Advertisement

Path-Based Mathematical Morphology on Tensor Fields

  • Jasper J. van de Gronde
  • Mikola Lysenko
  • Jos B. T. M. RoerdinkEmail author
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Traditional path-based morphology allows finding long, approximately straight, paths in images. Although originally applied only to scalar images, we show how this can be a very good fit for tensor fields. We do this by constructing directed graphs representing such data, and then modifying the traditional path opening algorithm to work on these graphs. Cycles are dealt with by finding strongly connected components in the graph. Some examples of potential applications are given, including path openings that are not limited to a specific set of orientations.

Keywords

Tangent Vector Directed Acyclic Graph Tensor Field Orientation Score Mathematical Morphology 
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.

Notes

Acknowledgements

The first and third author were funded by the Netherlands Organisation for Scientific Research (NWO), project no. 612.001.001. Also, we thank Remco Renken and Jelmer Kok from the NeuroImaging Center Groningen for supplying us with the diffusion MRI data (including regions of interest and tractography results), as well as for some inspiring discussions.

References

  1. 1.
    Angulo, J.: Supremum/infimum and nonlinear averaging of positive definite symmetric matrices. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 3–33. Springer, Berlin/Heidelberg (2013). doi:10.1007/978-3-642-30232-9_1CrossRefGoogle Scholar
  2. 2.
    Astola, J., Haavisto, P., Neuvo, Y.: Vector median filters. Proc. IEEE 78(4), 678–689 (1990). doi:10.1109/5.54807CrossRefGoogle Scholar
  3. 3.
    Bastiani, M., Shah, N.J., Goebel, R., Roebroeck, A.: Human cortical connectome reconstruction from diffusion weighted MRI: the effect of tractography algorithm. NeuroImage 62(3), 1732–1749 (2012). doi:10.1016/j.neuroimage.2012.06.002CrossRefGoogle Scholar
  4. 4.
    Bismuth, V., Vaillant, R., Talbot, H., Najman, L.: Curvilinear structure enhancement with the polygonal path image - application to guide-wire segmentation in X-ray fluoroscopy. In: Ayache, N., Delingette, H., Golland, P., Mori, K. (eds.) Medical Image Computing and Computer-Assisted Intervention. Lecture Notes in Computer Science, vol. 7511, pp. 9–16. Springer, Berlin/Heidelberg (2012). doi:10.1007/978-3-642-33418-4_2Google Scholar
  5. 5.
    Björklund, A., Husfeldt, T., Khanna, S.: Approximating longest directed paths and cycles. In: Díaz, J., Karhumäki, J., Lepistö, A., Sannella, D. (eds.) Automata, Languages and Programming. Lecture Notes in Computer Science, vol. 3142, pp. 222–233. Springer, Berlin/Heidelberg (2004). doi:10.1007/978-3-540-27836-8_21CrossRefGoogle Scholar
  6. 6.
    Booth, B.G., Hamarneh, G.: Multi-region competitive tractography via graph-based random walks. In: Proceedings IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, pp. 73–78 (2012). doi:10.1109/mmbia.2012.6164747Google Scholar
  7. 7.
    Bourbaki, N.: Algebra I. Elements of Mathematics. Springer, Berlin (1989)Google Scholar
  8. 8.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for matrix data: ordering versus PDE-based approach. Image Vis. Comput. 25(4), 496–511 (2007). doi:10.1016/j.imavis.2006.06.002CrossRefGoogle Scholar
  9. 9.
    Citti, G., Sarti, A.: A cortical based model of perceptual completion in the Roto-translation space. J. Math. Imaging Vision 24(3), 307–326 (2006). doi:10.1007/s10851-005-3630-2MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cokelaer, F., Talbot, H., Chanussot, J.: Efficient robust d-dimensional path operators. IEEE J. Sel. Top. Signal Process. 6(7), 830–839 (2012). doi:10.1109/jstsp.2012.2213578CrossRefGoogle Scholar
