Power Tree Filter: A Theoretical Framework Linking Shortest Path Filters and Minimum Spanning Tree Filters

  • Sravan Danda
  • Aditya Challa
  • B. S. Daya Sagar
  • Laurent Najman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10225)


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.


Short Path Minimum Span Tree Object Boundary Stereo Match Bilateral Filter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



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.


  1. 1.
    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)MathSciNetGoogle Scholar
  2. 2.
    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)Google Scholar
  3. 3.
    Couprie, C., Grady, L., Najman, L., Talbot, H.: Power watershed: a unifying graph-based optimization framework. IEEE PAMI 33(7), 1384–1399 (2011)CrossRefGoogle Scholar
  4. 4.
    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)CrossRefGoogle Scholar
  5. 5.
    Cousty, J., Bertrand, G., Najman, L., Couprie, M.: Watershed cuts: thinnings, shortest path forests, and topological watersheds. IEEE PAMI 32(5), 925–939 (2010)CrossRefGoogle Scholar
  6. 6.
    Falcao, A.X., Stolfi, J., de Alencar Lotufo, R.: The image foresting transform: theory, algorithms, and applications. IEEE PAMI 26(1), 19 (2004)CrossRefGoogle Scholar
  7. 7.
    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). ACMCrossRefGoogle Scholar
  8. 8.
    Floyd, R.W.: Algorithm 97: shortest path. Commun. ACM 5(6), 345 (1962)CrossRefGoogle Scholar
  9. 9.
    Grady, L.: Random walks for image segmentation. IEEE PAMI 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
  10. 10.
    Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods. Pattern Recogn. 42(10), 2306–2316 (2009)CrossRefzbMATHGoogle Scholar
  11. 11.
    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 CrossRefGoogle Scholar
  12. 12.
    Lerallut, R., Decencière, É., Meyer, F.: Image filtering using morphological amoebas. Image Vis. Comput. 25(4), 395–404 (2007)CrossRefGoogle Scholar
  13. 13.
    Najman, L.: Extending the PowerWatershed framework thanks to \(\Gamma \)-convergence. Technical report, Université Paris-Est, LIGM, ESIEE Paris (2017).
  14. 14.
    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)Google Scholar
  15. 15.
    Soille, P.: Constrained connectivity for hierarchical image partitioning and simplification. IEEE PAMI 30(7), 1132–1145 (2008)CrossRefGoogle Scholar
  16. 16.
  17. 17.
    Stawiaski, J., Meyer, F.: Minimum spanning tree adaptive image filtering. In: 2009 16th IEEE ICIP, pp. 2245–2248. IEEE (2009)Google Scholar
  18. 18.
    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)Google Scholar
  19. 19.
    Xu, L., Lu, C., Xu, Y., Jia, J.: Image smoothing via L 0 gradient minimization. ACM Trans. Graph. (TOG) 30, 174 (2011). ACMGoogle Scholar
  20. 20.
    Xu, L., Yan, Q., Xia, Y., Jia, J.: Structure extraction from texture via relative total variation. ACM Trans. Graph. (TOG) 31(6), 139 (2012)Google Scholar
  21. 21.
    Yang, Q.: Stereo matching using tree filtering. IEEE PAMI 37(4), 834–846 (2015)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Sravan Danda
    • 1
  • Aditya Challa
    • 1
  • B. S. Daya Sagar
    • 1
  • Laurent Najman
    • 2
  1. 1.Systems Science and Informatics UnitIndian Statistical InstituteBangaloreIndia
  2. 2.Université Paris-Est, LIGM, Equipe A3SI, ESIEEParisFrance

Personalised recommendations