Intrinsic Tensor Research Articles

Joint Projection Learning and Tensor Decomposition-Based Incomplete Multiview Clustering.

Incomplete multiview clustering (IMVC) has received increasing attention since it is often that some views of samples are incomplete in reality. Most existing methods learn similarity subgraphs from original incomplete multiview data and seek complete graphs by exploring the incomplete subgraphs of each view for spectral clustering. However, the graphs constructed on the original high-dimensional data may be suboptimal due to feature redundancy and noise. Besides, previous methods generally ignored the graph noise caused by the interclass and intraclass structure variation during the transformation of incomplete graphs and complete graphs. To address these problems, we propose a novel joint projection learning and tensor decomposition (JPLTD)-based method for IMVC. Specifically, to alleviate the influence of redundant features and noise in high-dimensional data, JPLTD introduces an orthogonal projection matrix to project the high-dimensional features into a lower-dimensional space for compact feature learning. Meanwhile, based on the lower-dimensional space, the similarity graphs corresponding to instances of different views are learned, and JPLTD stacks these graphs into a third-order low-rank tensor to explore the high-order correlations across different views. We further consider the graph noise of projected data caused by missing samples and use a tensor-decomposition-based graph filter for robust clustering. JPLTD decomposes the original tensor into an intrinsic tensor and a sparse tensor. The intrinsic tensor models the true data similarities. An effective optimization algorithm is adopted to solve the JPLTD model. Comprehensive experiments on several benchmark datasets demonstrate that JPLTD outperforms the state-of-the-art methods. The code of JPLTD is available at https://github.com/weilvNJU/JPLTD.

Local vector measures

Consider a BV function on a Riemannian manifold. What is its differential? And what about the Hessian of a convex function? These questions have clear answers in terms of (co)vector/matrix valued measures if the manifold is the Euclidean space. In more general curved contexts, the same objects can be perfectly understood via charts. However, charts are often unavailable in the less regular setting of metric geometry, where still the questions make sense.In this paper we propose a way to deal with this sort of problems and, more generally, to give a meaning to a concept of ‘measure acting in duality with sections of a given bundle’, loosely speaking. Despite the generality, several classical results in measure theory like Riesz's and Alexandrov's theorems have a natural counterpart in this setting. Moreover, as we are going to discuss, the notions introduced here provide a unified framework for several key concepts in nonsmooth analysis that have been introduced more than two decades ago, such as: Ambrosio-Kirchheim's metric currents, Cheeger's Sobolev functions and Miranda's BV functions.Not surprisingly, the understanding of the structure of these objects improves with the regularity of the underlying space. We are particularly interested in the case of RCD spaces where, as we will argue, the regularity of several key measures of the type we study nicely matches the known regularity theory for vector fields, resulting in a very effective theory.We expect that the notions developed here will help creating stronger links between differential calculus in Alexandrov spaces (based on Perelman's DC charts) and in RCD ones (based on intrinsic tensor calculus).

Enhanced Tensor Low-Rank and Sparse Representation Recovery for Incomplete Multi-View Clustering

Incomplete multi-view clustering (IMVC) has attracted remarkable attention due to the emergence of multi-view data with missing views in real applications. Recent methods attempt to recover the missing information to address the IMVC problem. However, they generally cannot fully explore the underlying properties and correlations of data similarities across views. This paper proposes a novel Enhanced Tensor Low-rank and Sparse Representation Recovery (ETLSRR) method, which reformulates the IMVC problem as a joint incomplete similarity graphs learning and complete tensor representation recovery problem. Specifically, ETLSRR learns the intra-view similarity graphs and constructs a 3-way tensor by stacking the graphs to explore the inter-view correlations. To alleviate the negative influence of missing views and data noise, ETLSRR decomposes the tensor into two parts: a sparse tensor and an intrinsic tensor, which models the noise and underlying true data similarities, respectively. Both global low-rank and local structured sparse characteristics of the intrinsic tensor are considered, which enhances the discrimination of similarity matrix. Moreover, instead of using the convex tensor nuclear norm, ETLSRR introduces a generalized non-convex tensor low-rank regularization to alleviate the biased approximation. Experiments on several datasets demonstrate the effectiveness of our method compared with the state-of-the-art methods.

