# Fuzzy Mathematical Morphology and Derived Spatial Relationships

• Isabelle Bloch
Part of the Studies in Fuzziness and Soft Computing book series (STUDFUZZ, volume 52)

## Summary

The joint importance of mathematical morphology in image processing and of fuzzy sets for representing imprecisely defined image objects has led to an increased interest in fuzzy mathematical morphology. In this chapter, we first review the main definitions of fuzzy mathematical morphology operators and show that the most important properties of classical mathematical morphology are kept. Then we use these operators, mainly fuzzy dilation, for defining some spatial relationships between fuzzy objects: distances, adjacency and boundary, directional relative position. Such spatial fuzzy relationships are important in pattern recognition (for instance model-based structural pattern recognition). Indeed, objects can be recognized in a scene using their own characteristics, but also using their relationships to other objects of the scene. Therefore the proposed operations and relationships between fuzzy objects lead to applications in structural pattern recognition under imprecision.

## Keywords

Membership Function Fuzzy Number Hausdorff Distance Reference Object Mathematical Morphology

## References

1. 1.
De Baets B., Idempotent Closing and Opening Operations in Fuzzy Mathematical Morphology, in: “Proceedings of ISUMA-NAFIPS’95” (College Park, MD), pp. 228–233, 1995Google Scholar
2. 2.
De Baets B. and Kerre E., An Introduction to Fuzzy Mathematical Morphology, in: “Proceedings of NAFIPS’93” (Allentown, Pennsylvania), pp. 129–133, 1993Google Scholar
3. 3.
Bandemer H. and Näther W., “Fuzzy Data Analysis”, Theory and Decision Library, Serie B: Mathematical and Statistical Methods, Kluwer Academic Publisher, Dordrecht, 1992Google Scholar
4. 4.
Bloch I., “About Properties of Fuzzy Mathematical Morphologies: Proofs of Main Results”, Technical report, Télécom Paris 93D023, December 1993Google Scholar
5. 5.
Bloch I., Triangular Norms as a Tool for Constructing Fuzzy Mathematical Morphologies, in: “Proceedings of the Int. Workshop on Mathematical Morphology and its Applications to Signal Processing” (Barcelona, Spain), pp. 157–161, 1993Google Scholar
6. 6.
Bloch I., Distances in Fuzzy Sets for Image Processing derived from Fuzzy Mathematical Morphology, in: “Proceedings of IPMU’96 — Information Processing and Management of Uncertainty in Knowledge-Based Systems” (Granada, Spain), pp. 1307–1312, 1996Google Scholar
7. 7.
Bloch I., Fuzzy Relative Position between Objects in Images: a Morphological Approach, in: “Proceedings of ICIP’96 — IEEE Int. Conf. on Image Processing” (Lausanne), Vol. 2, pp. 987–990, 1996
8. 8.
Bloch I., Information Combination Operators for Data Fusion: A Comparative Review with Classification, IEEE Trans. on Systems, Man, and Cybernetics, Vol. 26, No. 1, pp. 52–67, 1996Google Scholar
9. 9.
Bloch I., Using Fuzzy Mathematical Morphology in the Dempster-Shafer Framework for Image Fusion under Imprecision, in: “Proceedings of IFSA’97” (Prague), pp. 209–214, 1997Google Scholar
10. 10.
Bloch I., Fuzzy Morphology and Fuzzy Distances: New Definitions and Links in both Euclidean and Geodesic Cases, in: “Lecture Notes in Artificial Intelligence: Fuzzy Logic in Artificial Intelligence, towards Intelligent Systems (A. Ralescu, ed.)”, Springer Verlag, 1998Google Scholar
11. 11.
Bloch I., Fusion of Numerical and Structural Image Information in Medical Imaging in the Framework of Fuzzy Sets, in: “Fuzzy Systems in Medicine (P. Szczepaniak, ed.)”, Springer Verlag, 1999Google Scholar
12. 12.
