Abstract
The goal of this work is to describe an efficient algorithm for finding a binary segmentation of an image such that the indicated object satisfies a novel high-level prior, called local band, LB, constraint; the returned segmentation is optimal, with respect to an appropriate graph-cut measure, among all segmentations satisfying the given LB constraint. The new algorithm has two stages: expanding the number of edges of a standard edge-weighted graph of an image; applying to this new weighted graph an algorithm known as an oriented image foresting transform, OIFT. In our theoretical investigation, we prove that OIFT algorithm belongs to a class of general fuzzy connectedness algorithms and so has several good theoretical properties, like robustness for seed placement. The extension of the graph constructed in the first stage ensures, as we prove, that the resulted object indeed satisfies the given LB constraint. We also notice that this graph construction is flexible enough to allow combining it with other high-level constraints. Finally, we experimentally demonstrate that the LB constraint gives competitive results as compared to geodesic star convexity, boundary band, and hedgehog shape prior, all implemented within OIFT framework and applied to various scenarios involving natural and medical images.
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Notes
In fact, we can use \(h(\omega (s,t))\) in place of \(e^{-\omega (s,t)}\) when h is any strictly decreasing function from \({\mathbb {R}}\) into \([0,\infty )\).
Shortly, \(F_L:={\bar{\omega }}\cdot \chi _{X_L}\), where \(\chi _{X_L}:{{\mathcal {A}}}\rightarrow \{0,1\}\) is the characteristic function of \(X_L\).
References
Aho, A.V., Garey, M.R., Ullman, J.D.: The transitive reduction of a directed graph. SIAM J. Comput. 1(2), 131–137 (1972)
Bejar, H.H.C., Miranda, P.A.V.: Oriented relative fuzzy connectedness: theory, algorithms, and its applications in hybrid image segmentation methods. EURASIP J. Image Video Process. 2015(21) Jul (2015)
Boykov, Y., Funka-Lea, G.: Graph cuts and efficient N–D image segmentation. Int. J. Comput. Vis. 70(2), 109–131 (2006)
de Moraes Braz, C.: Segmentação de imagens pela transformada imagem-floresta com faixa de restrição geodésica. Master’s thesis, Instituto de Matemática e Estatística, Universidade de São Paulo, São Paulo, Brasil (2016)
Ciesielski, K.C., Udupa, J.K., Falcão, A.X., Miranda, P.A.V.: Fuzzy connectedness image segmentation in graph cut formulation: a linear-time algorithm and a comparative analysis. J. Math. Imaging Vis. 44(3), 375–398 (2012)
Ciesielski, K.C., Udupa, J.K., Falcão, A.X., Miranda, P.A.V.: A unifying graph-cut image segmentation framework: algorithms it encompasses and equivalences among them. In: Proceedings of SPIE on Medical Imaging: Image Processing, vol. 8314 (2012)
Ciesielski, K.C., Udupa, J.K., Saha, P.K., Zhuge, Y.: Iterative relative fuzzy connectedness for multiple objects with multiple seeds. Comput. Vis. Image Underst. 107(3), 160–182 (2007)
Ciesielski, K.C., Falcão, A.X., Miranda, P.A.V.: Path-value functions for which Dijkstra’s algorithm returns optimal mapping. J. Math. Imaging Vis. 60(7), 1025–1036 (2018)
Ciesielski, K.C., Herman, G.T., Yung Kong, T.: General theory of fuzzy connectedness segmentations. J. Math. Imaging Vis. 55(3), 304–342 (2016)
Ciesielski, K.C., Strand, R., Malmberg, F., Saha, P.K.: Efficient algorithm for finding the exact minimum barrier distance. Comput. Vis. Image Underst. 123, 53–64 (2014)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest path forests, and topological watersheds. Trans. Pattern Anal. Mach. Intell. 32, 925–939 (2010)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE Trans. Pattern Anal. Mach. Intell. 31(8), 1362–1374 (2008)
de Moraes Braz, C., Miranda, P.A.V.: Image segmentation by image foresting transform with geodesic band constraints. In: 2014 IEEE International Conference on Image Processing (ICIP), pp. 4333–4337 (2014)