Imbalanced low-rank tensor completion via latent matrix factorization

Tensor completion has been widely used in computer vision and machine learning. Most existing tensor completion methods empirically assume the intrinsic tensor is simultaneous low-rank in all over modes. However, tensor data recorded from real-world applications may conflict with these assumptions, e.g., face images taken from different subjects often lie in a union of low-rank subspaces, which may result in a quite high rank or even full rank structure in its sample mode. To this aim, in this paper, we propose an imbalanced low-rank tensor completion method, which can flexibly estimate the low-rank incomplete tensor via decomposing it into a mixture of multiple latent tensor ring (TR) rank components. Specifically, each latent component is approximated using low-rank matrix factorization based on TR unfolding matrix. In addition, an effective proximal alternating minimization algorithm is developed and theoretically proved to maintain the global convergence property, that is, the whole sequence of iterates is convergent and converges to a critical point. Extensive experiments on both synthetic and real-world tensor data demonstrate that the proposed method achieves more favorable completion results with less computational cost when compared to the state-of-the-art tensor completion methods.

Tensor-based plenoptic image denoising by integrating super-resolution

In this paper, we propose a novel tensor-based denoising method targeting at plenoptic images which contain 4D light field (2D angular + 2D spatial) and 5D hyperspectral light field (2D angular + 2D spatial + 1D spectral). In order to make use of the high-dimension structural property of plenoptic images, we first generalize the intrinsic tensor sparsity measure to plenoptic images by extending the nonlocal similarity from the spatial dimension to the angular dimension. Second, to eliminate the sub-pixel misalignment of different views, we integrate the spatial super-resolution into denoising and exploit the spatial-angular correlation by utilizing the nonlocal similarity of the refined high-resolution central view. In the procedure of super-resolution, we utilize an intensity consistency criterion and a coordinate rationality criterion to facilitate the process of projection. The denoising performance can be boosted after back-projection performed on the refined high-resolution central view. Experimental results validate the superior performance of the proposed method on several plenoptic image datasets in terms of both subjective and objective quality.

Atom-substituted tensor dictionary learning enhanced convolutional neural network for hyperspectral image classification

A novel sparse tensor dictionary learning algorithm and a convolutional neural network (CNN) classification method based on this algorithm are proposed for hyperspectral image (HSI) in this paper. The HSI is intuitively represented as a three-dimensional (3-D) cube, and utilizing the joint spatial-spectral information can significantly improve the accuracy of HSI classification. Therefore, the proposed atom-substituted tensor dictionary learning (ASTDL) algorithm utilizes tensor techniques to extract 3-D joint spatial-spectral features from HSI cubes directly. Sparsity constraint is enforced on the coefficient tensors, which obeys the sparsity attribute of HSI. The proposed ASTDL enhanced CNN (ASTDL-CNN) classification method utilizes a two-dimensional (2-D) CNN to extract deep features from the feature tensors obtained by the ASTDL, and to perform the pixel-wise classification. The use of ASTDL, which extracts intrinsic tensor features before CNN, alleviates within-class spatial-spectral variation and reduces the requirement of CNN for the labeled data. Whereas, the CNN works as a 3-D classifier, and provides the final classification results. Besides, we perform the majority vote on the classification map obtained by CNN to refine the classification. The performance of the proposed method is evaluated on three real HSI data sets. The competitive results to the compared state-of-the-art methods demonstrate that the ASTDL-CNN can provide accurate and robust classification results.

A Multispectral Denoising Framework for Seismic Random Noise Attenuation

Random noise attenuation plays an important role in the seismic data processing. The global coherency among different spectral segments is often neglected in the traditional denoising methods, even though the seismic data are naturally broadband in the frequency spectrum. We proposed a multispectral denoising framework (MDF) for seismic random noise attenuation. The MDF contains three key components. First, we use a series of narrow-band filters on the noisy data to construct the multispectral data. Second, we normalize the constructed multispectral data to mitigate the amplitude discrepancy between different spectral slices. Third, we apply the intrinsic tensor sparsity regularization method to denoise the multispectral data. Numerical experiments on synthetic and field data show that the proposed framework can reduce the random noise as well as the erratic noise, and it can achieve improved denoised results, compared with a traditional method (i.e., the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$f$ </tex-math></inline-formula> - <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$x$ </tex-math></inline-formula> deconvolution method) and the single-spectral version of the proposed method (i.e., the weighted nuclear norm minimization (WNNM) method). The proposed framework can be applied to both prestack and stacked data and generally well preserve the amplitude of useful signals, compared with the other two methods.

