International Journal of Computer Vision

, Volume 69, Issue 1, pp 93–107 | Cite as

Curvature-Driven PDE Methods for Matrix-Valued Images

  • Christian Feddern
  • Joachim Weickert
  • Bernhard Burgeth
  • Martin Welk


Matrix-valued data sets arise in a number of applications including diffusion tensor magnetic resonance imaging (DT-MRI) and physical measurements of anisotropic behaviour. Consequently, there arises the need to filter and segment such tensor fields. In order to detect edge-like structures in tensor fields, we first generalise Di Zenzo’s concept of a structure tensor for vector-valued images to tensor-valued data. This structure tensor allows us to extend scalar-valued mean curvature motion and self-snakes to the tensor setting. We present both two-dimensional and three-dimensional formulations, and we prove that these filters maintain positive semidefiniteness if the initial matrix data are positive semidefinite. We give an interpretation of tensorial mean curvature motion as a process for which the corresponding curve evolution of each generalised level line is the gradient descent of its total length. Moreover, we propose a geodesic active contour model for segmenting tensor fields and interpret it as a minimiser of a suitable energy functional with a metric induced by the tensor image. Since tensorial active contours incorporate information from all channels, they give a contour representation that is highly robust under noise. Experiments on three-dimensional DT-MRI data and an indefinite tensor field from fluid dynamics show that the proposed methods inherit the essential properties of their scalar-valued counterparts.


DT-MRI denoising segmentation edge detection structure tensor mean curvature motion self-snakes active contours 


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Copyright information

© Springer Science + Business Media, LLC 2006

Authors and Affiliations

  • Christian Feddern
    • 1
  • Joachim Weickert
    • 1
  • Bernhard Burgeth
    • 1
  • Martin Welk
    • 1
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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