CiteULike: Tag tensor

Reduction of noise in diffusion tensor images using anisotropic smoothing.

Z Ding — 2007-03-28T11:05:13-00:00

Magn Reson Med, Vol. 53, No. 2. (February 2005), pp. 485-490.

To improve the accuracy of tissue structural and architectural characterization with diffusion tensor imaging, a novel smoothing technique is developed for reducing noise in diffusion tensor images. The technique extends the traditional anisotropic diffusion filtering method by allowing isotropic smoothing within homogeneous regions and anisotropic smoothing along structure boundaries. This is particularly useful for smoothing diffusion tensor images in which direction information contained in the tensor needs to be restored following noise corruption and preserved around tissue boundaries. The effectiveness of this technique is quantitatively studied with experiments on simulated and human in vivo diffusion tensor data. Illustrative results demonstrate that the anisotropic smoothing technique developed can significantly reduce the impact of noise on the direction as well as anisotropy measures of the diffusion tensor images.

DtiStudio: resource program for diffusion tensor computation and fiber bundle tracking.

H Jiang — 2006-04-20T02:06:10-00:00

Comput Methods Programs Biomed, Vol. 81, No. 2. (February 2006), pp. 106-116.

A versatile resource program was developed for diffusion tensor image (DTI) computation and fiber tracking. The software can read data formats from a variety of MR scanners. Tensor calculation is performed by solving an over-determined linear equation system using least square fitting. Various types of map data, such as tensor elements, eigenvalues, eigenvectors, diffusion anisotropy, diffusion constants, and color-coded orientations can be calculated. The results are visualized interactively in orthogonal views and in three-dimensional mode. Three-dimensional tract reconstruction is based on the Fiber Assignment by Continuous Tracking (FACT) algorithm and a brute-force reconstruction approach. To improve the time and memory efficiency, a rapid algorithm to perform the FACT is adopted. An index matrix for the fiber data is introduced to facilitate various types of fiber bundles selection based on approaches employing multiple regions of interest (ROIs). The program is developed using C++ and OpenGL on a Windows platform.

Facial expression decomposition

H Wang — 2006-07-19T18:40:45-00:00

(2003)

In this paper, we propose a novel approach for facial expression decomposition - Higher-Order Singular Value Decomposition (HOSVD), a natural generalization of matrix SVD. We learn the expression subspace and person subspace from a corpus of images showing seven basic facial expressions, rather than resort to expert-coded facial expression parameters as in [3]. We propose a simultaneous face and facial expression recognition algorithm, which can classify the given image into one of the seven...

Multilinear Analysis of Image Ensembles: TensorFaces

MAO Vasilescu — 2007-03-17T17:25:33-00:00

(2002), pp. 511-514.

Natural images are the composite consequence of multiple factors related to scene structure, illumination, and imaging. For facial images, the factors include different facial geometries, expressions, head poses, and lighting conditions. We apply multilinear algebra, the algebra of higherorder tensors, to obtain a parsimonious representation of facial image ensembles which separates these factors. Our representation, called TensorFaces, yields improved facial recognition rates relative to...

A Multilinear Singular Value Decomposition

Lieven De Lathauwer — 2007-03-17T17:11:36-00:00

SIAM Journal on Matrix Analysis and Applications, Vol. 21, No. 4. (2000), pp. 1253-1278.

We discuss a multilinear generalization of the Singular Value Decomposition.

Multilinear Analysis of Image Ensembles: TensorFaces

MAO Vasilescu — 2006-09-19T09:23:54-00:00

(2002), pp. 447-460.

Image Classification Using Correlation Tensor Analysis

Yun Fu — 2008-06-04T02:19:43-00:00

Image Processing, IEEE Transactions on, Vol. 17, No. 2. (2008), pp. 226-234.

