Abstract
Basic mathematical morphology operations rely mainly on local information, based on the concept of structuring element. But mathematical morphology also deals with more global and structural information since several spatial relationships can be expressed in terms of morphological operations (mainly dilations). The aim of this paper is to show that this framework allows to represent in a unified way spatial relationships in various settings: a purely quantitative one if objects are precisely defined, a semi-quantitative one if objects are imprecise and represented as spatial fuzzy sets, and a qualitative one, for reasoning in a logical framework about space. This is made possible thanks to the strong algebraic structure of mathematical morphology, that finds equivalents in set theoretical terms, fuzzy operations and logical expressions.
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References
J. Allen. Maintaining Knowledge about Temporal Intervals. Comunications of the ACM, 26 (11): 832–843, 1983.
N. Asher and L. Vieu. Toward a Geometry of Common Sense: A Semantics and a Complete Axiomatization of Mereotopology. In IJCAI’95, pages 846–852, San Mateo, CA, 1995.
E. Bengoetxea, P. Larranaga, I. Bloch, and A. Perchant. Solving Graph Matching with EDAs Using a Permutation-Based Representation. In P. Larranaga and J. A. Lozano, editors, Estimation of Distribution Algorithms: A New Tool for Evolutionary Computation, chapter 12, pages 239–261. Kluwer Academic Publisher, Boston, Dordrecht, London, 2001.
B. Bennett. Modal Logics for Qualitative Spatial Reasoning. Bulletin of the IGPL, 4 (1): 23–45, 1995.
I. Bloch. About Properties of Fuzzy Mathematical Morphologies: Proofs of Main Results. Technical report, Télécom Paris 93D023, December 1993.
I. Bloch. Fuzzy Relative Position between Objects in Image Processing: a Morphological Approach. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21 (7): 657–664, 1999.
I. Bloch. Fuzzy Relative Position between Objects in Image Processing: New Definition and Properties based on a Morphological Approach. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 7 (2): 99–133, 1999.
I. Bloch. On Fuzzy Distances and their Use in Image Processing under Imprecision. Pattern Recognition, 32 (11): 1873–1895, 1999.
I. Bloch and J. Lang. Towards Mathematical Morpho-Logics. In 8th International Conference on Information Processing and Management of Uncertainty in Knowledge based Systems IPMU 2000, volume III, pages 1405–1412, Madrid, Spain, jul 2000.
I. Bloch and H. Maître. Fuzzy Mathematical Morphologies: A Comparative Study. Pattern Recognition, 28 (9): 1341–1387, 1995.
I. Bloch, H. Maître, and M. Anvari. Fuzzy Adjacency between Image Objects. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 5 (6): 615–653, 1997.
I. Bloch, C. Pellot, F. Sureda, and A. Herment. Fuzzy Modelling and Fuzzy Mathematical Morphology applied to 3D Reconstruction of Blood Vessels by Multi-Modality Data Fusion. In D. Dubois R. Yager and H. Prade, editors, Fuzzy Set Methods in Information Engineering: A Guided Tour of Applications, chapter 5, pages 93–110. John Wiley and Sons, New-York, 1996.
G. Borgefors. Distance Transforms in the Square Grid. In H. Maître, editor, Progress in Picture Processing, Les Houches, Session LVIII, 1992, chapter 1.4, pages 46–80. North-Holland, Amsterdam, 1996.
B. Chellas. Modal Logic, an Introduction. Cambridge University Press, Cambridge, 1980.
E. Clementini and O. Di Felice. Approximate Topological Relations. International Journal of Approximate Reasoning, 16: 173–204, 1997.
A. Cohn, B. Bennett, J. Gooday, and N. M. Gotts. Representing and Reasoning with Qualitative Spatial Relations about Regions. In O. Stock, editor, Spatial and Temporal Reasoning, pages 97–134. Kluwer, 1997.
B. de Baets. Fuzzy Morphology: a Logical Approach. In B. Ayyub and M. Gupta, editors, Uncertainty in Engineering and Sciences: Fuzzy Logic, Statistics and Neural Network Approach, pages 53–67. Kluwer Academic, 1997.
T.-Q. Deng and H. Heijmans. Grey-Scale Morphology Based on Fuzzy Logic. Technical Report PNA-R0012, CWI, Amsterdam, NL, 2000.
D. Dubois and H. Prade. Fuzzy Sets and Systems: Theory and Applications. Academic Press, New-York, 1980.
D. Dubois and H. Prade. A Review of Fuzzy Set Aggregation Connectives. Information Sciences, 36: 85–121, 1985.
D. Dubois and H. Prade. Weighted Fuzzy Pattern Matching. Fuzzy Sets and Systems, 28: 313–331, 1988.
S. Dutta. Approximate Spatial Reasoning: Integrating Qualitative and Quantitative Constraints. International Journal of Approximate Reasoning, 5: 307–331, 1991.
J. Freeman. The Modelling of Spatial Relations. Computer Graphics and Image Processing, 4 (2): 156–171, 1975.
K. P. Gapp. Basic Meanings of Spatial Relations: Computation and Evaluation in 3D Space. In 12th National Conference on Artificial Intelligence, AAAI-94, pages 1393–1398, Seattle, Washington, 1994.
J. Gasos and A. Ralescu. Using Imprecise Environment Information for Guiding Scene Interpretation. Fuzzy Sets and Systems, 88: 265–288, 1997.
J. Gasios and A. Saffiotti. Using Fuzzy Sets to Represent Uncertain Spatial Knowledge in Autonomous Robots. Journal of Spatial Cognition and Computation, 1: 205–226, 2000.
