12 Tensor Invariants and their Gradients

Second-order tensors may be described in terms of shape and orientation. Shape is quantified by tensor invariants, which are fixed with respect to coordinate system changes. This chapter describes an anatomically-motivated method of detecting edges in diffusion tensor fields based on the gradients of invariants. Three particular invariants (the mean, variance, and skewness of the tensor eigenvalues) are described in two ways: first, as the geometric parameters of an intuitive graphical device for representing tensor shape (the eigenvalue wheel), and second, in terms of their physical and anatomical significance in diffusion tensor MRI. Tensor-valued gradients of these invariants lead to an orthonormal basis for describing changes in tensor shape. The spatial gradient of the diffusion tensor field may be projected onto this basis, producing three different measures of edge strength, selective for different kinds of anatomical boundaries. The gradient measures are grounded in standard tensor analysis, and are demonstrated on synthetic data.