A novel approach for image sweeping functions using approximating scheme
Résumé
Using Kolmogorov's superposition theorem, complex N-dimensional signal can be expressed by simpler 1D functions. Precisely, Kolmogorov has demonstrated that any multivariate function can be decomposed into sums and compositions of monovariate functions, that are called inner and external functions. We present one of the most recent method of monovariate functions construction. The algorithm proposed by Igelnik approximates the monovariate functions. Different layers are constructed and superposed. A layer is constituted by a couple of internal and external functions, that realizes an approximation of the multivariate function with a given precision, which corresponds to a representation of the multidimensional function through a tilage. Each layer contains new pieces of information, which improves the whole network accuracy. A weight is associated to each layer. The network is trained, i.e., the monovariate functions and the weights associated to each layer are optimized to ensure the convergence of the network to the decomposed multivariate function. Sprecher has demonstrated that using internal monovariate functions, scanning functions can be constructed; i.e., a space filling curve connects every couple of the multidimensional space and uniquely matches corresponding values into [0,1]. Igelnik's construction produces a space filling curve per network layer: a unique path through the tiles of a layer. The contributions of this paper are the presentation and the analysis of an approximating scheme for images. Starting from an image traditionally represented as rows and columns of pixels, we extract two kinds of monovariate functions, one representing the scanning path through the image, and the second one being the core for the image reconstruction using Kolmogorov Superposition Theorem. We have applied the algorithm on images and presents the decomposition results as composition of monovariate functions. We also present compression results, taking advantage of the continuity of monovariate functions.