Images, as high-dimensional data, usually embody large variabilities. To classify images for versatile applications, an effective algorithm is necessarily designed by systematically considering the data structure, similarity metric, discriminant subspace, and classifier. In this paper, we provide evidence that, besides the Fisher criterion, graph embedding, and tensorization used in many existing methods, the correlation-based similarity metric embodied in supervised multilinear discriminant subspace learning can additionally improve the classification performance. In particular, a novel discriminant subspace learning algorithm, called correlation tensor analysis (CTA), is designed to incorporate both graph-embedded correlational mapping and discriminant analysis in a Fisher type of learning manner. The correlation metric can estimate intrinsic angles and distances for the locally isometric embedding, which can deal with the case when Euclidean metric is incapable of capturing the intrinsic similarities between data points. CTA learns multiple interrelated subspaces to obtain a low-dimensional data representation reflecting both class label information and intrinsic geometric structure of the data distribution. Extensive comparisons with most popular subspace learning methods on face recognition evaluation demonstrate the effectiveness and superiority of CTA. Parameter analysis also reveals its robustness.

Iterative techniques for blind source separation using only fourth order cumulants

J Cardoso — 2007-09-07T19:02:28-00:00

(1992)

. "Blind source separation" is an array processing problem without a priori information (no array manifold). This model can be identified resorting to 4th-order cumulants only via the concept of 4th-order signal subspace (FOSS) which is defined as a matrix space. This idea leads to a "Blind MUSIC" approach where identification is achieved by looking for the (approximate) intersections between the FOSS and the manifold of 1D projection matrices. Pratical implementations of these ideas are...

Tensor rank and the ill-posedness of the best low-rank approximation problem

Vin de Silva — 2008-02-17T21:06:33-00:00

(26 Jul 2006)

There has been continued interest in seeking a theorem describing optimal low-rank approximations to tensors of order 3 or higher, that parallels the Eckart-Young theorem for matrices. In this paper, we argue that the naive approach to this problem is doomed to failure because, unlike matrices, tensors of order 3 or higher can fail to have best rank-r approximations. The phenomenon is much more widespread than one might suspect: examples of this failure can be constructed over a wide range of dimensions, orders and ranks, regardless of the choice of norm (or even Bregman divergence). Moreover, we show that in many instances these counterexamples have positive volume: they cannot be regarded as isolated phenomena. In one extreme case, we exhibit a tensor space in which no rank-3 tensor has an optimal rank-2 approximation. The notable exceptions to this misbehavior are rank-1 tensors and order-2 tensors. In a more positive spirit, we propose a natural way of overcoming the ill-posedness of the low-rank approximation problem, by using weak solutions when true solutions do not exist. In our work we emphasize the importance of closely studying concrete low-dimensional examples as a first step towards more general results. To this end, we present a detailed analysis of equivalence classes of 2-by-2-by-2 tensors, and we develop methods for extending results upwards to higher orders and dimensions. Finally, we link our work to existing studies of tensors from an algebraic geometric point of view. The rank of a tensor can in theory be given a semialgebraic description; i.e., can be determined by a system of polynomial inequalities. In particular we make extensive use of the 2-by-2-by-2 hyperdeterminant.

Tensor rank is NP-complete

Johan H\aastad — 2007-08-08T04:45:35-00:00

J. Algorithms, Vol. 11, No. 4. (December 1990), pp. 644-654.

Local Discriminant Embedding with Tensor Representation

Jian Xia — 2008-03-12T08:25:47-00:00

Image Processing, 2006 IEEE International Conference on (2006), pp. 929-932.

We present a subspace learning method, called local discriminant embedding with tensor representation (LDET), that addresses simultaneously the generalization and data representation problems in subspace learning. LDET learns multiple interrelated subspaces for obtaining a lower-dimensional embedding by incorporating both class label information and neighborhood information. By encoding each object as a second- or higher-order tensor, LDET can capture higher-order structures in the data without requiring a large sample size. Extensive empirical studies have been performed to compare LDET with a second- or third-order tensor representation and the original LDE on their face recognition performance. Not only does LDET have a lower computational complexity than LDE, but LDET is also superior to LDE in terms of its recognition accuracy

The description of growth of plant organs: a continuous approach based on the growth tensor

Jerzy Nakielski — 2008-07-07T14:25:35-00:00

(2003), pp. 119-136.

A rigorous framework for diffusion tensor calculus

PG Batchelor — 2007-12-03T16:45:52-00:00

Magnetic Resonance in Medicine, Vol. 53, No. 1. (2005), pp. 221-225.

