Skip to main content
SpringerLink
Log in

Tropical Geometry, Mathematical Morphology and Weighted Lattices

  • Conference paper
  • First Online:
Book cover Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2019)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 11564))

Abstract

Mathematical Morphology and Tropical Geometry share the same max/min-plus scalar arithmetic and matrix algebra. In this paper we summarize their common ideas and algebraic structure, generalize and extend both of them using weighted lattices and a max-\(\star \) algebra with an arbitrary binary operation \(\star \) that distributes over max, and outline applications to geometry, image analysis, and optimization. Further, we outline the optimal solution of max-\(\star \) equations using weighted lattice adjunctions, and apply it to optimal regression for fitting max-\(\star \) tropical curves on arbitrary data.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Akian, M., Gaubert, S., Guterman, A.: Tropical polyhedra are equivalent to mean payoff games. Int. J. Algebra Comput. 22(1), 125001 (2012)

    Article  MathSciNet  Google Scholar 

  2. Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.P.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, Chichester (1992). web edn. 2001

    MATH  Google Scholar 

  3. Bellman, R., Karush, W.: On a new functional transform in analysis: the maximum transform. Bull. Am. Math. Soc. 67, 501–503 (1961)

    Article  Google Scholar 

  4. Birkhoff, G.: Lattice Theory. American Mathematical Society, Providence (1967)

    MATH  Google Scholar 

  5. Blyth, T.S., Janowitz, M.F.: Residuation Theory. Pergamon Press, Oxford (1972)

    MATH  Google Scholar 

  6. Butkovič, P.: Max-linear Systems: Theory and Algorithms. Springer, London (2010). https://doi.org/10.1007/978-1-84996-299-5

    Book  MATH  Google Scholar 

  7. Charisopoulos, V., Maragos, P.: Morphological perceptrons: geometry and training algorithms. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds.) ISMM 2017. LNCS, vol. 10225, pp. 3–15. Springer, Cham (2017). https://doi.org/10.1007/978-3-319-57240-6_1

    Chapter  MATH  Google Scholar 

  8. Cohen, G., Dubois, D., Quadrat, J., Viot, M.: A linear system theoretic view of discrete event processes and its use for performance evaluation in manufacturing. IEEE Trans. Autom. Control 30, 210–220 (1985)

    Article  MathSciNet  Google Scholar 

  9. Cohen, G., Gaubert, S., Quadrat, J.: Duality and separation theorems in idempotent semimodules. Linear Alegbra Appl. 379, 395–422 (2004)

    Article  MathSciNet  Google Scholar 

  10. Cuninghame-Green, R.: Minimax Algebra. Springer, Heidelberg (1979). https://doi.org/10.1007/978-3-642-48708-8

    Book  MATH  Google Scholar 

  11. Gaubert, S., Katz, R.D.: Minimal half-spaces and external representation of tropical polyhedra. J. Algebr. Comb. 33, 325–348 (2011)

    Article  MathSciNet  Google Scholar 

  12. Gondran, M., Minoux, M.: Graphs, Dioids and Semirings: New Models and Algorithms. Springer, Boston (2008). https://doi.org/10.1007/978-0-387-75450-5

    Book  MATH  Google Scholar 

  13. Heijmans, H.: Morphological Image Operators. Academic Press, Boston (1994)

    MATH  Google Scholar 

  14. Heijmans, H., Ronse, C.: The algebraic basis of mathematical morphology I. Dilations and erosions. Comput. Vis. Graph. Image Process. 50, 245–295 (1990)

    Article  Google Scholar 

  15. Perin, J.-E.: Tropical semirings. In: Gunawardena, J. (ed.) Idempotency, pp. 50–69. Cambridge University Press, Cambridge (1998). https://doi.org/10.1017/CBO9780511662508.004

    Chapter  Google Scholar 

  16. Litvinov, G.L., Maslov, V.P., Shpiz, G.B.: Idempotent functional analysis: an algebraic approach. Math. Notes 69(5), 696–729 (2001)

    Article  MathSciNet  Google Scholar 

  17. Lucet, Y.: What shape is your conjugate? A survey of computational convex analysis and its applications. SIAM Rev. 52(3), 505–542 (2010)

    Article  MathSciNet  Google Scholar 

  18. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. American Mathematical Society, Providence (2015)

    Book  Google Scholar 

  19. Maragos, P.: Lattice image processing: a unification of morphological and fuzzy algebraic systems. J. Math. Imaging Vis. 22, 333–353 (2005)

    Article  MathSciNet  Google Scholar 

  20. Maragos, P.: Representations for morphological image operators and analogies with linear operators. In: Hawkes, P. (ed.) Advances in Imaging and Electron Physics, vol. 177, pp. 45–187. Academic Press/Elsevier Inc. (2013)

    Google Scholar 

  21. Maragos, P.: Dynamical systems on weighted lattices: general theory. Math. Control Signals Syst. 29(21) (2017). https://doi.org/10.1007/s00498-017-0207-8

  22. Ritter, G.X., Wilson, J.N.: Handbook of Computer Vision Algorithms in Image Algebra, 2nd edn. CRC Press, Boca Raton (2001)

    MATH  Google Scholar 

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

    MATH  Google Scholar 

  24. Serra, J. (ed.): Image Analysis and Mathematical Morphology. Theoretical Advances, vol. 2. Academic Press, London (1988)

    Google Scholar 

  25. Sternberg, S.R.: Grayscale morphology. Comput. Vis. Graph. Image Process. 35, 333–355 (1986)

    Article  Google Scholar 

Download references

Acknowledgements

I wish to thank V. Charisopoulos and E. Theodosis for insightful discussions and help with the figures. This work was co-financed by the European Regional Development Fund of the EU and Greek national funds through the Operational Program Competitiveness, Entrepreneurship and Innovation, under the call ‘Research – Create – Innovate’ (T1EDK-01248, “i-Walk”).

Author information

Authors and Affiliations

  1. National Technical University of Athens, Athens, Greece

    Petros Maragos

Authors
  1. Petros Maragos
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Petros Maragos .

Editor information

Editors and Affiliations

  1. Universität des Saarlandes, Saarbrücken, Germany

    Bernhard Burgeth

  2. Forschungszentrum Jülich GmbH, Jülich, Germany

    Andreas Kleefeld

  3. Université de Strasbourg, Illkirch Cedex, France

    Benoît Naegel

  4. Université de Reims Champagne-Ardenne, Reims Cedex 2, France

    Nicolas Passat

  5. Université Paris-Est, Noisy le Grand Cedex, France

    Benjamin Perret

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Maragos, P. (2019). Tropical Geometry, Mathematical Morphology and Weighted Lattices. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2019. Lecture Notes in Computer Science(), vol 11564. Springer, Cham. https://doi.org/10.1007/978-3-030-20867-7_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-20867-7_1

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-20866-0

  • Online ISBN: 978-3-030-20867-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation