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Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning

  • Conference paper
Symbolic and Quantitative Approaches to Reasoning with Uncertainty (ECSQARU 2009)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 5590))

Abstract

Mathematical morphology is based on the algebraic framework of complete lattices and adjunctions, which endows it with strong properties and allows for multiple extensions. In particular, extensions to fuzzy sets of the main morphological operators, such as dilation and erosion, can be done while preserving all properties of these operators. Another, more recent, extension, concerns bipolar fuzzy sets. These extensions have numerous applications, two of each being presented here. The first one concerns the definition of spatial relations, for applications in spatial reasoning and model-based recognition of structures in images. The second one concerns the handling of the bipolarity feature of spatial information.

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References

  1. Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)

    MATH  Google Scholar 

  2. Ronse, C.: Why Mathematical Morphology Needs Complete Lattices. Signal Processing 21(2), 129–154 (1990)

    Article  MATH  Google Scholar 

  3. Heijmans, H.J.A.M., Ronse, C.: The Algebraic Basis of Mathematical Morphology – Part I: Dilations and Erosions. Computer Vision, Graphics and Image Processing 50, 245–295 (1990)

    Article  MATH  Google Scholar 

  4. Ronse, C., Heijmans, H.J.A.M.: The Algebraic Basis of Mathematical Morphology – Part II: Openings and Closings. Computer Vision, Graphics and Image Processing 54, 74–97 (1991)

    MATH  Google Scholar 

  5. Bloch, I., Lang, J.: Towards Mathematical Morpho-Logics. In: 8th International Conference on Information Processing and Management of Uncertainty in Knowledge based Systems IPMU 2000, Madrid, Spain, vol. III, pp. 1405–1412 (2000)

    Google Scholar 

  6. Bloch, I., Pino-Pérez, R., Uzcátegui, C.: Explanatory Relations based on Mathematical Morphology. In: Benferhat, S., Besnard, P. (eds.) ECSQARU 2001. LNCS, vol. 2143, pp. 736–747. Springer, Heidelberg (2001)

    Chapter  Google Scholar 

  7. Bloch, I., Pino-Pérez, R., Uzcategui, C.: A Unified Treatment of Knowledge Dynamics. In: International Conference on the Principles of Knowledge Representation and Reasoning, KR 2004, Canada, pp. 329–337 (2004)

    Google Scholar 

  8. Bloch, I., Pino-Pérez, R., Uzcategui, C.: Mediation in the Framework of Morphologic. In: European Conference on Artificial Intelligence ECAI 2006, Riva del Garda, Italy, pp. 190–194 (2006)

    Google Scholar 

  9. Bloch, I.: Modal Logics based on Mathematical Morphology for Spatial Reasoning. Journal of Applied Non Classical Logics 12(3-4), 399–424 (2002)

    Article  MATH  Google Scholar 

  10. Bloch, I., Heijmans, H., Ronse, C.: Mathematical Morphology. In: Aiello, M., Pratt-Hartman, I., van Benthem, J. (eds.) The Logic of Space, pp. 857–947. Kluwer, Dordrecht (2007)

    Google Scholar 

  11. Bandemer, H., Näther, W.: Fuzzy Data Analysis. Theory and Decision Library. Serie B: Mathematical and Statistical Methods. Kluwer Academic Publisher, Dordrecht (1992)

    Book  MATH  Google Scholar 

  12. Bloch, I.: Triangular Norms as a Tool for Constructing Fuzzy Mathematical Morphologies. In: Int. Workshop on ”Mathematical Morphology and its Applications to Signal Processing”, Barcelona, Spain, May 1993, pp. 157–161 (1993)

    Google Scholar 

  13. De Baets, B., Kerre, E., Gupta, M.: The Fundamentals of Fuzzy Mathematical Morphology Part 1: Basic Concepts. International Journal of General Systems 23(2), 155–171 (1995)

