Invariant and orthonormal scalar measures derived from magnetic resonance diffusion tensor imaging

A diffusion tensor is a mathematical construct used to describe water diffusion in complicated biological structures. It describes a process which occurs in all directions simultaneously, It is difficult to comprehend or graphically display the information in the diffusion tensor. This paper describes a coordinate system approach for producing scalar measures which characterize key aspects of the diffusion tensor. The eigenvalues of the diffusion tensor are introduced as the three elements of a point in a Cartesian coordinate system. The Cartesian coordinates are then expressed in cylindrical and spherical coordinates. The orthonormal coordinates of the spherical system are particularly useful scalar measures of attributes of the diffusion tensor: One coordinate contains all the information about the overall magnitude of diffusion. Another contains all of the anisotropy information. The third coordinate contains all of the information about skewness. No information is lost when transforming the original eigenvalues to spherical coordinates. (C) 1999 Academic Press.