In biological tissue, all eigenvalues of the diffusion tensor are assumed to be positive. Calculations in diffusion tensor MRI generally do not take into account this positive definiteness property of the tensor. Here, the space of positive definite tensors is used to construct a framework for diffusion tensor analysis. The method defines a distance function between a pair of tensors and the associated shortest path (geodesic) joining them. From this distance a method for computing tensor means, a new measure of anisotropy, and a method for tensor interpolation are derived. The method is illustrated using simulated and in vivo data. Magn Reson Med 53:221-225, 2005. © 2004 Wiley-Liss, Inc.

Log-Euclidean metrics for fast and simple calculus on diffusion tensors

Vincent Arsigny — 2007-12-03T16:44:40-00:00

Magnetic Resonance in Medicine, Vol. 56, No. 2. (2006), pp. 411-421.

Diffusion tensor imaging (DT-MRI or DTI) is an emerging imaging modality whose importance has been growing considerably. However, the processing of this type of data (i.e., symmetric positive-definite matrices), called ?tensors? here, has proved difficult in recent years. Usual Euclidean operations on matrices suffer from many defects on tensors, which have led to the use of many ad hoc methods. Recently, affine-invariant Riemannian metrics have been proposed as a rigorous and general framework in which these defects are corrected. These metrics have excellent theoretical properties and provide powerful processing tools, but also lead in practice to complex and slow algorithms. To remedy this limitation, a new family of Riemannian metrics called Log-Euclidean is proposed in this article. They also have excellent theoretical properties and yield similar results in practice, but with much simpler and faster computations. This new approach is based on a novel vector space structure for tensors. In this framework, Riemannian computations can be converted into Euclidean ones once tensors have been transformed into their matrix logarithms. Theoretical aspects are presented and the Euclidean, affine-invariant, and Log-Euclidean frameworks are compared experimentally. The comparison is carried out on interpolation and regularization tasks on synthetic and clinical 3D DTI data. Magn Reson Med 56, 2006. © 2006 Wiley-Liss, Inc.

A Riemannian Framework for Tensor Computing

Xavier Pennec — 2005-12-28T03:37:34-00:00

International Journal of Computer Vision, Vol. 66, No. 1. (January 2006), pp. 41-66.

Tensors and Manifolds: With Applications to Physics

Robert Wasserman — 2007-12-01T14:26:50-00:00

(22 June 2004)

This book is a new edition of "Tensors and Manifolds: With Applications to Mechanics and Relativity" which was published in 1992. It is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialised courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other's discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, as well as additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics. Mathematical rigour combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics.

Tensor Analysis

LP Lebedev — 2007-12-01T14:24:35-00:00

(24 April 2003)

Tensor analysis is an essential tool in any science (e.g. engineering, physics, mathematical biology) that employs a continuum description. This concise text offers a straightforward treatment of the subject suitable for the student or practicing engineer. The final chapter introduces the reader to differential geometry, including the elementary theory of curves and surfaces. A well-organized formula list, provided in an appendix, makes the book a very useful reference. A second appendix contains full hints and solutions for the exercises.

Tensor Geometry: The Geometric Viewpoint and its Uses (Graduate Texts in Mathematics)

Christopher Dodson — 2007-05-02T12:50:45-00:00

(07 December 2000)

This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique

Tensor Analysis on Manifolds

Samuel Goldberg — 2006-01-18T19:17:55-00:00

(01 December 1980)

<div>Proceeds from general to special, including chapters on vector analysis on manifolds and integration theory. </div>

Mathematical Physics

Sadri Hassani — 2007-12-01T14:17:12-00:00

(08 February 1999)

This book is for physics students interested in the mathematics they use and for mathematics students interested in seeing how some of the ideas of their discipline find realization in an applied setting. The presentation tries to strike a balance between formalism and application, between abstract and concrete. The interconnections among the various topics are clarified both by the use of vector spaces as a central unifying theme, recurring throughout the book, and by putting ideas into their historical context. Enough of the essential formalism is included to make the presentation self-contained. The book is divided into eight parts: The first covers finite- dimensional vector spaces and the linear operators defined on them. The second is devoted to infinite-dimensional vector spaces, and includes discussions of the classical orthogonal polynomials and of Fourier series and transforms. The third part deals with complex analysis, including complex series and their convergence, the calculus of residues, multivalued functions, and analytic continuation. Part IV treats ordinary differential equations, concentrating on second-order equations and discussing both analytical and numerical methods of solution. The next part deals with operator theory, focusing on integral and Sturm--Liouville operators. Part VI is devoted to Green's functions, both for ordinary differential equations and in multidimensional spaces. Parts VII and VIII contain a thorough discussion of differential geometry and Lie groups and their applications, concluding with Noether's theorem on the relationship between symmetries and conservation laws. Intended for advanced undergraduates or beginning graduate students, this comprehensive guide should also prove useful as a refresher or reference for physicists and applied mathematicians. Over 300 worked-out examples and more than 800 problems provide valuable learning aids.

