Approximating symmetric positive semi-definite tensors of even order

by Barmpoutis, A. | Ho, J. | Vemuri, B. C.

SIAM Journal on Imaging Sciences, vol. 5(1), 2012, pp. 434-464. https://dx.doi.org/10.1137/100801664

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Description

Tensors of various orders can be used for modeling physical quantities such as strain and diffusion as well as curvature and other quantities of geometric origin. Depending on the physical properties of the modeled quantity, the estimated tensors are often required to satisfy the positivity constraint, which can be satisfied only with tensors of even order. Although the space P^2m_0 of 2m^th- order symmetric positive semi-definite tensors is known to be a convex cone, enforcing positivity constraint directly on P^2m_0 is usually not straightforward computationally because there is no known analytic description of P^2m_0 for m > 1. In this paper, we propose a novel approach for enforcing the positivity constraint on even-order tensors by approximating the cone P^2m_0 for the cases 0 < m < 3, and presenting an explicit characterization of the approximation Sigma_2m < Omega_2m for m >= 1, using the subset Omega_2m < P^2m_0 of semi-definite tensors that can be written as a sum of squares of tensors of order m. Furthermore, we show that this approximation leads to a non-negative linear leastsquares (NNLS) optimization problem with the complexity that equals the number of generators in Sigma_2m. Finally, we experimentally validate the proposed approach and we present an application for computing 2mth-order diffusion tensors from Diffusion Weighted Magnetic Resonance Images.

Additional information

Author

Barmpoutis, A., Ho, J., Vemuri, B. C.
Journal

SIAM Journal on Imaging Sciences
Volume

5
Number

1
Year

2012
Pages

434-464
DOI

https://dx.doi.org/10.1137/100801664

Citation

Citation

Barmpoutis, A., Ho, J. and Vemuri, B., 2012. Approximating symmetric positive semi-definite tensors of even order. SIAM Journal on Imaging Sciences, vol. 5(1), pp. 434-464. https://dx.doi.org/10.1137/100801664

BibTex

@article{digitalWorlds:152,
doi = {https://dx.doi.org/10.1137/100801664},
author = {Barmpoutis, A.  and  Ho, J.  and  Vemuri, B. C.},
title = {Approximating symmetric positive semi-definite tensors of even order},
journal = {SIAM Journal on Imaging Sciences},
volume = {5},
number = {1},
year = {2012},
pages = {434-464}
}

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