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Tensors in Image Processing and Computer Vision pp 125–150Cite as

PDE-based Morphology for Matrix Fields: Numerical Solution Schemes

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Part of the book series: Advances in Pattern Recognition ((ACVPR))

Abstract

Tensor fields are important in digital imaging and computer vision. Hence there is a demand for morphological operations to perform e.g. shape analysis, segmentation or enhancement procedures. Recently, fundamental morphological concepts have been transferred to the setting of fields of symmetric positive definite matrices, which are symmetric rank two tensors. This has been achieved by a matrix-valued extension of the nonlinear morphological partial differential equations (PDEs) for dilation and erosion known for grey scale images. Having these two basic operations at our disposal, more advanced morphological operators such as top hats or morphological derivatives for matrix fields with symmetric, positive semidefinite matrices can be constructed. The approach realises a proper coupling of the matrix channels rather than treating them independently. However, from the algorithmic side the usual scalar morphological PDEs are transport equations that require special upwind-schemes or novel high-accuracy predictor-corrector approaches for their adequate numerical treatment. In this chapter we propose the non-trivial extension of these schemes to the matrix-valued setting by exploiting the special algebraic structure available for symmetric matrices. Furthermore we compare the performance and juxtapose the results of these novel matrix-valued high-resolution-type (HRT) numerical schemes by considering top hats and morphological derivatives applied to artificial and real world data sets.

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Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, Saarland University, Germany

    Bernhard Burgeth, Michael Breuß & Joachim Weickert

  2. Funded by the Excellence Cluster ‘Multimodal Computing and Interaction‘ at the Department of Mathematics and Computer Science, Saarland University, Germany

    Stephan Didas

Authors
  1. Bernhard Burgeth
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  2. Michael Breuß
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  3. Stephan Didas
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  4. Joachim Weickert
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Corresponding author

Correspondence to Bernhard Burgeth .

Editor information

Editors and Affiliations

  1. Escuela Técnica Superior de Ingenieros de Telecomunicacíon, Universidad de Valladolid, Campus Miguel Delibes, 47011 Valladolid, Spain

    Santiago Aja-Fernández  & Rodrigo de Luis García  & 

  2. Dept. Computing, HongKong Polytechnic University, Hong Kong

    Dacheng Tao

  3. University of London Birkbeck College School of Computer Science & Information Systems, Malet Street, London, WC1E 7HX, UK

    Xuelong Li

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Burgeth, B., Breuß, M., Didas, S., Weickert, J. (2009). PDE-based Morphology for Matrix Fields: Numerical Solution Schemes. In: Aja-Fernández, S., de Luis García, R., Tao, D., Li, X. (eds) Tensors in Image Processing and Computer Vision. Advances in Pattern Recognition. Springer, London. https://doi.org/10.1007/978-1-84882-299-3_6

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  • DOI: https://doi.org/10.1007/978-1-84882-299-3_6

  • Publisher Name: Springer, London

  • Print ISBN: 978-1-84882-298-6

  • Online ISBN: 978-1-84882-299-3

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