Approximation by Nonlinear Multivariate Sampling Kantorovich Type Operators and Applications to Image Processing

Abstract

In this article, we study a nonlinear version of the sampling Kantorovich type operators in a multivariate setting and we show applications to image processing. By means of the above operators, we are able to reconstruct continuous and uniformly continuous signals/images (functions). Moreover, we study the modular convergence of these operators in the setting of Orlicz spaces L ( n ) that allows us to deal the case of not necessarily continuous signals/images. The convergence theorems in L p ( n )-spaces, L log L( n )-spaces and exponential spaces follow as particular cases. Several graphical representations, for the various examples and image processing applications are included.

Neural network operators: Constructive interpolation of multivariate functions

“…Moreover, also not necessarily continuous signals can be reconstructed. In this case, sampling operators of the Kantorovich type seems to be the most appropriate to perform this task, see, e.g., [9,33,34,22,23,35,36,37]. As showed in Remark 2.1, the density functions φ s (x) dened in this paper satisfy all the typical properties of the approximate identities and then, can be used as kernels in the above sampling operators in the univariate case.…”

Multivariate sampling Kantorovich operators: from the theory to the Digital Image Processing algorithm

“…Moreover, we denote by I ϕ : M (IR n ) → [0, +∞] the corresponding modular functional associated to L ϕ (IR n ) (see e.g. [4,9,12]), where M (IR n ) denotes the space of all Lebesgue-measurable functions. We can obtain what follows.…”

“…The sampling Kantorovich operators S w (see also [9,14,15]) represent an L 1 -version of the generalized sampling operators and they revealed to be very suitable to reconstruct not necessarily continuous signals; thus applications to image reconstruction can be deduced.…”

Digital image processing algorithms for diagnosis in arterial diseases

“…Increasing the sampling rate and choosing an appropriate kernel it's possible to enhance the images under consideration [6,7]. We may consider a two-dimensional Jackson kernel as the result of the product of two one-dimensional Jackson kernels, i.e.,…”