A Course in Modern Mathematical Physics : Groups, Hilbert Space and Differential Geometry

Peter Szekeres — 2006-06-27T20:56:15-00:00

(16 December 2004)

Presenting an introduction to the mathematics of modern physics for advanced undergraduate and graduate students, this textbook introduces the reader to modern mathematical thinking within a physics context. Topics covered include tensor algebra, differential geometry, topology, Lie groups and Lie algebras, distribution theory, fundamental analysis and Hilbert spaces. The book also includes exercises and proofed examples to test the students' understanding of the various concepts, as well as to extend the text's themes.

Strategies for Direct Volume Rendering of Diffusion Tensor Fields

Gordon Kindlmann — 2007-09-22T07:11:18-00:00

IEEE Transactions on Visualization and Computer Graphics, Vol. 6, No. 2. (April 2000), pp. 124-138.

Regular representations and Huang-Lepowsky's tensor functors for vertex operator algebras

H Li — 2007-05-22T13:26:29-00:00

ArXiv Mathematics e-prints (March 2001)

This is the second paper in a series to study regular representations for vertex operator algebras. In this paper, given a module $W$ for a vertex operator algebra $V$, we construct, out of the dual space $W^*$, a family of canonical (weak) $V⊗ V$-modules called $\calD_Q(z)(W)$ parametrized by a nonzero complex number $z$. We prove that for $V$-modules $W,W_1$ and $W_2$, a $Q(z)$-intertwining map of type $W'\choose W_1W_2$ in the sense of Huang and Lepowsky exactly amounts to a $V⊗ V$-homomorphism from $W_1⊗ W_2$ to $\calD_Q(z)(W)$ and that a $Q(z)$-tensor product of $V$-modules $W_1$ and $W_2$ in the sense of Huang and Lepowsky amounts to a universal from $W_1⊗ W_2$ to the functor $\calF_Q(z)$, where $\calF_Q(z)$ is a functor from the category of $V$-modules to the category of weak $V⊗ V$-modules defined by $\calF_Q(z)(W)=\calD_Q(z)(W')$ for a $V$-module $W$. Furthermore, Huang-Lepowsky's $P(z)$ and $Q(z)$-tensor functors for the category of $V$-modules are extended to functors $T_P(z)$ and $T_Q(z)$ from the category of $V⊗ V$-modules to the category of $V$-modules. It is proved that functors $\calF_P(z)$ and $\calF_Q(z)$ are right adjoints of $T_P(z)$ and $T_Q(z)$, respectively.

Dimensionality reduction in higher-order signal processing and rank-(R1,R2,...,RN) reduction in multilinear algebra

Lieven De Lathauwer — 2007-06-27T16:06:41-00:00

Linear Algebra and its Applications, Vol. 391 (1 November 2004), pp. 31-55.

In this paper we review a multilinear generalization of the singular value decomposition and the best rank-(R1,R2,...,RN) approximation of higher-order tensors. We show that they are important tools for dimensionality reduction in higher-order signal processing. We discuss applications in independent component analysis, simultaneous matrix diagonalization and subspace variants of algorithms based on higher-order statistics.

On the Best Rank-1 and Rank-(R1,R2,. . .,RN) Approximation of Higher-Order Tensors

Lieven De Lathauwer — 2007-07-06T17:41:21-00:00

SIAM J. Matrix Anal. Appl., Vol. 21, No. 4. (2000), pp. 1324-1342.

Combining Effects: Sum and Tensor

Martin Hyland — 2005-10-05T18:09:00-00:00

We seek a unified account of modularity for computational effects. We begin by reformulating Moggi's monadic paradigm for modelling computational effects using the notion of enriched Lawvere theory, together with its relationship with strong monads; this emphasises the importance of the operations that produce the effects. Effects qua theories are then combined by appropriate bifunctors on the category of theories. We give a theory for the sum of computational effects, which in particular...

