Discrete Laplace operator, Fuzzy transform, Fuzzy partition, Basic functions, Inner product

Differential operators definitely play an important role in image processing tasks. One of the most frequently used operator for this purpose is the Laplace operator, usually defined as the divergence of a gradient of a function. Since images are considered as discrete objects, many applications require discrete version of this operator. But at the same time, these discrete Laplace operators should preserve certain properties known from the continuous case. Based on this fact, we present discrete variant of F-transform-based Laplace operator, satisfying such conditions. Moreover, Laplace operator can be also defined axiaomatically, throughout its certain properties. Therefore, we propose a construction of such operator from the other side, just with notion of the inner product of the current space and its desired properties.