Abstract
In this paper, we discuss a special type of fuzzy partitioned space generated by a fuzzy set that is used to enrich the data domain with a notion of closeness. We utilize this notion to sketch the solution to the denoising problem in the discrete, now only 1-D setting, where the Nyquist-Shannon-Kotelnikov sampling theorem in not applicable. The finite-dimensional space with closeness is described by a closeness matrix that transforms discrete one-dimensional signals (considered as functions defined on the space and identified with high-dimensional vectors) into a lower-dimensional vectors. On the basis of this and the corresponding pseudo-inverse transformation, we characterize the signal denoising problem as a type of inverse problem. This opens a new perspective on discrete data processing involving algebraic tools and singular value matrix decomposition. As there are many degrees of freedom in initializing parameters of the chosen model, we restrict ourselves on some special cases. The link between the generating function of the fuzzy partition and a fundamental subspace of the closeness matrix is expressed in terms of Euclidean orthogonality. The theoretical background as well as solutions in particular settings are illustrated by numerical examples.
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Notes
- 1.
X can be naturally connected with a topology that admits its homeomorphic embedding into \(\mathbb {R}\).
- 2.
Such as list of registered values of light frequencies produced by an ideal light source that creates only one frequency at a time.
- 3.
A particular occurrence of this phenomenon can be considered on a cassette player. The signal is stored in the tape but the device produces sound also based on the volume level set by a continuous rotation of volume button (dial knob that does not switch only between a few possible, distinguishable levels). We measure the produced signal but we would like to ignore the unknown volume setting – we consider it as a constant noise in the measured signal represented by a constant vector \(n_1\).
- 4.
In the sense of the Definition 11.
- 5.
Under sufficient conditions, reconstruction of u is equivalent with denoising \(u+v\).
- 6.
The points of the universe are indexed in the same way the components of these vectors are.
References
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Janeček, J., Perfiljeva, I.: Laplacian singular values. In: Joint Proceedings of the 19th World Congress of the International Fuzzy Systems Association (IFSA), the 12th Conference of the European Society for Fuzzy Logic and Technology (EUSFLAT), and the 11th International Summer School on Aggregation Operators (AGOP), pp. 142–146. Atlantis Press (2021)
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Acknowledgements
The support of the project SGS20/PřF-MF/2022 of the University of Ostrava is kindly announced.
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Janeček, J., Perfilieva, I. (2022). Noise Reduction as an Inverse Problem in F-Transform Modelling. In: Ciucci, D., et al. Information Processing and Management of Uncertainty in Knowledge-Based Systems. IPMU 2022. Communications in Computer and Information Science, vol 1602. Springer, Cham. https://doi.org/10.1007/978-3-031-08974-9_32
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DOI: https://doi.org/10.1007/978-3-031-08974-9_32
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