Infrared small target detection via incorporating spatial structural prior into intrinsic tensor sparsity regularization

Infrared small target detection is a crucial stage in many searching and tracking applications. Many tensor decomposition-based methods have achieved well performance in the scenes with uniform backgrounds and salient targets. However, the performance is potentially prone to be degraded when encountering highly complex scenes. It is mainly because the decomposition error caused by the sparse edge structures and imprecise tensor rank measures make the recovered background deviate from the real one. To mitigate these issues, we introduce an infrared sequence tensor decomposition-based method, which combines an intrinsic tensor rank measure (ITRM) and spatial structure prior to improve background recovery. With the inter-frame background correlation, ITRM is used to regularize the low-rank component of the observed tensor, which encodes the intrinsic tensor rank insights delivered by two most typical tensor decompositions to finely rectify the estimation bias of the tensor rank. To further suppress edge structures during the decomposition, we design a spatial descriptor with edge awareness by using a nonlocal structure tensor to delineate intra-frame structural edge. Furthermore, an adaptive indicator, which fuses spatial edge information and sparsity enhancing weight, is employed to replace the inflexibly and globally sparse penalty. The solution of the proposed model is addressed by an alternative direction minimization of multipliers (ADMM). Extensive experiments on real-world infrared sequences demonstrate the outperformance of the proposed method against other state-of-the-art ones, both quantitatively and qualitatively.

Multi-energy CT reconstruction using tensor nonlocal similarity and spatial sparsity regularization.

Multi-energy computed tomography (MECT) based on a photon-counting detector is an emerging imaging modality that collects projections at several energy bins with a single scan. However, the limited number of photons collected into the divided, narrow energy bins results in high quantum noise levels in reconstructed images. This study aims to improve MECT image quality by minimizing noise levels while retaining image details. A novel MECT reconstruction method was proposed by exploiting the nonlocal tensor similarity among interchannel images and spatial sparsity in single-channel images. Similar patches were initially extracted from the interchannel images in spectral and spatial domains, then stacked into a new three-order tensor. Intrinsic tensor sparsity regularization that combined the Tuker and canonical polyadic (CP) low-rank decomposition techniques were applied to exploit the nonlocal similarity of the formulated tensor. Spatial sparsity in single-channel images was modeled by total variation (TV) regularization that utilizes the compressibility of gradient image. A new MECT reconstruction model was established by simultaneously incorporating the intrinsic tensor sparsity and TV regularizations. The iterative alternating minimization method was utilized to solve the reconstruction model based on a flexible framework. The proposed method was applied to the digital phantom and real mouse data to assess its feasibility and reliability. The reconstruction and decomposition results in the mouse data were encouraging and demonstrated the ability of the proposed method in noise suppression while preserving image details, not observed with other methods. Imaging data from the digital phantom illustrated this method as achieving the best intuitive reconstruction and decomposition results among all compared methods. They reduced the root mean square error (RMSE) by 89.75%, 50.75%, and 36.54% on the reconstructed images compared with analytic, TV-based, and tensor-based methods, respectively. This phenomenon was also observed with decomposition results, where the RMSE was also reduced by 97.96%, 67.74%, 72.05%, respectively. In this study, we proposed a reconstruction method for photon counting detector-based MECT, using the intrinsic tensor sparsity and TV regularizations. Improvements in noise suppression and detail preservation in the digital phantom and real mouse data were validated by the qualitative and quantitative evaluations on the reconstruction and decomposition results, verifying the potential of the proposed method in MECT reconstruction.

Simplified approach to the derivation of the relationship between Hill polarization tensors of transformed problems and applications

This work aims at proposing a general framework based on an intrinsic tensor formalism to relate Hill polarization tensors of elastic problems associated by a linear transformation without detailing the resolution of the problems. Such an approach somehow casts a new light on the calculation of some Hill tensors already known in the literature by means of other methods or other presentations. In particular it allows to avoid the calculation of any Green tensor or derivatives and it boils down to very compact expressions which are rather easy to implement. The application of the method to the case of an arbitrary ellipsoid embedded in an elliptically orthotropic matrix recalls that the Hill tensor can be built from an associated isotropic case but also puts in evidence that the elliptically orthotropic case is the only one that can be derived from an isotropic counterpart. Another application of the method shows that the Hill tensor of a spheroid embedded in a coaxial transversely isotropic matrix can be derived from an associated problem involving a sphere in a modified transversely isotropic matrix. This approach provides new simple and compact expressions for the Hill tensor from which the known limits corresponding to flat spheroid or strongly prolate shapes are retrieved.