Bloch I., Fuzzy Relative Position between Objects in Image Processing: a Morphological Approach, IEEE Trans. on Pattern Analysis and Machine Intelligence, Vol. 21, No. 7, pp. 657–664, 1999
13. 13.
Bloch I., 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, Vol. 7, No. 2, pp. 99–133, 1999
14. 14.
Bloch I., Geodesic Balls in a Fuzzy Set and Fuzzy Geodesic Mathematical Morphology, Pattern Recognition, to appearGoogle Scholar
15. 15.
Bloch I., On Fuzzy Distances and their Use in Image Processing under Imprecision, Pattern Recognition, Vol. 32, No. 11, pp. 1873–1895, 1999
16. 16.
Bloch I., On Links between Mathematical Morphology and Rough Sets, Pattern Recognition, 1999Google Scholar
17. 17.
Bloch I. and Maître H., Constructing a Fuzzy Mathematical Morphology: Alternative Ways, in: “Proceedings of FUZZ-IEEE’93 — Second IEEE International Conference on Fuzzy Systems” (San Fransisco, California), pp. 1303–1308, 1993Google Scholar
18. 18.
Bloch I. and Maître H., Mathematical Morphology on Fuzzy Sets, in: “Proceedings of the Int. Workshop on Mathematical Morphology and its Applications to Signal Processing” (Barcelona, Spain), pp. 151–156, 1993Google Scholar
19. 19.
Bloch I. and Maître H., Fuzzy Mathematical Morphology, Annals of Mathematics and Artificial Intelligence, Vol. 10, pp. 55–84, 1994
20. 20.
Bloch I. and Maitre H., Fuzzy Mathematical Morphologies: A Comparative Study, Pattern Recognition, Vol. 28, No. 9, pp. 1341–1387, 1995
21. 21.
Bloch I., Maître H. and Anvari M., Fuzzy Adjacency between Image Objects, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol. 5, No. 6, pp. 615–653, 1997
22. 22.
Bloch I., Pellot C., Sureda F. and Herment A., Fuzzy Modelling and Fuzzy Mathematical Morphology applied to 3D Reconstruction of Blood Vessels by Multi-Modality Data Fusion, in: “Fuzzy Set Methods in Information Engineering: A Guided Tour of Applications (D. Dubois, R. Yager and H. Prade, eds.)”, John Wiley & Sons, New York, pp. 93–110, 1996Google Scholar
23. 23.
Borgefors G., Distance Transforms in the Square Grid, in: “Progress in Picture Processing, Les Houches, Session LVIII, 1992 (H. Maître, ed.)”, North-Holland, Amsterdam, pp. 46–80, 1996Google Scholar
24. 24.
Demko C. and Zahzah E.H., Image Understanding using Fuzzy Isomorphism of Fuzzy Structures, in: “Proceedings of the IEEE Int. Conf. on Fuzzy Systems” (Yokohama, Japan), pp. 1665–1672, 1995
25. 25.
di Gesu V., Mathematical Morphology and Image Analysis: A Fuzzy Approach, in: “Proceedings of the Workshop on Knowledge-Based Systems and Models of Logical Reasoning”, 1988Google Scholar
26. 26.
di Gesu V., Maccarone M.C. and Tripiciano M., Mathematical Morphology based on Fuzzy Operators, in: “Fuzzy Logic (R. Lowen and M. Roubens, eds.)”, Kluwer Academic, pp. 477–486, 1993
27. 27.
Dubois D. and Jaulent M.-C., A General Approach to Parameter Evaluation in Fuzzy Digital Pictures, Pattern Recognition Letters, Vol. 6, pp. 251–259, 1987
28. 28.
Dubois D. and Prade H., “Fuzzy Sets and Systems: Theory and Applications”, Academic Press, New York, 1980
29. 29.
Dubois D. and Prade H., Invers Operations for Fuzzy Numbers, in: “Fuzzy Information, Knowledge Representation and Decision Analysis (E. Sanchez and M. Gupta, eds.)”, IFAC Symposium (Marseille, France), pp. 391–396, 1983Google Scholar
30. 30.
Dubois D. and Prade H., A Review of Fuzzy Set Aggregation Connectives, Information Sciences, Vol. 36, pp. 85–121, 1985
31. 31.
Dubois D. and Prade H., Weighted Fuzzy Pattern Matching, Fuzzy Sets and Systems, Vol. 28, pp. 313–331, 1988
32. 32.
Dubois D. and Prade H., Fuzzy Sets in Approximate Reasoning, Part I: Inference with Possibility Distributions, Fuzzy Sets and Systems, Vol. 40, pp. 143–202, 1991
33. 33.
