Constructing the tree of shapes of an image by fusion of the trees of connected components of upper and lower level sets. (English) Zbl 1135.68609
Summary: 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 \({\mathbb{R}}^N, N \geq 2\). 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.
MSC:
| 68U10 | Computing methodologies for image processing |
| 68T10 | Pattern recognition, speech recognition |
| 06B23 | Complete lattices, completions |
References:
| [1] | C. Ballester, V. Caselles, The M-components of level sets of continuous functions in WBV, Publicacions Matemàtiques, 45 (2001), 477–527. · Zbl 0991.54013 |
| [2] | C. Ballester, V. Caselles, P. Monasse, The tree of shapes of an image. ESAIM: Control, Opt. Calc. Variations, 9 (2003) 1–18. · Zbl 1073.68094 |
| [3] | C. Ballester, V. Caselles, L. Igual, L. Garrido, Level lines selection with variational models for segmentation and encoding. J. Math. Imaging Vis., 27 (2007), 5–27. |
| [4] | F. Cao, Y. Gousseau, J.M. Morel, P. Musé, F. Sur, Shape recognition based on an a contrario methodology, statistics and analysis of shapes, 107–136, Model. Simul. Sci. Eng. Tecnol., Birkhäuser, Boston, MA, 2006. · Zbl 1188.68249 |
| [5] | V. Caselles, B. Coll, J.M. Morel, Topographic maps and local contrast changes in natural images, Int. J. Comp. Vis., 33 (1999), 5–27. |
| [6] | V. Caselles, P. Monasse, Geometric description of topographic maps and applications to image processing, in preparation. · Zbl 1191.68759 |
| [7] | A. Desolneux, L. Moisan, J.M. Morel, Edge detection by Helmholtz principle. J. Math. Imaging Vis., 14(3) (2001) 271–284. · Zbl 0988.68819 |
| [8] | F. Guichard, J.M. Morel, Image Iterative Smoothing and P.D.E.’s’. Book in preparation. |
| [9] | C. Kuratowski, Topologie I, II. Editions J. Gabay, Paris (1992). |
| [10] | J.L. Lisani, L. Moisan, P. Monasse, J.M. Morel, Affine invariant mathematical morphology applied to a generic shape recognition algorithm, Comp. Im. Vis. 18 (2000). · Zbl 1073.68774 |
| [11] | E. Meinhardt, V. Caselles, E. Zacur, A. Frangi, The tree of shapes of a 3D image. Applications to medical image segmentation, in preparation. |
| [12] | P. Monasse, Contrast invariant image registration, Proc. Int. Conf. Acoust. Speech Signal Proces. 6 (1999) 3221–3224. |
| [13] | P. Monasse, F. Guichard, Fast computation of a contrast invariant image representation, IEEE Trans. Image Proces., 9 (2000), 860–872. |
| [14] | P. Salembier, L. Garrido, Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval, IEEE Trans. Image Proces., 9(4) (2000), 561–576. |
| [15] | P. Salembier, J. Serra, Flat zones filtering, connected operators and filters by reconstruction. IEEE Trans. Image Proces., 4(8) (1995), 1153–1160. |
| [16] | H.H. Schaefer, Banach Lattices and Positive Operators, Springer (1974). · Zbl 0296.47023 |
| [17] | J. Serra, Image Analysis and Mathematical Morphology, Academic, New York (1982). · Zbl 0565.92001 |
| [18] | Y. Song, A. Zhang, Monotonic trees and its application to image processing, In: 10th International Conf. on Discrete Geometry for Computer Imagery, Bordeaux, France (2002). |
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