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Amoeba Techniques for Shape and Texture Analysis

  • Martin WelkEmail author
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

Morphological amoebas are image-adaptive structuring elements for morphological and other local image filters introduced by Lerallut et al. Their construction is based on combining spatial distance with contrast information into an image-dependent metric. Amoeba filters show interesting parallels to image filtering methods based on partial differential equations (PDEs), which can be confirmed by asymptotic equivalence results. In computing amoebas, graph structures are generated that hold information about local image texture. This chapter reviews and summarises the work of the author and his coauthors on morphological amoebas, particularly their relations to PDE filters and texture analysis. It presents some extensions and points out directions for future investigation on the subject.

Keywords

Median Filter Active Contour Texture Descriptor Level Line Initial Contour 
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.

References

  1. 1.
    Alvarez, L., Lions, P.-L., Morel, J.-M.: Image selective smoothing and edge detection by nonlinear diffusion. II. SIAM J. Numer. Anal. 29, 845–866 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Austin, T.L.: An approximation to the point of minimum aggregate distance. Metron 19, 10–21 (1959)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, Chichester (1992)zbMATHGoogle Scholar
  4. 4.
    Bonchev, D., Trinajstić, N.: Information theory, distance matrix, and molecular branching. J. Chem. Phys. 67 (10), 4517–4533 (1977)CrossRefGoogle Scholar
  5. 5.
    Braga-Neto, U.M.: Alternating sequential filters by adaptive neighborhood structuring functions. In: Maragos, P., Schafer, R.W., Butt, M.A. (eds.) Mathematical Morphology and Its Applications to Image and Signal Processing. Volume 5 of Computational Imaging and Vision, pp. 139–146. Kluwer, Dordrecht (1996)CrossRefGoogle Scholar
  6. 6.
    Burgeth, B., Kleefeld, A.: Morphology for color images via Loewner order for matrix fields. In: Luengo Hendriks, C.L., Borgefors, G., Strand, R. (eds.) Mathematical Morphology and Its Applications to Signal and Image Processing. Volume 7883 of Lecture Notes in Computer Science, pp. 243–254. Springer, Berlin (2013)Google Scholar
  7. 7.
    Burgeth, B., Bruhn, A., Papenberg, N., Welk, M., Weickert, J.: Mathematical morphology for matrix fields induced by the Loewner ordering in higher dimensions. Signal Process. 87 (2), 277–290 (2007)CrossRefzbMATHGoogle Scholar
  8. 8.
    Burgeth, B., Welk, M., Feddern, C., Weickert, J.: Morphological operations on matrix-valued images. In: Pajdla, T., Matas, J. (eds.) Computer Vision – ECCV 2004, Part IV. Volume 3024 of Lecture Notes in Computer Science, pp. 155–167. Springer, Berlin (2004)Google Scholar
  9. 9.
    Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. In: Proceedings of the Fifth International Conference on Computer Vision, Cambridge, June 1995, pp. 694–699. IEEE Computer Society Press.Google Scholar
  10. 10.
    Cohen, L.D.: On active contour models and balloons. CVGIP: Image Underst. 53 (2), 211–218 (1991)CrossRefzbMATHGoogle Scholar
  11. 11.
    Cootes, T.F., Taylor, C.J.: Statistical models of appearance for computer vision. Technical report, University of Manchester, Oct 2001Google Scholar
  12. 12.
    Dehmer, M.: Information processing in complex networks: graph entropy and information functionals. Appl. Math. Comput. 201, 82–94 (2008)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dehmer, M., Emmert-Streib, F., Mehler, A. (eds.): Towards an Information Theory of Complex Networks: Statistical Methods and Applications. Birkhäuser Publishing, Basel (2012)Google Scholar
  14. 14.
    Dehmer, M., Emmert-Streib, F., Tripathi, S.: Large-scale evaluation of molecular descriptors by means of clustering. PloS ONE 8 (12), e83956 (2013)CrossRefGoogle Scholar
  15. 15.
    Dehmer, M., Sivakumar, L.: Recent developments in quantitative graph theory: information inequalities for networks. PLoS ONE 7 (2), e31395 (2012)CrossRefGoogle Scholar
  16. 16.