    Article  MATH  Google Scholar 

  14. De Baets, B., Kerre, E., Gupta, M.: The Fundamentals of Fuzzy Mathematical Morphology Part 2: Idempotence, Convexity and Decomposition. International Journal of General Systems 23(4), 307–322 (1995)

    Article  MATH  Google Scholar 

  15. Sinha, D., Dougherty, E.R.: Fuzzification of Set Inclusion: Theory and Applications. Fuzzy Sets and Systems 55, 15–42 (1993)

    Article  MATH  Google Scholar 

  16. Bloch, I., Maître, H.: Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition 28(9), 1341–1387 (1995)

    Article  Google Scholar 

  17. De Baets, B.: Generalized Idempotence in Fuzzy Mathematical Morphology. In: Kerre, E., Nachtegael, M. (eds.) Fuzzy Techniques in Image Processing. Studies in Fuzziness and Soft Computing, vol. 52, pp. 58–75. Physica Verlag/Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  18. Deng, T.Q., Heijmans, H.: Grey-Scale Morphology Based on Fuzzy Logic. Journal of Mathematical Imaging and Vision 16, 155–171 (2002)

    Article  MATH  Google Scholar 

  19. Maragos, P.: Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems. Journal of Mathematical Imaging and Vision 22, 333–353 (2005)

    Article  Google Scholar 

  20. Nachtegael, M., Kerre, E.E.: Classical and Fuzzy Approaches towards Mathematical Morphology. In: Kerre, E.E., Nachtegael, M. (eds.) Fuzzy Techniques in Image Processing. Studies in Fuzziness and Soft Computing, pp. 3–57. Physica-Verlag/Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  21. Popov, A.T.: Morphological Operations on Fuzzy Sets. In: IEE Image Processing and its Applications, Edinburgh, UK, July 1995, pp. 837–840 (1995)

    Google Scholar 

  22. Dubois, D., Prade, H.: Inverse Operations for Fuzzy Numbers. In: Sanchez, E., Gupta, M. (eds.) Fuzzy Information, Knowledge Representation and Decision Analysis, IFAC Symposium, Marseille, France, July 1983, pp. 391–396 (1983)

    Google Scholar 

  23. Bloch, I.: Duality vs Adjunction and General Form for Fuzzy Mathematical Morphology. In: Bloch, I., Petrosino, A., Tettamanzi, A.G.B. (eds.) WILF 2005. LNCS, vol. 3849, pp. 354–361. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  24. Bloch, I.: Duality vs. Adjunction for Fuzzy Mathematical Morphology and General Form of Fuzzy Erosions and Dilations. Fuzzy Sets and Systems 160, 1858–1867 (2009)

    Google Scholar 

  25. Sussner, P., Valle, M.: Classification of Fuzzy Mathematical Morphologies based on Concepts of Inclusion Measure and Duality. Journal of Mathematical Imaging and Vision 21, 139–159 (2008)

    Article  Google Scholar 

  26. Dubois, D., Kaci, S., Prade, H.: Bipolarity in Reasoning and Decision, an Introduction. In: International Conference on Information Processing and Management of Uncertainty, IPMU 2004, Perugia, Italy, pp. 959–966 (2004)

    Google Scholar 

  27. Cornelis, C., Kerre, E.: Inclusion Measures in Intuitionistic Fuzzy Sets. In: Nielsen, T.D., Zhang, N.L. (eds.) ECSQARU 2003. LNCS (LNAI), vol. 2711, pp. 345–356. Springer, Heidelberg (2003)

    Chapter  Google Scholar 

  28. Bloch, I.: Dilation and Erosion of Spatial Bipolar Fuzzy Sets. In: Masulli, F., Mitra, S., Pasi, G. (eds.) WILF 2007. LNCS (LNAI), vol. 4578, pp. 385–393. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  29. Deschrijver, G., Cornelis, C., Kerre, E.: On the Representation of Intuitionistic Fuzzy t-Norms and t-Conorms. IEEE Transactions on Fuzzy Systems 12(1), 45–61 (2004)

