Representation and Estimation of Tensor-Pairs

  • Carl-Fredrik Westin
  • Hans Knutsson
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)


Over the years, several powerful models have been developed to represent specific elementary signal patterns, e.g. locally linear and planar structures. However, in real world problems there is often a need for handling more than one elementary pattern simultaneously. The straightforward approach of adaptive model selection has proven to be difficult and fragile. At the core of this problem is the vicious intractable search space created by having to simultaneously select models and corresponding samples. This calls for higher order models where multiple patterns are represented as one more complex pattern. In this work, we illustrate the advantages of this approach on data that has bi-modal tensor-valued distributions.The method uses first and second order invariants as a representation, and an eigenvector based solution for recovering the elementary tensor components. We show that this method allows estimation of the two tensors that best represent a given tensor distribution. This distribution can for example be samples from a local neighborhood. A bi-modal distribution will produce the two tensors corresponding to the peaks of the distribution. In addition, numbers indicating the amount of samples belonging to each sub distribution are produced. We demonstrate the potential of the approach by processing a number of simple tensor image examples. The results clearly show that new valuable information regarding the local tensor structure is revealed.


Tensor Field Outer Product Unordered Pair Fourth Order Tensor Vector Pair 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The NIH grants R01MH074794, R01MH092862, P41EB015902 and P41RR013218, and grants from the Swedish Research Council, Swedish e-Science Research Center, Linköpings Universitet are gratefully acknowledged for supporting this work. We also thank Lauren O’Donnell for insightful comments on the manuscript.


  1. 1.
    Herberthson, M., Brun, A., Knutsson, H.: Representing pairs of orientations in the plane. In: Proceedings of the 15th Scandinavian conference on image analysis (SCIA’07). Aalborg. Lecture Notes in Computer Science, vol. 4522, pp. 661–670. Springer, Berlin/New York (2007)Google Scholar
  2. 2.
    Knutsson, H., Westin, C.F.: Normalized and differential convolution: methods for interpolation and filtering of incomplete and uncertain data. In: Proceedings of Computer Vision and Pattern Recognition (CVPR’93), New York City, pp. 515–523. IEEE, Los Alamitos (1993)Google Scholar
  3. 3.
    Westin, C.F.: A tensor framework for multidimensional signal processing. Ph.D. thesis, Linkoping University, S-581 83 Linkoping (1994). Dissertation No 348, ISBN 91-7871-421-4Google Scholar

Copyright information

© Springer Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratory of Mathematics in ImagingBrigham and Women’s Hospital, Harvard Medical SchoolBostonUS
  2. 2.Department of Biomedical EngineeringLinköping University, Sweden and Center for Medical Image Science and Visualization (CMIV)LinköpingSweden

Personalised recommendations