Laplace operator, Proximity, Fuzzy transform

Laplace operator is a diverse concept throughout natural sciences. It appears in many research areas and every such area defines it accordingly based on underlying domain and plans on follow-up applications. This operator attracts a lot of attention e.g. in signal and image processing applications. However, signals, in general, can be defined not only on Euclidean domains such as regular grids (in case of images). There are cases when underlying space is considered to be e.g. a non-regular graph or even a manifold, but the Laplace operator is still closely bound to the space structure. Therefore, we investigated this operator from point of view of spaces, where distance may not be explicitly defined and thus is being replaced by more general, so-called, proximity. Our goal was to find such a representation, that would be simple for computations but at the same time applicable to more general domains, possibly to spaces without a notion of a classic distance. In this article, we will mention some of the various ways in which this operator can be introduced in relation to the corresponding space. Also, we will introduce the formula for the Laplace operator in the space whose structure is determined by a fuzzy partition. And we will investigate the properties of this kind of representation in parallelisms to standard well-known versions.