    Article  MATH  Google Scholar 

  30. Nachtegael, M., Sussner, P., Mélange, T., Kerre, E.: Some Aspects of Interval-Valued and Intuitionistic Fuzzy Mathematical Morphology. In: IPCV 2008 (2008)

    Google Scholar 

  31. Bloch, I.: A Contribution to the Representation and Manipulation of Fuzzy Bipolar Spatial Information: Geometry and Morphology. In: Workshop on Soft Methods in Statistical and Fuzzy Spatial Information, Toulouse, France, September 2008, pp. 7–25 (2008)

    Google Scholar 

  32. Bloch, I.: Bipolar Fuzzy Spatial Information: First Operations in the Mathematical Morphology Setting. In: De, R.K., Mandal, D.P., Ghosh, A. (eds.) Machine Interpretation of Patterns: Image Analysis, Data Mining and Bioinformatics. World Scientific Press, Singapore (2009)

    Google Scholar 

  33. Bloch, I.: Geometry of Spatial Bipolar Fuzzy Sets based on Bipolar Fuzzy Numbers and Mathematical Morphology. In: International Workshop on Fuzzy Logic and Applications WILF, Palermo, Italy (June 2009)

    Google Scholar 

  34. Bloch, I.: Bipolar Fuzzy Mathematical Morphology for Spatial Reasoning. In: International Symposium on Mathematical Morphology ISMM 2009, Groningen, The Netherlands (August 2009)

    Google Scholar 

  35. Bloch, I.: Fuzzy Spatial Relationships for Image Processing and Interpretation: A Review. Image and Vision Computing 23(2), 89–110 (2005)

    Article  Google Scholar 

  36. Freeman, J.: The Modelling of Spatial Relations. Computer Graphics and Image Processing 4(2), 156–171 (1975)

    Article  Google Scholar 

  37. Kuipers, B.J., Levitt, T.S.: Navigation and Mapping in Large-Scale Space. AI Magazine 9(2), 25–43 (1988)

    Google Scholar 

  38. Bloch, I.: Fuzzy Relative Position between Objects in Image Processing: a Morphological Approach. IEEE Transactions on Pattern Analysis and Machine Intelligence 21(7), 657–664 (1999)

    Article  Google Scholar 

  39. Bloch, I., Ralescu, A.: Directional Relative Position between Objects in Image Processing: A Comparison between Fuzzy Approaches. Pattern Recognition 36, 1563–1582 (2003)

    Article  MATH  Google Scholar 

  40. Dubois, D., Prade, H.: On Distance between Fuzzy Points and their Use for Plausible Reasoning. In: Int. Conf. Systems, Man, and Cybernetics, pp. 300–303 (1983)

    Google Scholar 

  41. Rosenfeld, A.: Distances between Fuzzy Sets. Pattern Recognition Letters 3, 229–233 (1985)

    Article  MATH  Google Scholar 

  42. Bloch, I.: On Fuzzy Distances and their Use in Image Processing under Imprecision. Pattern Recognition 32(11), 1873–1895 (1999)

    Article  Google Scholar 

  43. Bloch, I.: On Fuzzy Spatial Distances. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 128, pp. 51–122. Elsevier, Amsterdam (2003)

    Google Scholar 

  44. Bloch, I., Maître, H., Anvari, M.: Fuzzy Adjacency between Image Objects. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 5(6), 615–653 (1997)

    Article  MATH  Google Scholar 

  45. Mathet, Y.: Etude de l’expression en langue de l’espace et du déplacement : analyse linguistique, modélisation cognitive, et leur expérimentation informatique. PhD thesis, Université de Caen, France (December 2000)

    Google Scholar 

  46. Bloch, I., Colliot, O., Cesar, R.: On the Ternary Spatial Relation Between. IEEE Transactions on Systems, Man, and Cybernetics SMC-B 36(2), 312–327 (2006)

