Abstract
Braids of partitions have been introduced in a theoretical framework as a generalization of hierarchies of partitions, but practical guidelines to derive such structures remained an open question. In a previous work, we proposed a methodology to build a braid of partitions by experimentally composing cuts extracted from two hierarchies of partitions, notably paving the way for the hierarchical representation of multimodal images. However, we did not provide the formal proof that our proposed methodology was yielding a braid structure. We remedy to this point in the present paper and give a brief insight on the structural properties of the resulting braid of partitions.
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References
Angulo, J., Jeulin, D.: Stochastic watershed segmentation. In: Proceedings of the 8th International Symposium on Mathematical Morphology, pp. 265–276 (2007)
Beucher, S.: Watershed, hierarchical segmentation and waterfall algorithm. In: Serra, J., Soille, P. (eds.) Mathematical Morphology and its Applications to Image Processing, pp. 69–76. Springer, Dordrecht (1994). https://doi.org/10.1007/978-94-011-1040-2_10
Bosilj, P., Kijak, E., Lefèvre, S.: Partition and inclusion hierarchies of images: a comprehensive survey. J. Imaging 4(2), 33 (2018)
Carlinet, E., Géraud, T.: MToS: a tree of shapes for multivariate images. IEEE Trans. Image Process. 24(12), 5330–5342 (2015)
Cousty, J., Najman, L., Kenmochi, Y., Guimarães, S.: Hierarchical segmentations with graphs: quasi-flat zones, minimum spanning trees, and saliency maps. J. Math. Imaging Vis. 60(4), 479–502 (2018)
Kiran, B.R., Serra, J.: Braids of partitions. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) ISMM 2015. LNCS, vol. 9082, pp. 217–228. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18720-4_19
Monasse, P., Guichard, F.: Fast computation of a contrast-invariant image representation. IEEE Trans. Image Process. 9(5), 860–872 (2000)
Najman, L., Cousty, J.: A graph-based mathematical morphology reader. Pattern Recogn. Lett. 47, 3–17 (2014)
Palou, G., Salembier, P.: Hierarchical video representation with trajectory binary partition tree. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2099–2106 (2013)
Passat, N., Naegel, B.: Component-trees and multivalued images: structural properties. J. Math. Imaging Vis. 49(1), 37–50 (2014)
Perret, B., Cousty, J., Guimaraes, S.J.F., Maia, D.S.: Evaluation of hierarchical watersheds. IEEE Trans. Image Process. 27(4), 1676–1688 (2018)
Randrianasoa, J.F., Kurtz, C., Desjardin, E., Passat, N.: Binary partition tree construction from multiple features for image segmentation. Pattern Recogn. 84, 237–250 (2018)
Salembier, P., Garrido, L.: Binary partition tree as an efficient representation for image processing, segmentation, and information retrieval. IEEE Trans. Image Process. 9(4), 561–576 (2000)
Salembier, P., Oliveras, A., Garrido, L.: Antiextensive connected operators for image and sequence processing. IEEE Trans. Image Process. 7(4), 555–570 (1998)
Scharstein, D., et al.: High-resolution stereo datasets with subpixel-accurate ground truth. In: Jiang, X., Hornegger, J., Koch, R. (eds.) GCPR 2014. LNCS, vol. 8753, pp. 31–42. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-11752-2_3
Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE Trans. Pattern Anal. Mach. Intell. 30(7), 1132–1145 (2008)
Tochon, G.: Hierarchical analysis of multimodal images. Ph.D. thesis, Université Grenoble Alpes (2015)
Tochon, G., Dalla Mura, M., Chanussot, J.: Segmentation of multimodal images based on hierarchies of partitions. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) ISMM 2015. LNCS, vol. 9082, pp. 241–252. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-18720-4_21
Vilaplana, V., Marques, F., Salembier, P.: Binary partition trees for object detection. IEEE Trans. Image Process. 17(11), 2201–2216 (2008)
Xu, Y., Géraud, T., Najman, L.: Connected filtering on tree-based shape-spaces. IEEE Trans. Pattern Anal. Mach. Intell. 38(6), 1126–1140 (2016)
Xu, Y., Géraud, T., Najman, L.: Hierarchical image simplification and segmentation based on Mumford-Shah-salient level line selection. Pattern Recogn. Lett. 83, 278–286 (2016)
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Tochon, G., Dalla Mura, M., Chanussot, J. (2019). Constructing a Braid of Partitions from Hierarchies of Partitions. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_9
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