Higher Order Positive Semidefinite Diffusion Tensor Imaging

Due to the well-known limitations of diffusion tensor imaging (DTI), high angular resolution diffusion imaging (HARDI) is used to characterize non-Gaussian diffusion processes. One approach to analyze HARDI data is to model the apparent diffusion coefficient (ADC) with higher order diffusion tensors (HODT). The diffusivity function is positive semi-definite. In the literature, some methods have been proposed to preserve positive semi-definiteness of second order and fourth order diffusion tensors. None of them can work for arbitrary high order diffusion tensors. In this paper, we propose a comprehensive model to approximate the ADC profile by a positive semi-definite diffusion tensor of either second or higher order. We call this model PSDT (positive semi-definite diffusion tensor). PSDT is a convex optimization problem with a convex quadratic objective function constrained by the nonnegativity requirement on the smallest Z-eigenvalue of the diffusivity function. The smallest Z-eigenvalue is a computable measure of the extent of positive definiteness of the diffusivity function. We also propose some other invariants for the ADC profile analysis. Experiment results show that higher order tensors could improve the estimation of anisotropic diffusion and the PSDT model can ∗This work was partly supported by the Research Grant Council of Hong Kong, a postdoctoral fellowship from the Department of Applied Mathematics at the Hong Kong Polytechnic University, a grant from the Ph.D. Programs Foundation of Ministry of Education of China (No.200805581022) and the National Natural Science Foundation of China (No.10926029). †Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. E-mail: [email protected]. ‡Jiangxi Key laboratory of numerical simulation technology, School of Mathematics and Computer Sciences, GanNan Normal University, Ganzhou, 341000, China. E-mail: [email protected]. Present address: Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong. §Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong. E-mail: [email protected].