    Article  Google Scholar 

  47. Rosse, C., Mejino, J.L.V.: A Reference Ontology for Bioinformatics: The Foundational Model of Anatomy. Journal of Biomedical Informatics 36, 478–500 (2003)

    Article  Google Scholar 

  48. Hudelot, C., Atif, J., Bloch, I.: Fuzzy Spatial Relation Ontology for Image Interpretation. Fuzzy Sets and Systems 159, 1929–1951 (2008)

    Article  Google Scholar 

  49. Hudelot, C., Atif, J., Bloch, I.: A Spatial Relation Ontology Using Mathematical Morphology and Description Logics for Spatial Reasoning. In: ECAI 2008 Workshop on Spatial and Temporal Reasoning, Patras, Greece, July 2008, pp. 21–25 (2008)

    Google Scholar 

  50. Bloch, I., Géraud, T., Maître, H.: Representation and Fusion of Heterogeneous Fuzzy Information in the 3D Space for Model-Based Structural Recognition - Application to 3D Brain Imaging. Artificial Intelligence 148, 141–175 (2003)

    Article  MATH  Google Scholar 

  51. Colliot, O., Camara, O., Bloch, I.: Integration of Fuzzy Spatial Relations in Deformable Models - Application to Brain MRI Segmentation. Pattern Recognition 39, 1401–1414 (2006)

    Article  Google Scholar 

  52. Atif, J., Hudelot, C., Fouquier, G., Bloch, I., Angelini, E.: From Generic Knowledge to Specific Reasoning for Medical Image Interpretation using Graph-based Representations. In: International Joint Conference on Artificial Intelligence IJCAI 2007, Hyderabad, India, January 2007, pp. 224–229 (2007)

    Google Scholar 

  53. Fouquier, G., Atif, J., Bloch, I.: Local Reasoning in Fuzzy Attributes Graphs for Optimizing Sequential Segmentation. In: Escolano, F., Vento, M. (eds.) GbRPR. LNCS, vol. 4538, pp. 138–147. Springer, Heidelberg (2007)

    Chapter  Google Scholar 

  54. Fouquier, G., Atif, J., Bloch, I.: Sequential Spatial Reasoning in Images based on Pre-Attention Mechanisms and Fuzzy Attribute Graphs. In: European Conference on Artificial Intelligence ECAI, Patras, Greece, July 2008, pp. 611–615 (2008)

    Google Scholar 

  55. Bengoetxea, E., Larranaga, P., Bloch, I., Perchant, A., Boeres, C.: Inexact Graph Matching by Means of Estimation of Distribution Algorithms. Pattern Recognition 35, 2867–2880 (2002)

    Article  MATH  Google Scholar 

  56. Nempont, O., Atif, J., Angelini, E., Bloch, I.: Structure Segmentation and Recognition in Images Guided by Structural Constraint Propagation. In: European Conference on Artificial Intelligence ECAI, Patras, Greece, July 2008, pp. 621–625 (2008)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Télécom ParisTech (ENST), CNRS UMR 5141 LTCI, Paris, France

    Isabelle Bloch

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  1. Isabelle Bloch
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Editors and Affiliations

  1. ISIB-CNR, Corso Stati Uniti 4, 35127, Padova, Italy

    Claudio Sossai

  2. ISIB-CNR, Corso Stati Uniti, 4, 35127, Padova, Italy

    Gaetano Chemello

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Bloch, I. (2009). Fuzzy and Bipolar Mathematical Morphology, Applications in Spatial Reasoning. In: Sossai, C., Chemello, G. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2009. Lecture Notes in Computer Science(), vol 5590. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02906-6_1

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  • DOI: https://doi.org/10.1007/978-3-642-02906-6_1

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-02905-9

  • Online ISBN: 978-3-642-02906-6

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