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International Journal of Biomedical Imaging
Volume 2012 (2012), Article ID 192730, 9 pages
Research Article

Fast and Analytical EAP Approximation from a 4th-Order Tensor

ATHENA Research Team, INRIA Sophia Antipolis Méditerranée, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France

Received 20 July 2012; Accepted 2 December 2012

Academic Editor: Carl-Fredrik Westin

Copyright © 2012 Aurobrata Ghosh and Rachid Deriche. 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.


Generalized diffusion tensor imaging (GDTI) was developed to model complex apparent diffusivity coefficient (ADC) using higher-order tensors (HOTs) and to overcome the inherent single-peak shortcoming of DTI. However, the geometry of a complex ADC profile does not correspond to the underlying structure of fibers. This tissue geometry can be inferred from the shape of the ensemble average propagator (EAP). Though interesting methods for estimating a positive ADC using 4th-order diffusion tensors were developed, GDTI in general was overtaken by other approaches, for example, the orientation distribution function (ODF), since it is considerably difficult to recuperate the EAP from a HOT model of the ADC in GDTI. In this paper, we present a novel closed-form approximation of the EAP using Hermite polynomials from a modified HOT model of the original GDTI-ADC. Since the solution is analytical, it is fast, differentiable, and the approximation converges well to the true EAP. This method also makes the effort of computing a positive ADC worthwhile, since now both the ADC and the EAP can be used and have closed forms. We demonstrate our approach with 4th-order tensors on synthetic data and in vivo human data.