  11. 11.
    Comon, P., Golub, G., Lim, L.H., Mourrain, B.: Symmetric tensors and symmetric tensor rank. SIAM J. Matrix Anal. Appl. 30(3), 1254–1279 (2008). doi:10.1137/060661569MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Dell’Acqua, F., Scifo, P., Rizzo, G., Catani, M., Simmons, A., Scotti, G., Fazio, F.: A modified damped Richardson–Lucy algorithm to reduce isotropic background effects in spherical deconvolution. NeuroImage 49(2), 1446–1458 (2010). doi:10.1016/j.neuroimage.2009.09.033CrossRefGoogle Scholar
  13. 13.
    Dell’Acqua, F., Simmons, A., Williams, S.C.R., Catani, M.: Can spherical deconvolution provide more information than fiber orientations? Hindrance modulated orientational anisotropy, a true-tract specific index to characterize white matter diffusion. Hum. Brain Mapp. 34(10), 2464–2483 (2013). doi:10.1002/hbm.22080CrossRefGoogle Scholar
  14. 14.
    Descoteaux, M., Deriche, R., Knosche, T.R., Anwander, A.: Deterministic and probabilistic tractography based on complex fibre orientation distributions. IEEE Trans. Med. Imaging 28(2), 269–286 (2009). doi:10.1109/tmi.2008.2004424CrossRefGoogle Scholar
  15. 15.
    Duits, R.: Perceptual organization in image analysis: a mathematical approach based on scale, orientation and curvature. PhD thesis, Eindhoven University of Technology (2005)Google Scholar
  16. 16.
    Duits, R., Franken, E.: Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part I: Linear left-invariant diffusion equations on SE(2). Q. Appl. Math. 68(2), 255–292 (2010). doi:10.1090/s0033-569x-10-01172-0Google Scholar
  17. 17.
    Duits, R., Franken, E.: Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part II: Nonlinear left-invariant diffusions on invertible orientation scores. Q. Appl. Math. 68(2), 293–331 (2010). doi:10.1090/s0033-569x-10-01173-3MathSciNetGoogle Scholar
  18. 18.
    Duits, R., Dela Haije, T.C.J., Creusen, E., Ghosh, A.: Morphological and linear scale spaces for fiber enhancement in DW-MRI. J. Math. Imaging Vision 46(3), 326–368 (2013). doi:10.1007/s10851-012-0387-2MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Franken, E.M.: Enhancement of crossing elongated structures in images. PhD thesis, Eindhoven University of Technology (2008)Google Scholar
  20. 20.
    Franken, E., Duits, R.: Crossing-preserving coherence-enhancing diffusion on invertible orientation scores. Int. J. Comput. Vis. 85(3), 253–278 (2009). doi:10.1007/s11263-009-0213-5MathSciNetCrossRefGoogle Scholar
  21. 21.
    van de Gronde, J.J., Roerdink, J.B.T.M.: Frames for tensor field morphology. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information. Lecture Notes in Computer Science, vol. 8085, pp. 527–534. Springer, Berlin/Heidelberg (2013). doi:10.1007/978-3-642-40020-9_58CrossRefGoogle Scholar
  22. 22.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic, Boston (1994)zbMATHGoogle Scholar
  23. 23.
    Heijmans, H., Buckley, M., Talbot, H.: Path-based morphological openings. In: IEEE International Conference on Image Processing, vol. 5, pp. 3085–3088 (2004). doi:10.1109/icip.2004.1421765Google Scholar
  24. 24.
