It is Not Possible to Design a Rotationally Invariant Sampling Scheme for DT-MRI

INTRODUCTION: The choice of gradient sampling orientations for DT-MRI experiments has exercised many groups in the field. Of particular interest is the design of sampling schemes that will ensure that the error / variance in tensor estimates is independent of the relative orientation of the tensor to the reference frame established by the sampling vectors, which others have termed ‘statistical rotational invariance’ or ‘SRI’. Previous work has identified a relationship between the condition number of the quadratic encoding matrix, formed from the gradient sa mpling orientations, and the variance in the estima ted diffusion tensor. Batchelor et al. showed that certain sampling schemes (e.g., the dual-gradient scheme) had rotationally variant condition numbers while, those derived from the vertices of an icosahedron had rotationally invariant condition numbers. Thus, schemes based on vectors pointing to the vertices of an icosahedron (or tessellated icosahedrons) cause initial excitement. However, Jones showed that, in discord with this theory, not all icosahedral schemes are the same – and that rotationally invariant condition number is a necessary but insufficient requirement for rotational invariance. A more general framework has been proposed – which provides a template for sampling schemes in terms of the fourth order precision matrix. Previous theoretical considerations suggested that, under the linear framework, a statistically rotationally invariant sampling scheme could not be designed, except for the trivial case of isotropic tensors. Here, we address an important and long outstanding question in the DT-MRI literature, i.e. In the limit of an infinite number of sa mpling orientations, can one design a sampling scheme which is statistically rotationally invariant?