Closeness, Fuzzy partition, Denoising

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.