Abstract

The general classes of medical image processing and analysis procedures carried out over scalar images (viz. registration and visualization of X-ray, CT-scan, filtering, image segmentation, ultrasound, diffusion weighted MRI and others) need to be enlarged to diffusion tensor magnetic resonance imaging (DT-MRI) tensor fields in order to bring together qualitative and quantitative information, potentially advancing computer assisted diagnosis, following up best treatment and checking for major neuropsychiatric ailments and statistical analysis of structural and functional unpredictability of individual anatomy of human being. Moreover, DT-MRI offers a measurement of a symmetric second rank translational diffusion tensor Dof water molecules for each voxel within an imaging volume. Recently, various results concerning to noise in the estimate of diffusion tensor via an ideal DT-MRI experiment using Gaussian distribution have been established by Pajevic et al. ( 1999; 2003). In this analysis, second order symmetric diffusion tensors were arranged in the form of 6  QUOTE  1 vector random variables. Forming such a vector, random variable’s combination sometimes rise up the circumstances that do not preserve certain intrinsic algebraic relationships among the components of D and its geometric character with reference to laboratory co-ordinate systems in which it is measured. Here, our main object is to address the problem of applying spatial transformations (sometimes called image warps) to DT-MRI using certain geometric operations, viz. conformal collineation, affine collineation, isometric collineation and projective collineation which would most probably introduce some new dimensions in favor of DT-MRI studies. To study such spatial transformations, we put forward a natural interpretation of the “degree of connectivity” between two adjacent points of fiber or fibers in the manifold of human brain. This is because of the reason that diffusion operator  QUOTE   can naturally be associated with a Riemannian metric tensor G via the relation  QUOTE   and once we have the metric tensor G, we will be able to apply geometric operations of Riemannian geometry to DT-MRI study. Also, in the present article we shall discuss geodesic fibers of cortical brain manifold up to large extent.

Key words: Diffusion tensor, magnetic resonance, DT-MRI, spatial transformation, warp, Riemannian manifold, affine collineation, isometric, conformal, degree of connectivity.