Total Variation Regularization of Matrix-Valued Images

Christiansen, Oddvar; Lee, Tin-Man; Lie, Johan; Sinha, Usha; Chan, Tony F.

doi:https://doi.org/10.1155/2007/27432

International Journal of Biomedical Imaging

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Mathematics in Biomedical Imaging

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Research Article | Open Access

Volume 2007 | Article ID 027432 | https://doi.org/10.1155/2007/27432

Total Variation Regularization of Matrix-Valued Images

Oddvar Christiansen,¹Tin-Man Lee,²Johan Lie,¹Usha Sinha,²and Tony F. Chan³

Academic Editor: Hongkai Zhao

Received30 Oct 2006

Accepted13 Mar 2007

Published06 Jun 2007

Abstract

We generalize the total variation restoration model, introduced by Rudin, Osher, and Fatemi in 1992, to matrix-valued data, in particular, to diffusion tensor images (DTIs). Our model is a natural extension of the color total variation model proposed by Blomgren and Chan in 1998. We treat the diffusion matrix D implicitly as the product D=LLT, and work with the elements of L as variables, instead of working directly on the elements of D. This ensures positive definiteness of the tensor during the regularization flow, which is essential when regularizing DTI. We perform numerical experiments on both synthetical data and 3D human brain DTI, and measure the quantitative behavior of the proposed model.

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Copyright

Copyright © 2007 Oddvar Christiansen 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.

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