Abstract
Edge-preserving image filtering is an important pre-processing step in many filtering applications. In this article, we analyse the basis of edge-preserving filters and also provide theoretical links between the MST filter, which is a recent state-of-art edge-preserving filter, and filters based on geodesics. We define shortest path filters, which are closely related to adaptive kernel based filters, and show that MST filter is an approximation to the \(\varGamma -\)limit of the shortest path filters. We also propose a different approximation for the \(\varGamma -\)limit that is based on union of all MSTs and show that it yields better results than that of MST approximation by reducing the leaks across object boundaries. We demonstrate the effectiveness of the proposed filter in edge-preserving smoothing by comparing it with the tree filter.
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References
Bao, L., Song, Y., Yang, Q., Yuan, H., Wang, G.: Tree filtering: efficient structure-preserving smoothing with a minimum spanning tree. IEEE TIP 23(2), 555–569 (2014)
Chang, J.H.R., Wang, Y.C.F.: Propagated image filtering. In: 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 10–18. IEEE (2015)
Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: a unifying graph-based optimization framework. IEEE PAMI 33(7), 1384–1399 (2011)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: minimum spanning forests and the drop of water principle. IEEE PAMI 31(8), 1362–1374 (2009)
Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest path forests, and topological watersheds. IEEE PAMI 32(5), 925–939 (2010)
Falcao, A.X., Stolfi, J., de Alencar Lotufo, R.: The image foresting transform: theory, algorithms, and applications. IEEE PAMI 26(1), 19 (2004)
Farbman, Z., Fattal, R., Lischinski, D., Szeliski, R.: Edge-preserving decompositions for multi-scale tone and detail manipulation. ACM Trans. Graph. (TOG) 27, 67 (2008). ACM
Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)
Grady, L.: Random walks for image segmentation. IEEE PAMI 28(11), 1768–1783 (2006)
Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods. Pattern Recogn. 42(10), 2306–2316 (2009)
He, K., Sun, J., Tang, X.: Guided image filtering. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010. LNCS, vol. 6311, pp. 1–14. Springer, Heidelberg (2010). doi:10.1007/978-3-642-15549-9_1
Lerallut, R., Decencière, É., Meyer, F.: Image filtering using morphological amoebas. Image Vis. Comput. 25(4), 395–404 (2007)
Najman, L.: Extending the PowerWatershed framework thanks to \(\Gamma \)-convergence. Technical report, Université Paris-Est, LIGM, ESIEE Paris (2017). https://hal-upec-upem.archives-ouvertes.fr/hal-01428875
Sinop, A.K., Grady, L.: A seeded image segmentation framework unifying graph cuts and random walker which yields a new algorithm. In: 2007 IEEE 11th International Conference on Computer Vision, ICCV 2007, pp. 1–8. IEEE (2007)
Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE PAMI 30(7), 1132–1145 (2008)
StackOverFlow: Cut property. http://stackoverflow.com/questions/3327708/minimum-spanning-tree-what-exactly-is-the-cut-property. Accessed 05 Jan 2017
Stawiaski, J., Meyer, F.: Minimum spanning tree adaptive image filtering. In: 2009 16th IEEE ICIP, pp. 2245–2248. IEEE (2009)
Tomasi, C., Manduchi, R.: Bilateral filtering for gray and color images. In: 1998 Sixth International Conference on Computer Vision, ICCV 1998, pp. 839–846. IEEE (1998)
Xu, L., Lu, C., Xu, Y., Jia, J.: Image smoothing via L 0 gradient minimization. ACM Trans. Graph. (TOG) 30, 174 (2011). ACM
Xu, L., Yan, Q., Xia, Y., Jia, J.: Structure extraction from texture via relative total variation. ACM Trans. Graph. (TOG) 31(6), 139 (2012)
Yang, Q.: Stereo matching using tree filtering. IEEE PAMI 37(4), 834–846 (2015)
Acknowledgments
Sravan Danda and Aditya Challa are thankful for the financial support provided by the Indian Statistical Institute. B.S. Daya Sagar would like to acknowledge the partial support received from EMR/2015/000853 SERB and ISRO/SSPO/Ch-1/2016-17 ISRO research grants. Laurent Najman would like acknowledge the partial support received from ANR-15-CE40-0006 CoMeDiC and ANR-14-CE27-0001 GRAPHSIP research grants.
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Danda, S., Challa, A., Sagar, B.S.D., Najman, L. (2017). Power Tree Filter: A Theoretical Framework Linking Shortest Path Filters and Minimum Spanning Tree Filters. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2017. Lecture Notes in Computer Science(), vol 10225. Springer, Cham. https://doi.org/10.1007/978-3-319-57240-6_16
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DOI: https://doi.org/10.1007/978-3-319-57240-6_16
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