    Dijkstra, E.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Eckhardt, U.: Root images of median filters. J. Math. Imaging Vis. 19, 63–70 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Emmert-Streib, F., Dehmer, M.: Information theoretic measures of UHG graphs with low computational complexity. Appl. Math. Comput. 190, 1783–1794 (2007)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ferrer, M., Bunke, H.: Graph edit distance–theory, algorithms, and applications. In: Lezoray, O., Grady, L. (eds.) Image Processing and Analysis with Graphs: Theory and Practice, chapter 13, pp. 383–422. CRC Press, Boca Raton (2012)Google Scholar
  20. 20.
    Guichard, F., Morel, J.-M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. Number 63 in IMA Conference Series (New Series), pp. 525–562. Clarendon Press, Oxford (1997)Google Scholar
  21. 21.
    Haralick, R.: Statistical and structural approaches to texture. Proc. IEEE 67 (5), 786–804 (1979)CrossRefGoogle Scholar
  22. 22.
    Haralick, R., Shanmugam, K., Dinstein I.: Textural features for image classification. IEEE Trans. Syst. Man Cybern. 3 (6), 610–621 (1973)CrossRefGoogle Scholar
  23. 23.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic, Boston (1994)zbMATHGoogle Scholar
  24. 24.
    Heijmans, H.J.A.M., Ronse, C.: The algebraic basis of mathematical morphology. I: dilations and erosions. Comput. Vis. Graph. Image Process. 50, 245–295 (1990)zbMATHGoogle Scholar
  25. 25.
    Heijmans, H.J.A.M., van den Boomgaard, R.: Algebraic framework for linear and morphological scale-spaces. J. Vis. Commun. Image Represent. 13 (1/2), 269–301 (2002)CrossRefGoogle Scholar
  26. 26.
    Hosoya, H.: Topological index: a newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons. Bull. Chem. Soc. Jpn 44 (9), 2332–2339 (1971)CrossRefGoogle Scholar
  27. 27.
    Howarth, P., Rüger, S.: Evaluation of texture features for content-based image retrieval. In: Enser, P., Kompatsiaris, Y., O’Connor, N., Smeaton, A., Smeulders, A. (eds.) Image and Video Retrieval. Volume 3115 of Lecture Notes in Computer Science, pp. 326–334. Springer, Berlin (2004)Google Scholar
  28. 28.
    Huang, K., Murphy, R.: Automated classification of subcellular patterns in multicell images without segmentation into single cells. In: Proceedings of the 2004 IEEE International Symposium on Biomedical Imaging, Apr 2004, vol. 2, pp. 1139–1142Google Scholar
  29. 29.
    Ivanciuc, O., Balaban, T.-S., Balaban, A.: Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices. J. Math. Chem. 12 (1), 309–318 (1993)Google Scholar
  30. 30.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proceedings of the Fifth International Conference on Computer Vision, Cambridge, June 1995, pp. 810–815. IEEE Computer Society PressGoogle Scholar
  31. 31.
    Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Conformal curvature flows: from phase transitions to active vision. Arch. Ration. Mech. Anal. 134, 275–301 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Lerallut, R., Decencière, É., Meyer, F.: Image processing using morphological amoebas. In: Ronse, C., Najman, L., Decencière, E. (eds.) Mathematical Morphology: 40 Years on. Volume 30 of Computational Imaging and Vision, pp. 13–22. Springer, Dordrecht (2005)CrossRefGoogle Scholar
  33. 33.
    Lerallut, R., Decencière, É., Meyer, F.: Image filtering using morphological amoebas. Image Vis. Comput. 25 (4), 395–404 (2007)CrossRefGoogle Scholar
  34. 34.
    Leventon, M.E., Grimson, W.E.L., Faugeras, O.: Statistical shape influence in geodesic active contours. In: Proceedings of the 2000 IEEE International Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 316–323, Hilton Head Island, June 2000Google Scholar
  35. 35.
    Malladi, R., Sethian, J., Vemuri, B.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17, 158–175 (1995)CrossRefGoogle Scholar
  36. 36.
    Maragos, P.: Lattice image processing: a unification of morphological and fuzzy algebraic systems. J. Math. Imaging Vis. 22, 333–353 (2005)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Maragos, P., Vachier, C.: Overview of adaptive morphology: trends and perspectives. In: Proceedings of the 2009 IEEE International Conference on Image Processing, Cairo, Nov 2009, pp. 2241–2244Google Scholar
  38. 38.
    Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)Google Scholar
  39. 39.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  40. 40.
    Picard, R., Graczyk, C., Mann, S., Wachman, J., Picard, L., Campbell, L.: VisTex database. Online ressource, http://vismod.media.mit.edu/vismod/imagery/VisionTexture/vistex.html (1995). Retrieved 20 Nov 2013
