of (twisted) representations of a loop quiver for both tensors.We have introduced the concept of essentially finite loop quiver bundles based on the work of Nori, Borne, and Vistoli. As an application, we have given some examples of (semi-) Tannakian categories of equivariant bundles and Hitchin pairs. Additionally, we have defined the notion of H-nflat twisted loop quiver bundles and have established Tannakian category structures for certain classes of varieties.

Cyclic tensor singular value decomposition with applications in low-rank high-order tensor recovery

The rapid advancements in emerging technologies have increased the demand for recovery tasks involving high-dimensional data with complex structures. Effectively utilizing tensor decomposition techniques to capture the low-rank structure of such data is crucial. Recently, the high-order t-SVD has demonstrated strong adaptability. However, this decomposition approach can only capture the low-rank correlation of two modes along other modes individually, while disregarding the structural correlation between different modes. In this paper, we propose a novel cyclic tensor singular value decomposition (CTSVD) method that effectively characterizes the low-rank structures of high-order tensors along all modes. Specifically, our method decomposes an order-N tensor into N factor tensors and one core tensor, connecting them using a defined mode-k tensor-tensor product (t-product). Building upon this, we establish the corresponding tensor rank and its convex relaxation. To address the issue of dimensional imbalance between adjacent modes in high-dimensional data, we propose and integrate a square reshaping strategy into the recovery models for tensor completion (TC) and tensor principal component analysis (TRPCA) tasks. Effective alternating direction method of multipliers (ADMM)-based algorithms are designed to these tasks. Extensive experiments on both synthetic and real data demonstrate that our methods outperform state-of-the-art approaches.

Primal-dual hybrid gradient image denoising algorithm based on overlapping group sparsity and fractional-order total variation

This study introduces a non-convex fractional-order hyper-Laplacian variational model for Gaussian noise removal. It employs first the primal-dual hybrid gradient algorithm to solve problems involving overlapping group sparse structures. Additionally, this paper designs a new algorithm leveraging the framework of the Chambolle-Pock algorithm with convergence and aims to recover high-quality images. The model, integrating the overlapping group sparse structure of the hyper-Laplacian prior with the non-convex fractional-order total variation, exhibits superior performance in reducing the staircase effect and maintaining sharp edge contours. To further improve the performance of the algorithm, a semi-adaptive p(x) non-convex penalty weight assignment mechanism is designed by introducing the structure tensor, which according to the characteristics of each region of the image and the noise level. The effectiveness and superiority of the proposed algorithm in image denoising with simulation experiments and comparative analyses are fully verified.