A Local Structure Measure for Anisotropic Regularization of Tensor Fields

  • Suárez-Santana E. 
  • Rodriguez-Florido M. A. 
  • Castaño-Moraga C. 
  • Westin C.-F. 
  • Ruiz-Alzola J. 
Part of the Mathematics and Visualization book series (MATHVISUAL)


Acquisition systems are not fully reliable since any real sensor will provide noisy and possibly incomplete and degraded data. Therefore, in tensor measurements, all problems dealt with in conventional multidimensional statistical signal processing are present with tensor signals. In this chapter we describe some noniterative approaches to tensor signal processing. Our schemes are achieved by the estimation of a local structure tensor, which is used as a key element in regularization. A stochastic point of view as well as a phase-invariant implementation are presented. This work also covers tensor extensions for common scalar operations such as anisotropic interpolation and filtering. An application of the structure tensor for regularization of deformation fields in tensor image registration is also shown. The techniques presented in this chapter suppose an alternative to variational and PDEs schemes, and another point of view of the tensor signal processing.


Template Match Structure Measure Structure Tensor Anisotropic Tensor Tensor Data 
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  1. 1.
    D. C. Alexander, J.C. Gee, and R. K. Bajcsy. Similarity measures for matching diffusion tensor images. In Proc. British Machine Vision Conference, 1999.Google Scholar
  2. 2.
    Castaño-Moraga C.A., Rodriguez-Florido M.A., Alvarez L., Westin C.-F., Ruiz-Alzola J. Anisotropic interpolation of DT-MRI. In LNCS 3216, MICCAI. pp. 343–350, 2004.Google Scholar
  3. 3.
    C. Harris and M. Stephens. A combined corner and edge detector. In Proceedings of the Fourth Alvey Vision Conference, pp. 147–151, 1988.Google Scholar
  4. 4.
    Knutsson, H. Representing local structure using tensors. In 6th Scandinavian Conference on Image Analysis. Oulu, Finland, pp. 244–251, 1989.Google Scholar
  5. 5.
    Knutsson H., and Andersson M. What’s so good about quadrature filters? In 2003 IEEE International Conference on Image Processing, 2003.Google Scholar
  6. 6.
    Stanislav Kovacic and R.K. Bajcsy. Brain Warping, chapter Multiscale/Multiresolution Representations, pp. 45–65. Academic Press, 1999.Google Scholar
  7. 7.
    Pajevic S., Aldroubi A., and Basser P.J. A continuous tensor field approximation of discrete DT-mri data for extracting microstructural and architectural features of tissue. Journal of Magnetic Resonance, 154:85–100, 2002.CrossRefGoogle Scholar
  8. 8.
    Rodriguez-Florido M.A., Krissian K., Ruiz-Alzola J., Westin C.-F. Comparison between two restoration techniques in the context of 3d medical imaging. LNCS — Springer-Verlag, 2208:1031–1039, 2001.Google Scholar
  9. 9.
    Rodriguez-Florido M.A., Westin C.-F., and Ruiz-Alzola J. DT-MRI regularization using anisotropic tensor field filtering. In 2004 IEEE International Symposium on Biomedical Imaging, pp. 336–339, 2004.Google Scholar
  10. 10.
    K. Rohr. On 3D differential operators for detecting point landmarks. Image and Vision Computing, 15:219–233, 1997.CrossRefGoogle Scholar
  11. 11.
    Christian Ronse. On idempotence and related requirements in edge detection. IEEE Transactions on Pattern Analysis and Machine Intelligence, 15(5):484–490, 1993.CrossRefGoogle Scholar
  12. 12.
    J. Ruiz-Alzola, R. Kikinis, and C.-F. Westin. Detection of point landmarks in multidimensional tensor data. Signal Processing, 81:2243–2247, 2001.CrossRefGoogle Scholar
  13. 13.
    J. Ruiz-Alzola, C.F. Westin, S.K. Warfield, A. Nabavi, and R. Kikinis. Nonrigid registration of 3D scalar, vector and tensor medical data. In MICCAI, pp. 541–550, 2000.Google Scholar
  14. 14.
    Schriber, W.F. Wirephoto quality improvement by unsharp masking. J.Pattern Recognition, 2:117–121, 1970.CrossRefGoogle Scholar
  15. 15.
    Eduardo Suárez, Carl-Fredrik Westin, Eduardo Rovaris, and Juan Ruiz-Alzola. Nonrigid registration using regularized matching weighted by local structure. In MICCAI, number 2 in LNCS, pp. 581–589, Tokyo, Japan, September 2002.Google Scholar
  16. 16.
    C-F. Westin. A Tensor Framework for Multidimensional Signal Processing. PhD thesis, Linköoping University, Sweden, SE-581 83 Linköping, Sweden, 1994. Dissertation No 348, ISBN 91-7871-421-4.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Suárez-Santana E. 
    • 1
  • Rodriguez-Florido M. A. 
    • 1
  • Castaño-Moraga C. 
    • 1
  • Westin C.-F. 
    • 1
    • 2
  • Ruiz-Alzola J. 
    • 1
    • 2
  1. 1.Center for Technology in MedicineUniversity of Las Palmas de Gran Canaria, Campus de TafiraLas Palmas de Gran CanariaSpain
  2. 2.Laboratory of Mathematics in ImagingBrigham & Women’s Hospital and Harvard Medical SchoolBostonUSA

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