Abstract

Calculating the variance of a family of tensors, each represented by a symmetric positive semi-definite second order tensor/matrix, involves the formation of a fourth order tensor Rabcd. To form this tensor, the tensor product of each second order tensor with itself is formed, and these products are then summed, giving the tensor Rabcd the same symmetry properties as the elasticity tensor in continuum mechanics. This tensor has been studied with respect to many properties: representations, invariants, decomposition, the equivalence problem et cetera. In this paper we focus on the two-dimensional case where we give a set of invariants which ensures equivalence of two such fourth order tensors Rabcd and . In terms of components, such an equivalence means that components Rijkl of the first tensor will transform into the components of the second tensor for some change of the coordinate system.

Highlights

  • Positive semi-definite second order tensors arise in several applications

  • In particular we study the equivalence problem, namely, we ask the question: given the components Ri jkl and Ri jkl of two such tensors do they represent the same tensor in different coordinate systems?

  • We started with a family of symmetric positivedefinite tensors in two dimensions and considered its variance

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Summary

Introduction

Positive semi-definite second order tensors arise in several applications. In image processing, a structure tensor is computed from greyscale images that captures the local orientation of the image intensity variations [10, 17] and is employed to address a broad range of challenges. Diffusion tensor magnetic resonance imaging (DT-MRI) [1, 5] characterizes anisotropic water diffusion by enabling the measurement of the apparent diffusion tensor, which makes it possible to delineate the fibrous structure of the tissue. Recent work has shown that diffusion MR measurements of

Herberthson (B)
Outline
Preliminaries
Tensor Notation and Representations
The Vector Space of Symmetric Two-Tensors
The Tensor Rabcd and the Equivalence Problem
The Voigt/Kelvin Notation
Visualization in R3
Invariants, Traces and Decompositions
Natural Traces and Invariants
A Canonical Decomposition
Rabcd as a Quadratic Form on R3
Representation of the Canonically Derived Parts of Rabcd
The Behaviour of Mi j Under a Rotation of the Coordinate System in V a
The Equivalence Problem for Rabcd
Different Ways to Characterize the Equivalence of Rabcd and Rabcd
Orientation of the Ellipsoid in R3
Components in a Canonical Coordinate System
Equivalence Through (algebraic) Invariants of Rabcd
Discussion

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