The design of an optimal gradient encoding scheme (GES) is a fundamental problem in diffusion MRI. It iswell studied for the case of second-order tensor imaging (Gaussian diffusion). However, it has not been investigatedfor the wide range of non-Gaussian diffusion models. The optimal GES is the one that minimizes the variance ofthe estimated parameters. Such a GES can be realized by minimizing the condition number of the design matrix(K-optimal design). In this paper, we propose a new approach to solve theK-optimal GES design problem for fourth-order tensor-based diffusion profile imaging. The problem is a nonconvex experiment design problem. Using convexrelaxation, we reformulate it as a tractable semidefinite programming problem. Solving this problem leads to severaltheoretical properties ofK-optimal design: (i) the odd moments of theK-optimal design must be zero; (ii) the evenmoments of theK-optimal design are proportional to the total number of measurements; (iii) theK-optimal design isnot unique, in general; and (iv) the proposed method can be used to compute theK-optimal design for an arbitrarynumber of measurements. Our Monte Carlo simulations support the theoretical results and show that, in comparisonwith existing designs, theK-optimal design leads to the minimum signal deviation.
展开▼