Improved hydrodynamic interaction in macromolecular bead models

B Carrasco — 2007-08-29T20:49:24-00:00

The Journal of Chemical Physics, Vol. 111, No. 10. (1999), pp. 4817-4826.

The calculation of hydrodynamic properties of macromolecules in terms of bead models requires an adequate description of the hydrodynamic interaction between the spherical elements. For this purpose, the original or modified Oseen tensor are customarily used, although it has been shown that this simple description may lead to erroneous results, particularly for rotational coefficients. In this paper we study several more elaborate theories for multisphere systems. We apply those treatments to our problem of rigid bead models, implementing them in computer programs, and making calculations for various test structures. The comparison of the results from the various theories, and from other, presumably very accurate procedures, allow us to give some guidelines to improve the treatment of hydrodynamic interactions in macromolecular bead models. These advances are introduced in new versions of our public-domain computer software. ©1999 American Institute of Physics.

A Common Framework for Distributed Representation Schemes for Compositional Structure

Tony Plate — 2008-03-18T04:12:05-00:00

Over the last few years a number of schemes for encoding compositional structure in distributed representations have been proposed, e.g., Smolensky's tensor products, Pollack's RAAMs, Plate's HRRs, Halford et al's STAR model, and Kanerva's binary spatter codes. All of these schemes can placed in a general framework involving superposition and binding of patterns. Viewed in this way, it is often simple to decide whether what can be achieved within one scheme will be able to be achieved in...

Holographic reduced representations

TA Plate — 2008-03-18T02:26:02-00:00

Neural Networks, IEEE Transactions on, Vol. 6, No. 3. (1995), pp. 623-641.

Associative memories are conventionally used to represent data with very simple structure: sets of pairs of vectors. This paper describes a method for representing more complex compositional structure in distributed representations. The method uses circular convolution to associate items, which are represented by vectors. Arbitrary variable bindings, short sequences of various lengths, simple frame-like structures, and reduced representations can be represented in a fixed width vector. These representations are items in their own right and can be used in constructing compositional structures. The noisy reconstructions extracted from convolution memories can be cleaned up by using a separate associative memory that has good reconstructive properties

Curvature of n-dimensional space curves in grey-value images

B Rieger — 2006-11-07T09:51:47-00:00

Image Processing, IEEE Transactions on, Vol. 11, No. 7. (2002), pp. 738-745.

Local curvature represents an important shape parameter of space curves which are well described by differential geometry. We have developed an estimator for local curvature of space curves embedded in n-dimensional (n-D) grey-value images. There is neither a segmentation of the curve needed nor a parametric model assumed. Our estimator works on the orientation field of the space curve. This orientation field and a description of local structure is obtained by the gradient structure tensor. The orientation field has discontinuities; walking around a closed contour yields two such discontinuities in orientation. This field is mapped via the Knutsson (1985) mapping to a continuous representation; from a n-D vector to a symmetric n/sup 2/-D tensor field. The curvature of a space curve, a coordinate invariant property, is computed in this tensor field representation. An extensive evaluation shows that our curvature estimation is unbiased even in the presence of noise, independent of the scale of the object and furthermore the relative error stays small.

On Diffusion Tensor Estimation

Marc Niethammer — 2008-07-09T08:47:03-00:00

Engineering in Medicine and Biology Society, 2006. EMBS '06. 28th Annual International Conference of the IEEE (2006), pp. 2622-2625.

In this paper we propose a formal formulation for the estimation of Diffusion Tensors in the space of symmetric positive semidefinite (PSD) tensors. Traditionally, diffusion tensor model estimation has been carried out imposing tensor symmetry without constraints for negative eigenvalues. When diffusion weighted data does not follow the diffusion model, due to noise or signal drop, negative eigenvalues may arise. An estimation method that accounts for the positive definiteness is desirable to respect the underlying principle of diffusion. This paper proposes such an estimation method and provides a theoretical interpretation of the result. A closed-form solution is derived that is the optimal data-fit in the matrix 2-norm sense, removing the need for optimization-based tensor estimation.

A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images

Olivier Coulon — 2008-07-09T09:30:16-00:00

Information Processing in Medical Imaging (2001), pp. 92-105.