    Heijmans, H., Buckley, M., Talbot, H.: Path openings and closings. J. Math. Imaging Vision 22(2), 107–119 (2005). doi:10.1007/s10851-005-4885-3MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Iturria-Medina, Y., Canales-Rodríguez, E.J., Melie-García, L., Valdés-Hernández, P.A., Martínez-Montes, E., Alemán-Gómez, Y., Sánchez-Bornot, J.M.: Characterizing brain anatomical connections using diffusion weighted MRI and graph theory. NeuroImage 36(3), 645–660 (2007). doi:10.1016/j.neuroimage.2007.02.012CrossRefGoogle Scholar
  26. 26.
    Kahn, A.B.: Topological sorting of large networks. Commun. ACM 5(11), 558–562 (1962). doi:10.1145/368996.369025CrossRefzbMATHGoogle Scholar
  27. 27.
    Karger, D., Motwani, R., Ramkumar, G.D.S.: On approximating the longest path in a graph. Algorithmica 18(1), 82–98 (1997). doi:10.1007/bf02523689MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Kingsley, P.B.: Introduction to diffusion tensor imaging mathematics: Part I. Tensors, rotations, and eigenvectors. Concepts Magn. Reson. 28A(2), 101–122 (2006). doi:10.1002/cmr.a.20048Google Scholar
  29. 29.
    Knutsson, H., Westin, C.F., Andersson, M.: Structure tensor estimation: introducing monomial quadrature filter sets. In: Laidlaw, D.H., Vilanova, A. (eds.) New Developments in the Visualization and Processing of Tensor Fields. Mathematics and Visualization, pp. 3–28. Springer, Berlin/Heidelberg (2012). doi:10.1007/978-3-642-27343-8_1CrossRefGoogle Scholar
  30. 30.
    Kofidis, E., Regalia, P.A.: On the best rank-1 approximation of higher-order supersymmetric tensors. SIAM J. Matrix Anal. Appl. 23(3), 863–884 (2002). doi:10.1137/s0895479801387413MathSciNetCrossRefGoogle Scholar
  31. 31.
    Kostrikin, A.I., Manin, I.I.: Linear Algebra and Geometry. Algebra, Logic and Applications, vol. 1. Gordon and Breach, Amsterdam (1997)Google Scholar
  32. 32.
    Köthe, U.: Edge and junction detection with an improved structure tensor. In: Michaelis, B., Krell, G. (eds.) Pattern Recognition. Lecture Notes in Computer Science, vol. 2781, pp. 25–32. Springer, Berlin/Heidelberg (2003). doi:10.1007/978-3-540-45243-0_4Google Scholar
  33. 33.
    Leemans, A., Jeurissen, B., Sijbers, J., Jones, D.K.: ExploreDTI: a graphical toolbox for processing, analyzing, and visualizing diffusion MR data. In: ISMRM 17th Scientific Meeting & Exhibition, p. 3537 (2009)Google Scholar
  34. 34.
    Luengo Hendriks, C.L.: Constrained and dimensionality-independent path openings. IEEE Trans. Image Process. 19(6), 1587–1595 (2010). doi:10.1109/tip.2010.2044959MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    McLoughlin, T., Laramee, R.S., Peikert, R., Post, F.H., Chen, M.: Over two decades of integration-based, geometric flow visualization. Comput. Graph. Forum 29(6), 1807–1829 (2010). doi:10.1111/j.1467-8659.2010.01650.xCrossRefGoogle Scholar
  36. 36.
    Melhem, E.R., Mori, S., Mukundan, G., Kraut, M.A., Pomper, M.G., van Zijl, P.C.M.: Diffusion tensor MR imaging of the brain and white matter tractography. Am. J. Roentgenol. 178(1), 3–16 (2002). doi:10.2214/ajr.178.1.1780003CrossRefGoogle Scholar
  37. 37.
    Morard, V., Dokladal, P., Decencière, E.: One-dimensional openings, granulometries and component trees in O(1) per pixel. IEEE J. Sel. Top. Signal Process. 6(7), 840–848 (2012). doi:10.1109/jstsp.2012.2201694CrossRefGoogle Scholar
  38. 38.
    Morard, V., Dokládal, P., Decencière, E.: Parsimonious path openings and closings. IEEE Trans. Image Process. 23(4), 1543–1555 (2014). doi:10.1109/tip.2014.2303647MathSciNetCrossRefGoogle Scholar
  39. 39.
    Mori, S., van Zijl, P.C.M.: Fiber tracking: principles and strategies – a technical review. NMR Biomed. 15(7–8), 468–480 (2002). doi:10.1002/nbm.781CrossRefGoogle Scholar
  40. 40.
