Kolmogorov Superposition Theorem and Wavelet Decomposition for Image Compression

Abstract

Kolmogorov Superposition Theorem stands that any multivariate function can be decomposed into two types of monovariate functions that are called inner and external functions: each inner function is associated to one 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]. These intermediate values are then associated by external functions to the corresponding value of the multidimensional function. Thanks to the decomposition into monovariate functions, our goal is to apply this decomposition to images and obtain image compression.

We propose a new algorithm to decompose images into continuous monovariate functions, and propose a compression approach: thanks to the decomposition scheme, the quantity of information taken into account to define the monovariate functions can be adapted: only a fraction of the pixels of the original image have to be contained in the network used to build the correspondence between monovariate functions. To improve the reconstruction quality, we combine KST and multiresolution approach, where the low frequencies will be represented with the highest accuracy, and the high frequencies representation will benefit from the adaptive aspect of our method to achieve image compression.

Our main contribution is the proposition of a new compression scheme: we combine KST and multiresolution approach. Taking advantage of the KST decomposition scheme, the low frequencies will be represented with the highest accuracy, and the high frequencies representation will be replaced by a decomposition into simplified monovariate functions, preserving the reconstruction quality. We detail our approach and our results on different images and present the reconstruction quality as a function of the quantity of pixels contained in monovariate functions.