de Moraes Braz, C., Miranda, P.A.V., Ciesielski, K.C., Cappabianco, F.A.M.: Graph-based segmentation with local band constraints. In: Couprie, M., Cousty, J., Kenmochi, Y., Mustafa, N. (eds.) Discrete Geometry for Computer Imagery, pp. 155–166. Springer, Cham (2019)
Falcão, A.X., Stolfi, J., Lotufo, R.A.: The image foresting transform: theory, algorithms, and applications. IEEE TPAMI 26(1), 19–29 (2004)
Falcão, A.X., Udupa, J.K., Samarasekera, S., Sharma, S., Hirsch, B.E., Lotufo, R.A.: User-steered image segmentation paradigms: live-wire and live-lane. Graph. Models Image Proc. 60, 233–260 (1998)
Freedman, D., Zhang, T.: Interactive graph cut based segmentation with shape priors. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2005. CVPR 2005, vol. 1, pp. 755–762. IEEE (2005)
Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)
Gulshan, V., Rother, C., Criminisi, A., Blake, A., Zisserman, A.: Geodesic star convexity for interactive image segmentation. In: Proceedings of Computer Vision and Pattern Recognition, pp. 3129–3136 (2010)
Isack, H., Veksler, O., Sonka, M.,Boykov, Y.: Hedgehog shape priors for multi-object segmentation. In: 2016 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 2434–2442 (2016)
Isack, H.N., Boykov, Y., Veksler, O.: A-expansion for multiple “hedgehog” shapes. CoRR, arXiv:1602.01006 (2016)
Leon, L.M.C., Miranda, P.A.V.D.: Multi-object segmentation by hierarchical layered oriented image foresting transform. In: 2017 30th SIBGRAPI Conference on Graphics, Patterns and Images (SIBGRAPI), pp. 79–86 (2017)
Lézoray, O., Grady, L.: Image Processing and Analysis with Graphs: Theory and Practice. CRC Press, California (2012)
Li, X., Chen, J., Fan, H.: Interactive image segmentation based on grow cut of two scale graphs. In: Zhang, W., Yang, X., Xu, Z., An, P., Liu, Q., Lu, Y. (eds.) Advances on Digital Television and Wireless Multimedia Communications, pp. 90–95. Springer, Berlin (2012)
Madabhushi, A., Udupa, J.K.: Interplay between intensity standardization and inhomogeneity correction in MR image processing. IEEE Trans. Med. Imaging 24(5), 561–576 (2005)
Mansilla, L.A.C., Miranda, P.A.V.: Oriented image foresting transform segmentation: connectivity constraints with adjustable width. In: 29th SIBGRAPI Conference on Graphics, Patterns and Images, pp. 289–296 (2016)
Mansilla, L.A.C., Miranda, P.A.V., Cappabianco, F.A.M.: Oriented image foresting transform segmentation with connectivity constraints. In: 2016 IEEE International Conference on Image Processing (ICIP), pp. 2554–2558 (2016)
Mansilla, L.A.C., Miranda, P.A.V.: Image segmentation by oriented image foresting transform: handling ties and colored images. In 18th International Conference on Digital Signal Processing, Greece, pp. 1–6 (2013)
Mansilla, L.A.C., Miranda, P.A.V.: Image segmentation by oriented image foresting transform with geodesic star convexity. In: 15th International Conference on Computer Analysis of Images and Patterns (CAIP), York, UK, vol. 8047, pp. 572–579 (2013)
Miranda, P.A.V., Mansilla, L.A.C.: Oriented image foresting transform segmentation by seed competition. IEEE Trans. Image Process. 23(1), 389–398 (2014)
Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. USA 93(4), 1591–5 (1996)
Xu, Y., Géraud, T., Najman, L.: Context-based energy estimator: Application to object segmentation on the tree of shapes. In: 2012 19th IEEE International Conference on Image Processing, pp. 1577–1580 (2012)
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Thanks to CNPq (313554/2018-8, 486988/2013-9, FINEP 1266/13), FAPESP (2014/12236-1, 2016/21591-5), Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001, and NAP eScience - PRP - USP for funding.
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de Moraes Braz, C., Miranda, P.A., Ciesielski, K.C. et al. Optimum Cuts in Graphs by General Fuzzy Connectedness with Local Band Constraints. J Math Imaging Vis 62, 659–672 (2020). https://doi.org/10.1007/s10851-020-00953-w
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DOI: https://doi.org/10.1007/s10851-020-00953-w