A second order tensor is usually used to describe the diffusion of water for each voxel within a Diffusion Tensor Magnetic Resonance (DT-MR) images. However, a second order tensor approximation fails to accurately represent complex local tissue structures such as crossing fibers. Therefore, higher order tensors are used to represent more complex diffusivity profiles. In this work we examine and compare segmentations of both second order and fourth order DT-MR images using the Random Walker segmentation algorithm with the emphasis of pointing the shortcomings of second order tensor model in segmenting regions with complex fiber structures. We first adopt the Random Walker algorithm for segmenting diffusion tensor data by using appropriate tensor distance metrics and then demonstrate the advantages of performing segmentation on higher order DT-MR data. The approach proposed takes advantage of all the information provided by the tensors by using suitable tensor distance metrics. The distance metrics used are: the Log-Euclidean for the second order tensors and the normalized L 2 distance for the fourth order tensors. The segmentation is carried out on a weighted graph that represents the image where, the tensors are the nodes and the edge weights are computed using the tensor distance metrics. Applying the approach to both synthetic and real DT-MRI data yields segmentations that are both robust and qualitatively accurate.
Fractional anisotropy (FA), defined as the distance of a diffusion tensor from its closest isotropic tensor, has been extensively studied as quantitative anisotropy measure for diffusion tensor magnetic resonance images (DT-MRI). It has been used to reveal the white matter profile of brain images, as guiding feature for seeding and stopping in fiber tractography and for the diagnosis and assessment of degenerative brain diseases. Despite its extensive use in DT-MRI community, however, not much attention has been given to the mathematical correctness of its derivation from diffusion tensors which is achieved using Euclidean dot product in 9D space. But, recent progress in DT-MRI have shown that the space of diffusion tensors does not form a Euclidean vector space and thus Euclidean dot product is not appropriate for tensors. In this paper, we propose a novel and robust rotationally invariant diffusion anisotropy measure derived using the recently proposed Log-Euclidean and Jdivergence tensor distance measures. An interesting finding of our work is that given a diffusion tensor, its closest isotropic tensor is different for different tensor distance used. We demonstrate qualitatively that our new anisotropy measure reveals superior white matter profile of DT-MR brain images and analytically show that it has a higher signal to noise ratio than FA.
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