Dutta S., Approximate Spatial Reasoning: Integrating Qualitative and Quantitative Constraints, International Journal of Approximate Reasoning, Vol. 5, pp. 307–331, 1991
34. 34.
Géraud T., Bloch I. and Maître H., Atlas-guided Recognition of Cerebral Structures in MRI using Fusion of Fuzzy Structural Information, in: “Proceedings of CIMAF’99 Symposium on Artificial Intelligence” (La Havana, Cuba), pp. 99–106, 1999Google Scholar
35. 35.
Giardina C.R. and Sinha D., Image Processing using Pointed Fuzzy Sets, in: SPIE Intelligent Robots and Computer Vision VIII: Algorithms and Techniques, Vol. 1192, pp. 659–668, 1989Google Scholar
36. 36.
Goetcherian V., From Binary to Grey Tone Image Processing using Fuzzy Logic Concepts, Pattern Recognition, Vol. 12, pp. 7–15, 1980
37. 37.
Kaufmann A. and Gupta M.M., “Fuzzy Mathematical Models in Engineering and Management Science, North-Holland, Amsterdam, 1988
38. 38.
Koczy L.T., On the Description of Relative Position of Fuzzy Patterns, Pattern Recognition Letters, Vol. 8, pp. 21–28, 1988
39. 39.
Krishnapuram R., Keller J.M. and Ma Y., Quantitative Analysis of Properties and Spatial Relations of Fuzzy Image Regions, IEEE Transactions on Fuzzy Systems, Vol. 1, No. 3, pp. 222–233, 1993
40. 40.
Laplante P.A. and Giardina C.R., Fast Dilation and Erosion of Time Varying Grey Valued Images with Uncertainty, in: SPIE Image Algebra and Morphological Image Processing II, Vol. 1568, pp. 295–302, 1991Google Scholar
41. 41.
Nakatsuyama M., Fuzzy Mathematical Morphology for Image Processing, in: “Proceedings of ANZIIS-93” (Perth, Western Australia), pp. 75–79, 1993Google Scholar
42. 42.
Perchant A. and Bloch I., A New Definition for Fuzzy Attributed Graph Homomorphism with Application to Structural Shape Recognition in Brain Imaging, in: “Proceedings of IMTC’99 — 16th IEEE Instrumentation and Measurement Technology Conference” (Venice, Italy), pp. 1801–1806, 1999
43. 43.
Perchant A., Boeres C., Bloch I., Roux M. and Ribeiro C., Model-based Scene Recognition Using Graph Fuzzy Homomorphism Solved by Genetic Algorithm, in: “Proceedings of GbR’99 — 2nd International Workshop on Graph-Based Representations in Pattern Recognition” ( Castle of Haindorf, Austria ), 1999Google Scholar
44. 44.
Popov A.T., Convexity Indicators based on Fuzzy Morphology, Pattern Recognition Letters, Vol. 18, pp. 259–267, 1997
45. 45.
Popov A.T., Morphological Operations on Fuzzy Sets, in: IEE Image Processing and its Applications, Edinburgh, UK, pp. 837–840, 1995
46. 46.
Rosenfeld A., Fuzzy Digital Topology, Information and Control, Vol. 40, pp. 76–87, 1979
47. 47.
Rosenfeld A., The Fuzzy Geometry of Image Subsets, Pattern Recognition Letters, Vol. 2, pp. 311–317, 1984
48. 48.
Rosenfeld A. and Kak A.C., “Digital Picture Processing, Academic Press, New York, 1976Google Scholar
49. 49.
Serra J., “Image Analysis and Mathematical Morphology, Academic Press, London, 1982
50. 50.
Sinha D. and Dougherty E., Fuzzy Mathematical Morphology, Journal of Visual Communication and Image Representation, Vol. 3, No. 3, pp. 286–302, 1992
51. 51.
Sinha D. and Dougherty E.R., Fuzzification of Set Inclusion: Theory and Applications, Fuzzy Sets and Systems, Vol. 55, pp. 15–42, 1993
52. 52.
Sinha D., Sinha P., Dougherty E.R. and Batman S., Design and Analysis of Fuzzy Morphological Algorithms for Image Processing, IEEE Trans. on Fuzzy Systems, Vol. 5, No. 4, pp. 570–584, 1997
53. 53.
Yager R.R., Connectives and Quantifiers in Fuzzy Sets, Fuzzy Sets and Systems, Vol. 40, pp. 39–75, 1991
54. 54.
Zadeh L.A., Fuzzy Sets, Information and Control, Vol. 8, pp. 338–353, 1965
55. 55.
Zadeh L.A., The Concept of a Linguistic Variable and its Application to Approximate Reasoning, Information Sciences, Vol. 8, pp. 199–249, 1975