Abstract
In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
An online demo for [3] is available: https://bit.ly/anisop_demo.
- 2.
Note that \(\varDelta (W)\) is a metric, not exactly between vertices, but between their equivalence classes induced by Definition 5, as all vertices are eigen-nodes when W is symmetric.
References
Akian, M., Bapat, R., Gaubert, S.: Max-plus algebra. In: Handbook of Linear Algebra (Discrete Mathematics and its Applications), vol. 39, pp. 10–14 (2006)
Blusseau, S., Velasco-Forero, S., Angulo, J., Bloch, I.: Tropical and morphological operators for signal processing on graphs. In: 25th IEEE International Conference on Image Processing, pp. 1198–1202 (2018)
Blusseau, S., Velasco-Forero, S., Angulo, J., Bloch, I.: Adaptive anisotropic morphological filtering based on co-circularity of local orientations. Image Process. Line 12, 111–141 (2022). https://doi.org/10.5201/ipol.2022.397
Bouaynaya, N., Charif-Chefchaouni, M., Schonfeld, D.: Theoretical foundations of spatially-variant mathematical morphology part I: binary images. IEEE Trans. Pattern Anal. Mach. Intell. 30(5), 823–836 (2008)
Carré, B.A.: An algebra for network routing problems. IMA J. Appl. Math. 7(3), 273–294 (1971)
Cuninghame-Green, R.A.: Minimax Algebra, vol. 166. Springer-Verlag, Berlin Heidelberg (1979). https://doi.org/10.1007/978-3-642-48708-8
Debayle, J., Pinoli, J.C.: General adaptive neighborhood image processing: Part I: Introduction and theoretical aspects. J. Math. Imaging Vis. 25(2), 245–266 (2006)
Heijmans, H., Buckley, M., Talbot, H.: Path openings and closings. J. Math. Imaging Vis. 22(2), 107–119 (2005). https://doi.org/10.1007/s10851-005-4885-3
Heijmans, H., Ronse, C.: The algebraic basis of mathematical morphology I. Dilations and erosions. Comput. Vis. Graph. Image Process. 50(3), 245–295 (1990)
Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image Vis. Comput. 25(4), 395–404 (2007)
Maragos, P.: Chapter two - representations for morphological image operators and analogies with linear operators. Adv. Imag. Electron Phys. 177, 45–187 (2013)
Ronse, C.: Why mathematical morphology needs complete lattices. Sig. Process. 21(2), 129–154 (1990)
Salembier, P.: Study on nonlocal morphological operators. In: 16th IEEE International Conference on Image Processing, pp. 2269–2272 (2009)
Velasco-Forero, S., Angulo, J.: On nonlocal mathematical morphology. In: Hendriks, C.L.L., Borgefors, G., Strand, R. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 219–230. Springer, Heidelberg (2013). https://doi.org/10.1007/978-3-642-38294-9_19
Velasco-Forero, S., Angulo, J.: Nonlinear operators on graphs via stacks. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2015. LNCS, vol. 9389, pp. 654–663. Springer, Cham (2015). https://doi.org/10.1007/978-3-319-25040-3_70
Verdú-Monedero, R., Angulo, J., Serra, J.: Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields. IEEE Trans. Image Process. 20(1), 200–212 (2011)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 Springer Nature Switzerland AG
About this paper
Cite this paper
Blusseau, S., Velasco-Forero, S., Angulo, J., Bloch, I. (2022). Morphological Adjunctions Represented by Matrices in Max-Plus Algebra for Signal and Image Processing. In: Baudrier, É., Naegel, B., Krähenbühl, A., Tajine, M. (eds) Discrete Geometry and Mathematical Morphology. DGMM 2022. Lecture Notes in Computer Science, vol 13493. Springer, Cham. https://doi.org/10.1007/978-3-031-19897-7_17
Download citation
DOI: https://doi.org/10.1007/978-3-031-19897-7_17
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-19896-0
Online ISBN: 978-3-031-19897-7
eBook Packages: Computer ScienceComputer Science (R0)