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BioMed Research International
Volume 2015, Article ID 760230, 10 pages
Research Article

K-Optimal Gradient Encoding Scheme for Fourth-Order Tensor-Based Diffusion Profile Imaging

1Department of Signals and Systems, Chalmers University of Technology, 41296 Gothenburg, Sweden
2Centre for Microscopy, Characterisation and Analysis, The University of Western Australia, Perth, WA 6009, Australia
3Department of Radiology, Sahlgrenska University Hospital, Gothenburg University, 41345 Gothenburg, Sweden
4Department of Radiation Physics, Institute of Clinical Sciences, University of Gothenburg, 41345 Gothenburg, Sweden
5Department of Medical Physics and Biomedical Engineering, Sahlgrenska University Hospital, 41345 Gothenburg, Sweden

Received 22 June 2015; Revised 26 August 2015; Accepted 27 August 2015

Academic Editor: Sergio Murgia

Copyright © 2015 Mohammad Alipoor et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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