  41. 41.
    Plavšić, D., Nikolić, S., Trinajstić, N.: On the Harary index for the characterization of chemical graphs. J. Math. Chem. 12 (1), 235–250 (1993)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Quadrat, J.-P.: Max-plus algebra and applications to system theory and optimal control. In: Chatterji, S.D. (ed.) Proceedings of the International Congress of Mathematicians, pp. 1511–1522. Birkhäuser, Basel (1995)CrossRefGoogle Scholar
  43. 43.
    Sanfeliu, A., Fu, K.-S.: A distance measure between attributed relational graphs for pattern recognition. IEEE Trans. Syst. Man Cybern. 13 (3), 353–362 (1983)CrossRefzbMATHGoogle Scholar
  44. 44.
    Sapiro, G.: Vector (self) snakes: a geometric framework for color, texture and multiscale image segmentation. In: Proceedings of the 1996 IEEE International Conference on Image Processing, Lausanne, Sept 1996, vol. 1, pp. 817–820Google Scholar
  45. 45.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 1. Academic, London (1982)zbMATHGoogle Scholar
  46. 46.
    Serra, J.: Image Analysis and Mathematical Morphology, vol. 2. Academic, London (1988)Google Scholar
  47. 47.
    Shannon, C.: A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423; 623–656 (1948)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Spence, C., Fancourt, C.: An iterative method for vector median filtering. In: Proceedings of the 2007 IEEE International Conference on Image Processing, vol. 5, pp. 265–268 (2007)Google Scholar
  49. 49.
    Tesař, L., Shimizu, A., Smutek, D., Kobatake, H., Nawano, S.: Medical image analysis of 3D CT images based on extension of Haralick texture features. Comput. Med. Imaging Graph. 32 (6), 513–520 (2008)CrossRefGoogle Scholar
  50. 50.
    Tukey, J.W.: Exploratory Data Analysis. Addison–Wesley, Menlo Park (1971)zbMATHGoogle Scholar
  51. 51.
    Verly, J.G., Delanoy, R.L.: Adaptive mathematical morphology for range imagery. IEEE Trans. Image Process. 2 (2), 272–275 (1993)CrossRefGoogle Scholar
  52. 52.
    Weiszfeld, E.: Sur le point pour lequel la somme des distances de n points donnés est minimum. Tôhoku Mathematics Journal 43, 355–386 (1937)zbMATHGoogle Scholar
  53. 53.
    Welk, M.: Amoeba active contours. In: Bruckstein, A.M., ter Haar Romeny, B., Bronstein, A.M., Bronstein, M.M. (eds.) Scale Space and Variational Methods in Computer Vision. Volume 6667 of Lecture Notes in Computer Science, pp. 374–385. Springer, Berlin (2012)Google Scholar
  54. 54.
    Welk, M.: Relations between amoeba median algorithms and curvature-based PDEs. In: Kuijper, A., Pock, T., Bredies, K., Bischof, H. (eds.) Scale Space and Variational Methods in Computer Vision. Volume 7893 of Lecture Notes in Computer Science, pp. 392–403. Springer, Berlin (2013)Google Scholar
  55. 55.
    Welk, M.: Discrimination of image textures using graph indices. In: Dehmer, M., Emmert-Streib, F. (eds.) Quantitative Graph Theory: Mathematical Foundations and Applications, chapter 12, pp. 355–386. CRC Press, Boca Raton (2014)Google Scholar
  56. 56.
    Welk, M.: Analysis of amoeba active contours. J. Math. Imaging Vis. 52, 37–54 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Welk, M., Breuß, M.: Morphological amoebas and partial differential equations. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 185, pp. 139–212. Elsevier/Academic, Amsterdam (2014)Google Scholar
  58. 58.
    Welk, M., Breuß, M., Vogel, O.: Morphological amoebas are self-snakes. J. Math. Imaging Vis. 39, 87–99 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Welk, M., Feddern, C., Burgeth, B., Weickert, J.: Median filtering of tensor-valued images. In: Michaelis, B., Krell, G. (eds.) Pattern Recognition. Volume 2781 of Lecture Notes in Computer Science, pp. 17–24. Springer, Berlin (2003)Google Scholar
  60. 60.
    Welk, M., Weickert, J., Becker, F., Schnörr, C., Feddern, C., Burgeth, B.: Median and related local filters for tensor-valued images. Signal Process. 87, 291–308 (2007)CrossRefzbMATHGoogle Scholar
  61. 61.
    Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69 (1), 17–20 (1947)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Biomedical Image Analysis Division, Department of Biomedical Computer Science and MechatronicsUniversity for Health Sciences Medical Informatics and TechnologyHall/TyrolAustria

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