Kolmogorov superposition theorem and its application to wavelet image decompositions

This paper deals with the decomposition of multivariate functions into sums and compositions of monovariate functions. The global purpose of this work is to find a suitable strategy to express complex multivariate functions using simpler functions that can be analyzed using well know techniques, instead of developing complex N-dimensional tools. More precisely, most of signal processing techniques are applied in 1D or 2D and cannot easily be extended to higher dimensions. We recall that such a decomposition exists in the Kolmogorov’s superposition theorem. According to this theorem, any multivariate function can be decomposed into two types of univariate functions, that are called inner and external functions. Inner functions are associated to each dimension and linearly combined to construct a hash-function that associates every point of a multidimensional space to a value of the real interval [0, 1]. Every inner function is the argument for one external function. The external functions associate real values in [0, 1] to the image by the multivariate function of the corresponding point of the multidimensional space. Sprecher, in Ref. 1, has proved that internal functions can be used to construct space filling curves, i.e. there exists a curve that sweeps the multidimensional space and uniquely matches corresponding values into [0, 1]. Our goal is to obtain both a new decomposition algorithm for multivariate functions (at least bi-dimensional) an