A method for regularizing diffusion tensor magnetic resonance images (DT-MRI) is presented. The scheme is divided into two main parts: a restoration of the principal diffusion direction, and a regularization of the 3 eigenvalue maps. The former make use of recent variational methods for restoring direction maps, while the latter makes use of the strong structural information embedded in the diffusion tensor image to drive a non-linear anisotropic diffusion process. The whole process is illustrated on synthetic and real data, and possible improvements are discussed.

A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI

Zhizhou Wang — 2008-05-02T14:52:57-00:00

Medical Imaging, IEEE Transactions on, Vol. 23, No. 8. (2004), pp. 930-939.

In this paper, we present a novel constrained variational principle for simultaneous smoothing and estimation of the diffusion tensor field from complex valued diffusion-weighted images (DWI). The constrained variational principle involves the minimization of a regularization term of L/sup p/ norms, subject to a nonlinear inequality constraint on the data. The data term we employ is the original Stejskal-Tanner equation instead of the linearized version usually employed in literature. The complex valued nonlinear form leads to a more accurate (when compared to the linearized version) estimate of the tensor field. The inequality constraint requires that the nonlinear least squares data term be bounded from above by a known tolerance factor. Finally, in order to accommodate the positive definite constraint on the diffusion tensor, it is expressed in terms of Cholesky factors and estimated. The constrained variational principle is solved using the augmented Lagrangian technique in conjunction with the limited memory quasi-Newton method. Experiments with complex-valued synthetic and real data are shown to depict the performance of our tensor field estimation and smoothing algorithm.

Derivatives and Rates of the Stretch and Rotation Tensors

Luciano Rosati — 2007-04-10T12:39:10-00:00

Journal of Elasticity, Vol. 56, No. 3. (1999), pp. 213-230.

A novel approach to the solution of the tensor equation AX+XA=H

L Rosati — 2007-04-10T12:23:53-00:00

International Journal of Solids and Structures, Vol. 37, No. 25. (June 2000), pp. 3457-3477.

A systematic approach to the solution of the tensor equation AX+XA=H, where A is symmetric, is presented. It is based upon the reformulation of the original equation in the form X=H where is the fourth-order tensor obtained from the square tensor product of the second-order tensors A and 1. It is shown that the solution X, which is known to be an isotropic function of A and H, can be effectively obtained either by providing explicit formulas for -1 or by reconverting to the format X=H the well-known representation formulas for tensor-valued isotropic functions. The final form of the solution can thus be established a priori by suitably choosing a set of independent generators for -1. The coefficients of the expansion of -1 with respect to the assigned generators are then obtained by means of basic composition rules for square tensor products. In this way it is possible to provide new expressions of the solution as well as to derive the existing ones in a simpler way. Both three-dimensional and two-dimensional cases are addressed in detail.

On the frame invariance of linear elasticity theory

Steigmann — 2007-01-19T13:07:36-00:00

Zeitschrift für angewandte Mathematik und Physik ZAMP, Vol. 58, No. 1. (January 2007), pp. 121-136.

On the gradients of the principal invariants of a second-order tensor

Jovo Jaric — 2007-08-29T14:44:53-00:00

Journal of Elasticity, Vol. 44, No. 3. (1996), pp. 285-287.

In this paper the gradients of the principal invariants of an arbitrary second-order tensor are derived in a very concise way.

Twirl tensors and the tensor equation AX−XA=C

Zhong-Heng Guo — 2007-04-15T13:52:04-00:00

Journal of Elasticity, Vol. 27, No. 3. (1 March 1992), pp. 227-245.

Principal axis intrinsic method and the high dimensional tensor equation AX−XA=C

Liang Haoyun — 2007-04-15T13:49:04-00:00

Applied Mathematics and Mechanics, Vol. 17, No. 10. (1 October 1996), pp. 945-951.

Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor

Guan-Suo Dui — 2006-04-12T12:36:32-00:00

Journal of Elasticity, Vol. 83, No. 2. (May 2006), pp. 113-151.

Representing Tensor Functions with a Cholesky Transform

Alan Freed — 2007-11-08T19:48:56-00:00

Mathematics and Mechanics of Solids, Vol. 4, No. 2. (1 June 1999), pp. 169-181.