Unified Graph and Low-Rank Tensor Learning for Multi-View Clustering

Multi-view clustering aims to take advantage of multiple views information to improve the performance of clustering. Many existing methods compute the affinity matrix by low-rank representation (LRR) and pairwise investigate the relationship between views. However, LRR suffers from the high computational cost in self-representation optimization. Besides, compared with pairwise views, tensor form of all views' representation is more suitable for capturing the high-order correlations among all views. Towards these two issues, in this paper, we propose the unified graph and low-rank tensor learning (UGLTL) for multi-view clustering. Specifically, on the one hand, we learn the view-specific affinity matrix based on projected graph learning. On the other hand, we reorganize the affinity matrices into tensor form and learn its intrinsic tensor based on low-rank tensor approximation. Finally, we unify these two terms together and jointly learn the optimal projection matrices, affinity matrices and intrinsic low-rank tensor. We also propose an efficient algorithm to iteratively optimize the proposed model. To evaluate the performance of the proposed method, we conduct extensive experiments on multiple benchmarks across different scenarios and sizes. Compared with the state-of-the-art approaches, our method achieves much better performance.

Full-Spectrum-Knowledge-Aware Tensor Model for Energy-Resolved CT Iterative Reconstruction.

Energy-resolved computed tomography (ErCT) with a photon counting detector concurrently produces multiple CT images corresponding to different photon energy ranges. It has the potential to generate energy-dependent images with improved contrast-to-noise ratio and sufficient material-specific information. Since the number of detected photons in one energy bin in ErCT is smaller than that in conventional energy-integrating CT (EiCT), ErCT images are inherently more noisy than EiCT images, which leads to increased noise and bias in the subsequent material estimation. In this work, we first deeply analyze the intrinsic tensor properties of two-dimensional (2D) ErCT images acquired in different energy bins and then present a F ull- S pectrum-knowledge-aware Tensor analysis and processing (FSTensor) method for ErCT reconstruction to suppress noise-induced artifacts to obtain high-quality ErCT images and high-accuracy material images. The presented method is based on three considerations: (1) 2D ErCT images obtained in different energy bins can be treated as a 3-order tensor with three modes, i.e., width, height and energy bin, and a rich global correlation exists among the three modes, which can be characterized by tensor decomposition. (2) There is a locally piecewise smooth property in the 3-order ErCT images, and it can be captured by a tensor total variation regularization. (3) The images from the full spectrum are much better than the ErCT images with respect to noise variance and structural details and serve as external information to improve the reconstruction performance. We then develop an alternating direction method of multipliers algorithm to numerically solve the presented FSTensor method. We further utilize a genetic algorithm to tackle the parameter selection in ErCT reconstruction, instead of manually determining parameters. Simulation, preclinical and synthesized clinical ErCT results demonstrate that the presented FSTensor method leads to significant improvements over the filtered back-projection, robust principal component analysis, tensor-based dictionary learning and low-rank tensor decomposition with spatial-temporal total variation methods.

FD-MIMO via Pilot-Data Superposition: Tensor-Based DOA Estimation and System Performance

Increased pilot overhead is one of the major issues for 5G full-dimension MIMO (FD-MIMO) systems. In this paper, we introduce the FD-MIMO system using pilot-data superposition to reduce the uplink pilot overhead and investigate the impact of superimposed pilots on the overall network performance. The pilot-data superposition can relatively increase the resources allocated to the uplink data transmission. However, in time division duplex (TDD) systems, it also negatively impacts the downlink throughput since the relatively reduced pilot power can affect the estimation of the channel state information (CSI) which is used for the downlink precoding. To improve the CSI estimation, the intrinsic tensor feature of the FD-MIMO channel is exploited in our method. The introduced CSI estimation algorithm is designed through the expectation-maximization (EM) framework via tensor as the processing data structure. Furthermore, Cramer-Rao lower bound (CRLB) is utilized as a metric in our evaluation. The overall system achievable rate which is a weighted sum of the uplink and downlink throughput is investigated to reveal the fundamental trade-off between the uplink and downlink transmission in the TDD-based FD-MIMO system. In our simulation, the results demonstrate the superior performance of the introduced strategy as opposed to the conventional orthogonal pilot-data approach. Meanwhile, the pilot power allocation, as well as different transmitting and receiving strategies are investigated to offer various trade-off points between the uplink and downlink transmission.

