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Structure Tensor Estimation: Introducing Monomial Quadrature Filter Sets

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New Developments in the Visualization and Processing of Tensor Fields

Part of the book series: Mathematics and Visualization ((MATHVISUAL))

Abstract

Description and estimation of local spatial structure has a long history and numerous analysis tools have been developed. A concept that is widely recognized as fundamental in the analysis is the structure tensor. It has, however, a fairly broad and unspecific meaning. This chapter is intended to provide a framework for displaying the differences and similarities of existing structure estimation approaches. A new method for structure tensor estimation, which is a generalization of many of it’s predecessors, is presented. The method uses pairs of filter sets having Fourier directional responses in the form of monomials, one odd order set and one even order set. It is shown that such filter sets allow for a particularly simple way of attaining phase invariant, positive semi-definite, local structure tensor estimates. In addition, we show that the chosen filter sets directly links order, scale and the gradient operator. We continue to compare a number of known structure tensor algorithms by formulating them in monomial filter set terms. In conclusion we show how higher order tensors can be estimated using a generalization of the same simple formulation.

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Acknowledgements

The Swedish Research Council, Swedish e-Science Research Center, Linköpings Universitet and NIH (grants R01MH074794, R01MH092862, and P41RR013218) are gratefully acknowledged for supporting this work.

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Authors and Affiliations

  1. Department of Biomedical Engineering, Linköping University, Linköping, Sweden

    Hans Knutsson, Carl-Fredrik Westin & Mats Andersson

  2. Center for Medical Image Science and Visualization (CMIV), Linköping, Sweden

    Hans Knutsson & Mats Andersson

  3. Laboratory of Mathematics in Imaging, Brigham and Women’s Hospital, Harvard Medical School, Boston, MA, USA

    Carl-Fredrik Westin

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  1. Hans Knutsson
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Correspondence to Hans Knutsson .

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Editors and Affiliations

  1. Dept. Computer Science, Brown University, Providence, Rhode Island, USA

    David H. Laidlaw

  2. Department of Biomedical Engineering, Einhoven University of Technology, Eindhoven, 5600 MB, Netherlands

    Anna Vilanova

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Knutsson, H., Westin, CF., Andersson, M. (2012). Structure Tensor Estimation: Introducing Monomial Quadrature Filter Sets. In: Laidlaw, D., Vilanova, A. (eds) New Developments in the Visualization and Processing of Tensor Fields. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27343-8_1

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