The tensor representation theorem provides a powerful tool for evaluating isotropic tensor-valued tensor functions. Efficient computational techniques exist when an analysis is being done in rectangular Cartesian coordinates. Introducing a Cholesky decomposition of the metric tensor permits the tensor polynomial of the tensor representation theorem, when expressed in curvilinear coordinates, to be transformed into a pseudo-Cartesian frame where these efficient algorithms apply. It also leads to accurate descriptions for the physical components of a tensor. 10.1177/108128659900400202

A closed-form representation for the derivative of non-symmetric tensor power series

Mikhail Itskov — 2007-06-18T18:24:41-00:00

International Journal of Solids and Structures, Vol. 39, No. 24. (December 2002), pp. 5963-5978.

Fourth-order tensors - tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis

O Kintzel — 2007-06-18T18:18:57-00:00

ZAMM, Vol. 86, No. 4. (2006), pp. 291-311.

The present contribution provides a tensor formalism for fourth-order tensors in the so-called absolute notation and focusses in particular on the use of this notation in the process of tensor differentiation with respect to a second-order tensor. Three tensor products, two new double contraction rules and a set of well-defined notations are introduced which in combination with the tensor differentiation rules simplify analytical derivation procedures considerably and provide significant advantages for various tasks in continuum mechanics. The suitability of the proposed rules and definitions is demonstrated in a number of relevant problems of continuum mechanics such as linearization of the generalized midpoint-rule and the exponential function. Special attention is given to the differentiation with respect to symmetric, skew-symmetric and inverse second-order tensors.

The tensor equation AX+XA=Φ(A,H), with applications to kinematics of continua

Mike Scheidler — 2007-04-24T17:18:36-00:00

Journal of Elasticity, Vol. 36, No. 2. (1 January 1994), pp. 117-153.

On the representation of the elasticity tensor for isotropic materials

James Knowles — 2007-08-29T14:30:04-00:00

Journal of Elasticity, Vol. 39, No. 2. (1 May 1995), pp. 175-180.

The Derivative with respect to a Tensor: some Theoretical Aspects and Applications

M Itskov — 2007-06-18T18:12:44-00:00

ZAMM, Vol. 82, No. 8. (2002), pp. 535-544.

The object of the paper is the derivative with respect to a second-order tensor. Basis-free as well as basis-related definitions of the derivative are addressed and compared. Special attention is focused on the derivative of a tensor function defined on a subset of all linear mappings within the real vector space, symmetric second-order tensors representing a most important example of such a subset. Due to a special definition of the simple contraction of fourth- and second-order tensors the well-known product rule of differentiation valid for scalar functions is proved to hold also for second-order tensor functions. Introducing a new tensor product of second-order tensors, another tensor differentiation rules as well as some useful tensor algebra operations are formulated. Applying this formalism the derivative of non-symmetric tensor power series is obtained in a closed form. This closed-formula solution is finally illustrated by some examples of continuum mechanics. As such, simple expressions for the derivative of the stretch and rotation tensor with respect to the deformation gradient and for the stresses conjugate to the logarithmic and more general Hill's strains are presented.

Explicit formulations of tangent stiffness tensors for isotropic materials

Guansuo Dui — 2007-09-02T16:14:38-00:00

International Journal for Numerical Methods in Engineering, Vol. 69, No. 4. (2007), pp. 665-675.

A compact explicit expression for the tangent stiffness tensor is presented. Throughout the analysis, the formulation holds for general isotropic elastic materials and does not require solving eigenvector problems. On the theoretical side, a very simple solution of a tensor equation is obtained. Then the expressions for the derivatives of general symmetric isotropic tensor functions of a symmetric tensor are developed. On the computational side, particular attention is given to the consideration of the special case, Green elastic materials, in which the strain energy does not admit a closed-form expression in terms of principal invariants. Finally, a simple formulation of the tangent stiffness tensor for Ogden material model is supplied. Copyright © 2006 John Wiley & Sons, Ltd.

Constraint groups

Alexandru Danescu — 2007-08-29T14:16:39-00:00

Journal of Elasticity, Vol. 37, No. 1. (1 January 1994), pp. 91-92.

The complete list of constraint manifolds having the group structure is given.

On the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor

Zhi-Qiao Wang — 2007-09-02T16:13:30-00:00

International Journal of Solids and Structures, Vol. 44, No. 16. (1 August 2007), pp. 5369-5379.