Precision β − ν correlation measurements with the Beta-decay Paul Trap

The Beta-decay Paul Trap (BPT) at Argonne National Laboratory has proven to be an extremely effective tool for high-precision tests of the Standard Model via measurements of the β − ν correlation in mass-8 isotopes. Using four double-sided silicon strip detectors (DSSDs) backed by plastic scintillators and surrounding the ions confined by the BPT, the kinematics of the decays of the mirror nuclei lithium-8 and boron-8 are overdetermined when all charged decay products are measured. The most stringent low-energy limit on an intrinsic tensor current in the weak interaction was set using the BPT in 2015 (Sternberg, M.G., et al., Phys. Rev. Lett. 115, 182501 2015) utilizing trapped lithium-8. Since then, similar data for boron-8 and higher statistics data for lithium-8 have been collected and are currently being analyzed. With the elimination of radio-frequency (RF) pickup from the DSSDs and a detailed investigation of experimental systematic errors, the uncertainty is now dominated by the contribution from recoil-order terms in the decay rate. Our eventual goal is to limit tensor currents in the weak interaction with relative precision at or below 0.1%.

Video reconstruction based on Intrinsic Tensor Sparsity model

The natural images have self-similarities which can be used to improve the image reconstruction. However, the existing video reconstruction algorithms pay more attention to modeling and ignore the importance of priors in the reconstruction. In this paper, the self-similarities are involved in the modeling when the video is reconstructed from temporally compressed video measurements. The proposed reconstruction model includes two parts: First, the video tensor sparsity model is formulated by using a spatial–temporal tensor sparse penalty for similar patches. The Intrinsic Tensor Sparsity (ITS) measure is used as the sparsity measure, which encodes both sparsity insights delivered by the Tucker and CANDECOMP/PARAFAC (CP) decomposition for tensors. Second, 3D video patches are modeled as the Gaussian Joint Sparsity (GJS) by exploiting the temporal similarity to obtain an initial image which has distinct direction structure. GJS is a combination of statistical distribution and joint sparsity model. The experimental results show that both the reconstruction models based on ITS and GJS contribute to improving the quality of the video reconstruction.

Sparse tensor neighbor embedding based pan-sharpening via N-way block pursuit

Most of the available pan-sharpening methods use vector or matrix based detail injection to enhance the resolution of MultiSpectral (MS) image, which may result in unavoidable spectral and spatial distortions. In this paper we explore the intrinsic tensor structure and local sparsity of MS images, to develop a novel Sparse Tensor Neighbor Embedding (STNE) based pan-sharpening method that reduces the distortions in the fused images. First, MS images are formulated as some spectral tensors, and each tensor and its nearest neighbor tensors are assumed to lie in a low-dimensional manifold. Then the tensor is sparsely coded under its neighbor tensors, and a joint sparse coding assumption is cast on bands to develop an N-way Block Pursuit algorithm for solving sparse tensor coefficients. Finally high resolution MS tensor can be obtained by weighting Panchromatic image with the sparse tensor coefficients. Tensors are higher order generalizations of vectors and matrices, and taking advantage of high-order structure of multi-dimensional data can help us understand them. The proposed method first combines a sparse tensor with neighbor embedding, to construct a new high-dimensional sparse tensor embedding for efficient pan-sharpening. Because tensor formulation can exploit the structural correlations in high-dimensional MS data, the proposed method can well preserve spectral correlation among different bands simultaneously and capture the underlying high-order statistical properties of MS image. Some experiments are performed on several real QuickBird and GeoEye datasets, and experimental results show that STNE is superior to its counterparts in reducing spectral and spatial distortions.

