Gordon Kindlman

"Tensor invariants, their gradients, and their fallings"

Diffusion tensor MRI (DT-MRI) is a powerful method of non-invasively measuring in vivo tissue microstructure. The visualization and analysis of DT-MRI is a challenging task due to the multi-variate nature of the tensor data and complexity of the three-dimensional structures in question. Basic ingredients of the visualization and analysis are tensor invariants-- tensor metrics independent of the coordinate frame in which the tensor is expressed. The spatial-domain gradients of these invariants enable improved visualization by providing an approximate surface normal for shading purposes. On the other hand, the value-domain gradients of the invariants provide means of characterizing the degrees of freedom in tensor shape. An interesting aspect of the invariants is their failure to characterize tensor variations near points of isotropy, at which two or three eigenvalues are equal. A framework for overcoming this limitation is described, and is used in an application to PDE-based filtering of tensor values.