Abstract

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. Performance of PSDT is depicted on synthetic data as well as MRI data. PSDT can also be regarded as a conic linear programming (CLP) problem. Yinyu Ye and I investigated PSDT from the viewpoint of CLP. We characterize the dual cone of the positive semi-definite space tensor cone, and study the CLP formulation and duality of the positive semi-definite space tensor programming (STP) problem.