Study of the numerical simulation of tight sandstone gas molecular diffusion based on digital core technology

Diffusion is an important mass transfer mode of tight sandstone gas. Since nano-pores are extensively developed in the interior of tight sandstone, a considerable body of research indicates that the type of diffusion is mainly molecular diffusion based on Fick’s law. However, accurate modeling and understanding the physics of gas transport phenomena in nano-porous media is still a challenge for researchers and traditional investigation (analytical and experimental methods) have many limitations in studying the generic behavior. In this paper, we used Nano-CT to observe the pore structures of samples of the tight sandstone of western of Sichuan. Combined with advanced image processing technology, three-dimensional distributions of the nanometer-sized pores were reconstructed and a tight sandstone digital core model was built, as well the pore structure parameters were analyzed quantitatively. Based on the digital core model, the diffusion process of methane molecules from a higher concentration area to a lower concentration area was simulated by a finite volume method. Finally, the reservoir’s concentration evolution was visualized and the intrinsic molecular diffusivity tensor which reflects the diffusion capabilities of this rock was calculated. Through comparisons, we found that our calculated result was in good agreement with other empirical results. This study provides a new research method for tight sandstone digital rock physics. It is a foundation for future tight sandstone gas percolation theory and numerical simulation research.

Overview + Detail Visualization for Ensembles of Diffusion Tensors

AbstractA Diffusion Tensor Imaging (DTI) group study consists of a collection of volumetric diffusion tensor datasets (i.e., an ensemble) acquired from a group of subjects. The multivariate nature of the diffusion tensor imposes challenges on the analysis and the visualization. These challenges are commonly tackled by reducing the diffusion tensors to scalar‐valued quantities that can be analyzed with common statistical tools. However, reducing tensors to scalars poses the risk of losing intrinsic information about the tensor. Visualization of tensor ensemble data without loss of information is still a largely unsolved problem. In this work, we propose an overview + detail visualization to facilitate the tensor ensemble exploration. We define an ensemble representative tensor and variations in terms of the three intrinsic tensor properties (i.e., scale, shape, and orientation) separately. The ensemble summary information is visually encoded into the newly designed aggregate tensor glyph which, in a spatial layout, functions as the overview. The aggregate tensor glyph guides the analyst to interesting areas that would need further detailed inspection. The detail views reveal the original information that is lost during aggregation. It helps the analyst to further understand the sources of variation and formulate hypotheses. To illustrate the applicability of our prototype, we compare with most relevant previous work through a user study and we present a case study on the analysis of a brain diffusion tensor dataset ensemble from healthy volunteers.

A macroscopic model for slightly compressible gas slip-flow in homogeneous porous media

The study of gas slip-flow in porous media is relevant in many applications ranging from nanotechnology to enhanced oil recovery and in any situation involving low-pressure gas-transport through structures having sufficiently small pores. In this paper, we use the method of volume averaging for deriving effective-medium equations in the framework of a slightly compressible gas flow. The result of the upscaling process is an effective-medium model subjected to time- and length-scale constraints, which are clearly identified in our derivation. At the first order in the Knudsen number, the macroscopic momentum transport equation corresponds to a Darcy-like model involving the classical intrinsic permeability tensor and a slip-flow correction tensor that is also intrinsic. It generalizes the Darcy-Klinkenberg equation for ideal gas flow, and exhibits a more complex form for dense gas. The component values of the two intrinsic tensors were computed by solving the associated closure problems on two- and three-dimensional periodic unit cells. Furthermore, the dependence of the slip-flow correction with the porosity was also verified to agree with approximate analytical results. Our predictions show a power-law relationship between the permeability and the slip-flow correction that is consistent with other works. Nevertheless, the generalization of such a relationship to any configuration requires more analysis.

A distributional approach to 2D Volterra dislocations at the continuum scale

We develop a theory to represent dislocations and disclinations in single crystals at the continuum (or mesoscopic) scale by directly modelling the defect densities as concentrated effects governed by the distribution theory. The displacement and rotation multi-valuedness is resolved by introducing the intrinsic and single-valued Frank and Burgers tensors from the distributional gradients of the strain field. Our approach provides a new understanding of the theory of line defects as developed by Kröner [10] and other authors [6, 9]. The fundamental identity relating the incompatibility tensor to the Frank and Burgers vectors (and which is a cornerstone of the theory of dislocations in single crystals) is proved in the 2D case under appropriate assumptions on the strain and strain curl growth in the vicinity of the assumed isolated defect lines. In general, our theory provides a rigorous framework for the treatment of crystal line defects at the mesoscopic scale and a basis to strengthen the theory of homogenisation from mesoscopic to macroscopic scale.