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Combinatorial Maps for 2D and 3D Image Segmentation

  • Chapter
Digital Geometry Algorithms

Part of the book series: Lecture Notes in Computational Vision and Biomechanics ((LNCVB,volume 2))

Abstract

This chapter shows how combinatorial maps can be used for 2D or 3D image segmentation. We start by introducing combinatorial maps and we show how they can be used to describe image partitions. Then, we present a generic segmentation algorithm that uses and modifies the image partition represented by a combinatorial map. One advantage of this algorithm is that one can mix different criteria and use different image features which can be associated with the cells of the partition. In particular, it is interesting that the topological properties of the image partition can be controlled through this approach. This property is illustrated by the computation of classical topological invariants, known as Betti numbers, which are then used to control the number of cavities or the number of tunnels of regions in the image partition. Finally, we present some experimental results of 2D and 3D image segmentation using different criteria detailed in this chapter.

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Notes

  1. 1.

    A permutation on a set D is a one to one mapping from D onto D.

  2. 2.

    An involution f on a set D is a one to one mapping from D onto D such that f=f −1.

  3. 3.

    β i ∘β j is the composition of both permutations: (β i ∘β j )(x)=β i (β j (x)).

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Author information

Authors and Affiliations

  1. LIRIS, UMR5205, Université de Lyon, CNRS, 69622, Lyon, France

    Guillaume Damiand

  2. Unit 698, Inserm, 75018, Paris, France

    Alexandre Dupas

Authors
  1. Guillaume Damiand
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  2. Alexandre Dupas
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Correspondence to Guillaume Damiand .

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Editors and Affiliations

  1. Buffalo State College, Department of Mathematics, State University of New York, Elmwood Avenue 1300, Buffalo, 14222, New York, USA

    Valentin E. Brimkov

  2. Department of Computer Science, State University of New York at Fredonia, Central Ave. 280, Fredonia, 14063, New York, USA

    Reneta P. Barneva

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Damiand, G., Dupas, A. (2012). Combinatorial Maps for 2D and 3D Image Segmentation. In: Brimkov, V., Barneva, R. (eds) Digital Geometry Algorithms. Lecture Notes in Computational Vision and Biomechanics, vol 2. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4174-4_12

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  • DOI: https://doi.org/10.1007/978-94-007-4174-4_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-4173-7

  • Online ISBN: 978-94-007-4174-4

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