Constructing the Tree of Shapes of an Image by Fusion of the Trees of Connected Components of Upper and Lower Level Sets

Constructing the Tree of Shapes of an Image by Fusion of the Trees of Connected Components of... The tree of shapes of an image is an ordered structure which permits an efficient manipulation of the level sets of an image, modeled as a real continuous function defined on a rectangle of $${\mathbb{R}}^N$$ , N ≥ 2. In this paper we construct the tree of shapes of an image by fusing both trees of connected components of upper and lower level sets. We analyze the branch structure of both trees and we construct the tree of shapes by joining their branches in a suitable way. This was the algorithmic approach for 2D images introduced by F. Guichard and P. Monasse in their initial paper, though other efficient approaches were later developed in this case. In this paper, we prove the well-foundedness of this approach for the general case of multidimensional images. This approach can be effectively implemented in the case of 3D images and can be applied for segmentation, but this is not the object of this paper. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

Constructing the Tree of Shapes of an Image by Fusion of the Trees of Connected Components of Upper and Lower Level Sets

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Publisher
Springer Journals
Copyright
Copyright © 2007 by Birkhäuser Verlag Basel/Switzerland
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-007-2150-2
Publisher site
See Article on Publisher Site

Abstract

The tree of shapes of an image is an ordered structure which permits an efficient manipulation of the level sets of an image, modeled as a real continuous function defined on a rectangle of $${\mathbb{R}}^N$$ , N ≥ 2. In this paper we construct the tree of shapes of an image by fusing both trees of connected components of upper and lower level sets. We analyze the branch structure of both trees and we construct the tree of shapes by joining their branches in a suitable way. This was the algorithmic approach for 2D images introduced by F. Guichard and P. Monasse in their initial paper, though other efficient approaches were later developed in this case. In this paper, we prove the well-foundedness of this approach for the general case of multidimensional images. This approach can be effectively implemented in the case of 3D images and can be applied for segmentation, but this is not the object of this paper.

Journal

PositivitySpringer Journals

Published: Oct 29, 2007

References

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