Diffusion tensor magnetic resonance imaging (DT-MRI) is a powerful non-invasive imaging modality whose processing, analysis and visualization has become a strong focus in medical imaging research. In this modality, the direction of the water diffusion is locally modeled by a Gaussian probability density function whose covariance matrix is a second order 3 X 3 symmetric positive definite matrix, also called the tensor here. The manifold-valued nature of the data as well as its high dimensionality makes the computational analysis of DT images complex. Very often, the data dimensionality is reduced to a single scalar derived from the tensors. Another common approach has been to ignore the restriction to the manifold of symmetric second-order tensors and, instead, treat the data as a multi-valued image. In this thesis, we try to address the above challenges posed by DT data using two different approaches. Our first contribution employs a geometric approach for representing DT data as low dimensional manifold embedded in higher dimensional space and then applying mathematical tools traditionally used in the study of Riemannian geometry for formulating first order and second order differential operators for DT images. Our second contribution is an algebraic one, where the key novel idea is to represent the DT data using the 8 dimensional hypercomplex algebra-biquaternions. This approach enables the processing of DT images in a holistic manner and facilitates the seamless introduction of traditional signal processing methodologies from biquaternion theory such as computing the Fourier transform, convolution, and edge detection for DT images. The preliminary results on synthetic and real DT data show great promise in both our approaches for DT image processing. In particular, we demonstrate greater detection ability of our features over scalar based approaches such as fractional anisotropy and show novel applications of our new biquaternion tools that have not been possible before for DT images.Applied Science, Faculty ofElectrical and Computer Engineering, Department ofGraduat
