The algebra of fourth-order tensors with application to diffusion MRI

Abstract

In this chapter we give different descriptions of fourth-order tensors. We show that under certain symmetries it is possible to describe a fourth-order tensor in three-dimensional space by a second-order tensor in a six-dimensional space. Such a representation makes the manipulation of fourth-order tensors similar to that of the more familiar second-order tensors. We discuss the algebra of the space of fourth-order symmetric tensors and describe different metrics on this space. Special emphasis is placed on totally symmetric tensors and on orientation tensors. Applications to high angular resolution diffusion imaging are discussed.