On Characterizing and Analyzing Diffusion Tensor Images by Learning their Underlying Manifold Structure

The growing importance of diffusion tensor imaging (DTI) in studying the white matter architecture in normal and pathologic states necessitates the development of tools for comprehensive analysis of diffusion tensor data. Operations such as multivariate statistical analysis and hypothesis testing, interpolation and filtering, must now be performed on tensor data, and must overcome challenges introduced by the non-linearity and high dimensionality of the tensors. In this paper, we present a novel approach to performing these computations by modeling the underlying manifold structure of the tensors, using a combination of two manifold learning techniques, isometric mapping (ISOMAP) and local tangent space alignment (LTSA). While ISOMAP identifies the dimensionality of the manifold of the tensors and embeds the tensors into a linear space, facilitating statistical computations therein, operations like interpolation and filtering, integral to the process of normalization, require the reconstruction of the tensor in the tensor domain. To obtain this reverse mapping from the linear space to the tensor domain, i.e. to the domain of the original tensor data, we use LTSA. The modeling of the underlying manifold structure renders our approach better applicable to tensor data than existing methods that may not always be able to capture the non-linearity present in the tensors under consideration. In various simulations with known ground truth, we demonstrate the effectiveness of our framework based on ISOMAP and LTSA in performing a comprehensive analysis of DTI data.