Skip to main content
SpringerLink
Log in
Book cover

Advances in Low-Level Color Image Processing pp 223–239Cite as

Vector Ordering and Multispectral Morphological Image Processing

  • Chapter
  • First Online:

Part of the book series: Lecture Notes in Computational Vision and Biomechanics ((LNCVB,volume 11))

Abstract

This chapter illustrates the suitability of recent multivariate ordering approaches to morphological analysis of colour and multispectral images working on their vector representation. On the one hand, supervised ordering renders machine learning notions and image processing techniques, through a learning stage to provide a total ordering in the colour/multispectral vector space. On the other hand, anomaly-based ordering, automatically detects spectral diversity over a majority background, allowing an adaptive processing of salient parts of a colour/multispectral image. These two multivariate ordering paradigms allow the definition of morphological operators for multivariate images, from algebraic dilation and erosion to more advanced techniques as morphological simplification, decomposition and segmentation. A number of applications are reviewed and implementation issues are discussed in detail.

This is a preview of subscription content, 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
Hardcover Book
USD   54.99
Price excludes VAT (USA)
  • Durable hardcover 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

Learn about institutional subscriptions

Notes

  1. 1.

    Theoretically, a partial ordering is enough but to make easier the presentation we analyse the case of total ordering.

  2. 2.

    Adaptive in the sense that the mapping depend on the information contained in a multivariate image \({\mathbf {I}}\). The correct notation should be \(h(\cdot ;{\mathbf {I}})\). However, in order to make easier the understanding of the section we use \(h\) for adaptive mapping.

  3. 3.

    In this case the sense of the inequality change, i.e., \({\mathbf {x}}_1 \le _{h_\mathtt{REF}} {\mathbf {x}}_2 \iff ||{\mathbf {x}}_1-{\mathbf {t}}||^2 \ge ||{\mathbf {x}}_2-{\mathbf {t}}||^2\).

  4. 4.

    It is important to note that any adjunction based morphological transformations as openings, closings, levelings and so on, can be implemented in similar way, i.e., by changing the function \(\mathtt {Erode}\) by another grey scale morphological transformation.

References

  1. Angulo J (2007) Morphological colour operators in totally ordered lattices based on distances: application to image filtering, enhancement and analysis. Comput Vis Image Underst 107(1–2):56–73

    Article  Google Scholar 

  2. Aptoula E, Lefèvre S (2007) A comparative study on multivariate mathematical morphology. Pattern Recogn 40(11):2914–2929

    Article  MATH  Google Scholar 

  3. Barnett V (1976) The ordering of multivariate data (with discussion). J Roy Stat Soc: Ser A 139(3):318–354

    Google Scholar 

  4. Bennett KP, Bredensteiner EJ (2000) Duality and geometry in svm classifiers. In: Proceedings 17th international conference on machine learning, Morgan Kaufmann, pp 57–64

    Google Scholar 

  5. Beucher S, Meyer F (1993) The morphological approach to segmentation: the watershed transformation. Mathematical morphology in image processing. Opt Eng 34:433–481

    Google Scholar 

  6. Brown M, Süsstrunk S (2011) Multispectral SIFT for scene category recognition. In: Computer vision and pattern recognition (CVPR11). Colorado Springs, Colorado, pp 177–184

    Google Scholar 

  7. Cristianini N, Shawe-Taylor J (2000) An introduction to support vector machines and other kernel based learning methods. Cambridge University Press, Cambridge

    Google Scholar 

  8. Goutsias J, Heijmans HJAM, Sivakumar K (1995) Morphological operators for image sequences. Comput Vis Image Underst 62(3):326–346

    Article  Google Scholar 

  9. Heijmans HJAM, Ronse C (1990) The algebraic basis of mathematical morphology—Part I: dilations and erosions. Comput Vision Graph Image Process 50:245–295

    Google Scholar 

  10. Jolliffe IT (1986) Principal component analysis. Springer-Verlag, New York

    Google Scholar 

  11. Kambhatla N, Leen TK (1997) Dimension reduction by local principal component analysis. Neural Comp 9(7):1493–1516

    Article  Google Scholar 

  12. Lezoray O, Charrier C, Elmoataz A (2009) Learning complete lattices for manifold mathematical morphology. In: Proceedings of the ISMM’09, pp 1–4