This paper studies a subclass of isotropic tensor-valued functions of a nonsymmetric tensor, which satisfy the commutative condition, and their derivatives. This subclass of tensor functions includes tensor power series, exponential tensor function, etc., and is more general than those investigated before. In the case of three distinct eigenvalues, the derivatives of these tensor functions are constructed by solving a tensor equation, which is acquired by differentiating the commutative condition. By taking limits, the results are extended to the cases of repeated eigenvalues.

Lamb's problem for solids of general anisotropy

Wang — 2006-11-07T18:55:38-00:00

Wave Motion, Vol. 24, No. 3. (November 1996), pp. 227-242.

A method to construct solutions for elastic waves generated in a half-space has been developed, based on representing the wave field by a superposition of time-transient plane waves. By the use of this method, the classical Lamb's problem has been solved for a solid of general anisotropy. New expressions for the 2-D and 3-D solutions to Lamb's problem in both the time-domain and the frequency-domain have been obtained. These expressions, which have been given in terms of integrals defined in a finite domain, have a simple structure which is attractive for numerical applications. The usefulness and computability of the integral expressions have been demonstrated by numerical examples.

CiteULike: Tag tensor

Reduction of noise in diffusion tensor images using anisotropic smoothing.

DtiStudio: resource program for diffusion tensor computation and fiber bundle tracking.

Facial expression decomposition

Multilinear Analysis of Image Ensembles: TensorFaces

A Multilinear Singular Value Decomposition

Multilinear Analysis of Image Ensembles: TensorFaces

Image Classification Using Correlation Tensor Analysis

Iterative techniques for blind source separation using only fourth order cumulants

Tensor rank and the ill-posedness of the best low-rank approximation problem

Tensor rank is NP-complete

Local Discriminant Embedding with Tensor Representation

The description of growth of plant organs: a continuous approach based on the growth tensor

A rigorous framework for diffusion tensor calculus

Log-Euclidean metrics for fast and simple calculus on diffusion tensors

A Riemannian Framework for Tensor Computing

Tensors and Manifolds: With Applications to Physics

Tensor Analysis

Tensor Geometry: The Geometric Viewpoint and its Uses (Graduate Texts in Mathematics)

Tensor Analysis on Manifolds

Mathematical Physics

A Course in Modern Mathematical Physics : Groups, Hilbert Space and Differential Geometry

Strategies for Direct Volume Rendering of Diffusion Tensor Fields

Regular representations and Huang-Lepowsky's tensor functors for vertex operator algebras

Dimensionality reduction in higher-order signal processing and rank-(R1,R2,...,RN) reduction in multilinear algebra

On the Best Rank-1 and Rank-(<i>R</i><sub>1</sub>,<i>R</i><sub>2</sub>,. . .,<i>R<sub>N</sub></i>) Approximation of Higher-Order Tensors

Combining Effects: Sum and Tensor

Improved hydrodynamic interaction in macromolecular bead models

A Common Framework for Distributed Representation Schemes for Compositional Structure

Holographic reduced representations

Curvature of n-dimensional space curves in grey-value images

On Diffusion Tensor Estimation

A Regularization Scheme for Diffusion Tensor Magnetic Resonance Images

A constrained variational principle for direct estimation and smoothing of the diffusion tensor field from complex DWI

Derivatives and Rates of the Stretch and Rotation Tensors

A novel approach to the solution of the tensor equation AX+XA=H

On the frame invariance of linear elasticity theory

On the gradients of the principal invariants of a second-order tensor

Twirl tensors and the tensor equation AX−XA=C

Principal axis intrinsic method and the high dimensional tensor equation AX−XA=C

Some Basis-Free Formulae for the Time Rate and Conjugate Stress of Logarithmic Strain Tensor

Representing Tensor Functions with a Cholesky Transform

A closed-form representation for the derivative of non-symmetric tensor power series

Fourth-order tensors - tensor differentiation with applications to continuum mechanics. Part I: Classical tensor analysis

The tensor equation AX+XA=Φ(A,H), with applications to kinematics of continua

On the representation of the elasticity tensor for isotropic materials

The Derivative with respect to a Tensor: some Theoretical Aspects and Applications

Explicit formulations of tangent stiffness tensors for isotropic materials

Constraint groups

On the derivatives of a subclass of isotropic tensor functions of a nonsymmetric tensor

Lamb's problem for solids of general anisotropy