T. Géraud, I. Bloch, and H. Maître. Atlas-guided Recognition of Cerebral Structures in MRI using Fusion of Fuzzy Structural Information. In CIMAF’99 Symposium on Artificial Intelligence, pages 99–106, La Havana, Cuba, March 1999.
H. J. A. M. Heijmans and C. Ronse. The Algebraic Basis of Mathematical Morphology–Part I: Dilations and Erosions. Computer Vision, Graphics and Image Processing, 50: 245–295, 1990.
G. E. Hughes and M. J. Cresswell. An Introduction to Modal Logic. Methuen, London, UK, 1968.
J. M. Keller and X. Wang. Comparison of Spatial Relation Definitions in Computer Vision. In ISUMA-NAFIPS’95, pages 679–684, College Park, MD, September 1995.
L. T. Koczy. On the Description of Relative Position of Fuzzy Patterns. Pattern Recognition Letters, 8: 21–28, 1988.
R. Krishnapuram, J. M. Keller, and Y. Ma. Quantitative Analysis of Properties and Spatial Relations of Fuzzy Image Regions. IEEE Transactions on Fuzzy Systems, 1 (3): 222–233, 1993.
B. Kuipers. Modeling Spatial Knowledge. Cognitive Science, 2: 129–153, 1978.
B. J. Kuipers and T. S. Levitt. Navigation and Mapping in Large-Scale Space. AI Magazine, 9 (2): 25–43, 1988.
J. Liu. A Method of Spatial Reasoning based on Qualitative Trigonometry. Artificial Intelligence, 98: 137–168, 1998.
J.-F. Mangin, I. Bloch, J. Lopez-Krahe, and V. Frouin. Chamfer Distances in Anisotropic 3D Images. In EUSIPCO 94, pages 975–978, Edinburgh, UK, September 1994.
G. Matheron. Eléments pour une théorie des milieux poreux. Masson, Paris, 1967.
G. Matheron. Random Sets and Integral Geometry. Wiley, New-York, 1975.
P. Matsakis and L. Wendling. A New Way to Represent the Relative Position between Areal Objects. IEEE Trans. on Pattern Analysis and Machine Intelligence, 21 (7): 634–642, 1999.
K. Miyajima and A. Ralescu. Spatial Organization in 2D Segmented Images: Representation and Recognition of Primitive Spatial Relations. Fuzzy Sets and Systems, 65: 225–236, 1994.
A. Perchant and I. Bloch. Fuzzy Morphisms between Graphs. Fuzzy Sets and Systems, 2001.
A. Perchant, C. Boeres, I. Bloch, M. Roux, and C. Ribeiro. Model-based Scene Recognition Using Graph Fuzzy Homomorphism Solved by Genetic Algorithm. In GbR’99 2nd International Workshop on Graph-Based Representations in Pattern Recognition, pages 61–70, Castle of Haindorf, Austria, 1999.
D. J. Peuquet. Representations of Geographical Space: Toward a Conceptual Synthesis. Annals of the Association of American Geographers, 78 (3): 375–394, 1988.
D. Pullar and M. Egenhofer. Toward Formal Definitions of Topological Relations Among Spatial Objects. In Third Int. Symposium on Spatial Data Handling, pages 225–241, Sydney, Australia, August 1988.
D. Randell, Z. Cui, and A. Cohn. A Spatial Logic based on Regions and Connection. In Principles of Knowledge Representation and Reasoning KR’92, pages 165–176, San Mateo, CA, 1992.
A. Rosenfeld. Fuzzy Digital Topology. Information and Control, 40: 76–87, 1979.
A. Rosenfeld. The Fuzzy Geometry of Image Subsets. Pattern Recognition Letters, 2: 311–317, 1984.
A. Rosenfeld and A. C. Kak. Digital Picture Processing. Academic Press, New-York, 1976.
A. Rosenfeld and R. Klette. Degree of Adjacency or Surroundness. Pattern Recognition, 18 (2): 169–177, 1985.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.
J. Serra. Image Analysis and Mathematical Morphology, Part II: Theoretical Advances. Academic Press (J. Serra Ed. ), London, 1988.
D. Sinha and E. Dougherty. Fuzzy Mathematical Morphology. Journal of Visual Communication and Image Representation, 3 (3): 286–302, 1992.
J. K. Udupa and S. Samarasekera. Fuzzy Connectedness and Object Definition: Theory, Algorithms, and Applications in Image Segmentation. Graphical Models and Image Processing, 58 (3): 246–261, 1996.
A. Varzi. Parts, Wholes, and Part-Whole Relations: The Prospects of Mereotopology. Data and Knowledge Engineering, 20 (3): 259–286, 1996.
L. Vieu. Spatial Representation and Reasoning in Artificial Intelligence. In O. Stock, editor, Spatial and Temporal Reasoning, pages 5–41. Kluwer, 1997.
L. A. Zadeh. Fuzzy Sets. Information and Control, 8: 338–353, 1965.
L. A. Zadeh. The Concept of a Linguistic Variable and its Application to Approximate Reasoning. Information Sciences, 8: 199–249, 1975.
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Bloch, I. (2002). Mathematical Morphology and Spatial Relationships: Quantitative, Semi-Quantitative and Symbolic Settings. In: Matsakis, P., Sztandera, L.M. (eds) Applying Soft Computing in Defining Spatial Relations. Studies in Fuzziness and Soft Computing, vol 106. Physica, Heidelberg. https://doi.org/10.1007/978-3-7908-1752-2_4
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DOI: https://doi.org/10.1007/978-3-7908-1752-2_4
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