    Google Scholar 

  13. Lorand R (2000) Aesthetic order: a philosophy of order, beauty and art, Routledge studies in twentieth century philosophy. Routledge, London

    Google Scholar 

  14. Meyer F (1998) The levelings. In: Proceedings of the ISMM’98, Kluwer Academic Publishers, Massachusetts, pp 199–206

    Google Scholar 

  15. Muller K, Mika S, Ritsch G, Tsuda K, Scholkopf B (2001) An introduction to kernel-based learning algorithms. IEEE Trans on Neural Networks 12:181–201

    Article  Google Scholar 

  16. Najman L, Talbot H (2010) Mathematical morphology: from theory to applications. ISTE-Wiley, London

    Google Scholar 

  17. Ronse C (2011) Idempotent block splitting on partial partitions, I: isotone operators. Order 28:273–306

    Article  MATH  MathSciNet  Google Scholar 

  18. Salembier P, Serra J (1995) Flat zones filtering, connected operators, and filters by reconstruction. IEEE Trans Image Process 4(8):1153–1160

    Article  Google Scholar 

  19. Serra J (1982) Image analysis and mathematical morphology. Academic Press, Massachusetts

    MATH  Google Scholar 

  20. Serra J (1988) Image analysis and mathematical morphology. In: Theoretical advances, Vol. 2. Academic Press, Massachusetts

    Google Scholar 

  21. Serra J (2012) Tutorial on connective morphology. IEEE J Sel Top Sign Proces 6(7):739–752

    Google Scholar 

  22. Soille P (2003) Morphological image analysis. Springer-Verlag, New York

    Google Scholar 

  23. Talbot H, Evans C, Jones R (1998) Complete ordering and multivariate mathematical morphology. In: Proceedings of the ISMM’98, Kluwer Academic Publishers, Massachusetts, pp 27–34

    Google Scholar 

  24. Velasco-Forero S, Angulo J (2011) Mathematical morphology for vector images using statistical depth. Mathematical morphology and its applications to image and signal processing, vol. 6671 of Lecture notes in computer science. Springer, Berlin, pp 355–366

    Google Scholar 

  25. Velasco-Forero S, Angulo J (2011) Supervised ordering in \({\mathbb{R}}^p\): application to morphological processing of hyperspectral images. IEEE Trans Image Process 20(11):3301–3308

    Google Scholar 

  26. Velasco-Forero S, Angulo J (2012) Random projection depth for multivariate mathematical morphology. J Sel Top Sign Proces 6(7):753–763

    Article  Google Scholar 

  27. Velasco-Forero S, Angulo J (2013) Classification of hyperspectral images by tensor modeling and additive morphological decomposition. Pattern Recogn 46(2):566–577

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. ITWM - Fraunhofer Institute, Kaiserslautern, Germany

    Santiago Velasco-Forero

  2. Mathématiques et Systèmes, CMM-Centre de Morphologie Mathématique, MINES ParisTech; 35, rue Saint-Honoré, 77305 , Fontainebleau CEDEX, France

    Jesus Angulo

Authors
  1. Santiago Velasco-Forero
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jesus Angulo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Santiago Velasco-Forero .

Editor information

Editors and Affiliations

  1. Computer Science dept., Louisiana State University, Shreveport, Louisiana, USA

    M. Emre Celebi

  2. Automatic Control, Silesian University of Technology, Gliwice, Poland

    Bogdan Smolka

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer Science+Business Media Dordrecht

About this chapter

Cite this chapter

Velasco-Forero, S., Angulo, J. (2014). Vector Ordering and Multispectral Morphological Image Processing. In: Celebi, M., Smolka, B. (eds) Advances in Low-Level Color Image Processing. Lecture Notes in Computational Vision and Biomechanics, vol 11. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-7584-8_7

Download citation

  • DOI: https://doi.org/10.1007/978-94-007-7584-8_7

  • Published:

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-007-7583-1

  • Online ISBN: 978-94-007-7584-8

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation