Abstract
The appropriate representation of high-dimensional data is the main focus of machine learning, pattern recognition and computer vision. With the same motivation, the F-transform uses fuzzy partitions in order to establish a compressed representation of data. Two distinguished properties of the F-transform: the best approximation in a local sense and dimensionality reduction contributed to the fact that the F-transform has many successful applications. We show that the technique of F-transform fully agrees with the technique of dimensionality reduction, based on Laplacian eigenmaps. To justify this claim, we characterize the processed by the F-transform data in terms of the adjacency graph that reflects their (data) intrinsic geometry. An application to the problem of image restoration is given.
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
References
M. Belkin, P. Niyogi, Laplacian eigenmaps for dimensionality reduction and data representation. Neural Comput. 15(6), 1373–1396 (2003)
M. Bertalmio, G. Sapiro, V. Caselles, C. Ballester, Image inpainting, in Proceedings of 27th Annual Conference on Computer Graphics and Interactive Techniques (ACM Press/Addison-Wesley Publishing Co., 2000), pp. 417–424
S. Haykin, Neural Networks: A Comprehensive Foundation, Upper Saddle River (Prentice Hall, NJ, 1999)
V. Novák, I. Perfilieva, A. Dvořák, Insight into Fuzzy Modeling (Wiley, Hoboken, New Jersey, 2016)
I. Perfilieva, Fuzzy transform: application to the Reef growth problem, in Fuzzy Logic in Geology, ed. by R.V. Demicco, G.J. Klir (Academic Press, Amsterdam, 2003), pp. 275–300
I. Perfilieva, Fuzzy transform: theory and application. Fuzzy Sets Syst. 157, 993–1023 (2006)
I. Perfilieva, M. Danková, B. Bede, Towards a higher degree F-transform. Fuzzy Sets Syst. 180, 3–19 (2011)
I. Perfilieva, P. Števuliáková, R. Valášek, F-transform-based shooting method for nonlinear boundary value problems. Soft Comput. 21, 3493–3502 (2017)
I. Perfilieva, P. Števuliáková, R. Valášek, F-transform for numerical solution of two-point boundary value problem. Iran. J. Fuzzy Syst. 14(6), 1–13 (2017)
I. Perfilieva, P. Vlašánek, Total variation with nonlocal FT-Laplacian for patch-based inpainting. Soft Computing 23, 1833–1841(2019) https://doi.org/10.1007/s00500-018-3589-8
L. Rudin, S. Osher, E. Fatemi, Non linear total variation based noise removal algorithms. Physica 60, 259–268 (1992)
B. Scholkopf, A. Smola, K.-R. Mulller, Nonlinear component analysis as a Kernel eigenvalue problem. Neural Comput. 10, 1299–1319 (1998)
L.A. Zadeh, Toward a theory of fuzzy information granulation and its centrality in human reasoning and fuzzy logic. Fuzzy Sets Syst. 90, 111–127 (1997)
Acknowledgements
This work was supported by the project LQ1602 IT4Innovations excellence in science. The additional support was also provided by the Czech Science Foundation (GAČR) through the project of No.18-06915S. The SW and the implementation of the technique of F-transforms to image restoration was performed by Dr. Pavel Vlašánek.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Perfilieva, I. (2021). Dimensionality Reduction: From Fuzzy Partitions to F-Transforms. In: Shahbazova, S.N., Kacprzyk, J., Balas, V.E., Kreinovich, V. (eds) Recent Developments and the New Direction in Soft-Computing Foundations and Applications. Studies in Fuzziness and Soft Computing, vol 393. Springer, Cham. https://doi.org/10.1007/978-3-030-47124-8_32
Download citation
DOI: https://doi.org/10.1007/978-3-030-47124-8_32
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-47123-1
Online ISBN: 978-3-030-47124-8
eBook Packages: Intelligent Technologies and RoboticsIntelligent Technologies and Robotics (R0)