    Nolet, G.: A Breviary of Seismic Tomography. Cambridge University Press, New York (2008)CrossRefzbMATHGoogle Scholar
  41. 41.
    Serra, J. (ed.): Theoretical Advances, Image Analysis and Mathematical Morphology, vol. 2. Academic, London (1988)Google Scholar
  42. 42.
    Serra, J.: Anamorphoses and function lattices. In: Dougherty, E.R., Gader, P.D., Serra, J.C. (eds.) Image Algebra and Morphological Image Processing IV, SPIE Proceedings, vol. 2030, pp. 2–11 (1993). doi:10.1117/12.146650MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sha, F., Lin, Y., Saul, L.K., Lee, D.D.: Multiplicative updates for nonnegative quadratic programming. Neural Comput. 19(8), 2004–2031 (2007). doi:10.1162/neco.2007.19.8.2004MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Sotiropoulos, S.N., Bai, L., Morgan, P.S., Constantinescu, C.S., Tench, C.R.: Brain tractography using Q-ball imaging and graph theory: improved connectivities through fibre crossings via a model-based approach. NeuroImage 49(3), 2444–2456 (2010). doi:10.1016/j.neuroimage.2009.10.001CrossRefzbMATHGoogle Scholar
  45. 45.
    Talbot, H., Appleton, B.: Efficient complete and incomplete path openings and closings. Image Vis. Comput. 25(4), 416–425 (2007). doi:10.1016/j.imavis.2006.07.021CrossRefGoogle Scholar
  46. 46.
    Tarjan, R.: Depth-first search and linear graph algorithms. SIAM J. Comput. 1(2), 146–160 (1972). doi:10.1137/0201010MathSciNetCrossRefGoogle Scholar
  47. 47.
    Terajima, K., Nakada, T.: EZ-tracing: a new ready-to-use algorithm for magnetic resonance tractography. J. Neurosci. Methods 116(2), 147–155 (2002). doi:10.1016/s0165-0270(02)00039-0Google Scholar
  48. 48.
    Tournier, J.D., Mori, S., Leemans, A.: Diffusion tensor imaging and beyond. Magn. Reson. Med. 65(6), 1532–1556 (2011). doi:10.1002/mrm.22924CrossRefGoogle Scholar
  49. 49.
    Valero, S., Chanussot, J., Benediktsson, J.A., Talbot, H., Waske, B.: Advanced directional mathematical morphology for the detection of the road network in very high resolution remote sensing images. Pattern Recognit. Lett. 31(10), 1120–1127 (2010). doi:10.1016/j.patrec.2009.12.018CrossRefGoogle Scholar
  50. 50.
    van de Gronde, J.J., Roerdink, J.B.T.M.: Frames, the Loewner order and eigendecomposition for morphological operators on tensor fields. Pattern Recognit. Lett. (2014). doi:10.1016/j.patrec.2014.03.013Google Scholar
  51. 51.
    van de Gronde, J.J., Roerdink, J.B.T.M.: Group-invariant colour morphology based on frames. IEEE Trans. Image Process. 23(3), 1276–1288 (2014). doi:10.1109/tip.2014.2300816MathSciNetCrossRefGoogle Scholar
  52. 52.
    Vincent, L.: Minimal path algorithms for the robust detection of linear features in gray images. In: Heijmans, H.J.A.M., Roerdink, J.B.T.M. (eds.) Mathematical Morphology and Its Applications to Image and Signal Processing, ISMM ’98, pp. 331–338. Kluwer Academic Publishers, Norwell, MA (1998)Google Scholar
  53. 53.
    Wilkinson, M.H.F.: Hyperconnectivity, attribute-space connectivity and path openings: theoretical relationships. In: Wilkinson, M.H.F., Roerdink, J.B.T.M. (eds.) Mathematical Morphology and Its Application to Signal and Image Processing. Lecture Notes in Computer Science, vol. 5720, Chap. 5, pp. 47–58. Springer, Berlin/Heidelberg (2009). doi:10.1007/978-3-642-03613-2_5Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jasper J. van de Gronde
    • 1
  • Mikola Lysenko
    • 2
  • Jos B. T. M. Roerdink
    • 1
    Email author
  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of Computer SciencesUniversity of Wisconsin - MadisonMadisonUSA

Personalised recommendations