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26 January 2010 Kolmogorov superposition theorem and wavelets for image compression
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Abstract
We propose a new compression approached based on the decomposition of images into continuous monovariate functions, which provide adaptability over the quantity of information taken into account to define the monovariate functions: 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. The Kolmogorov Superposition Theorem (KST) stands that any multivariate functions can be decomposed into sums and compositions of monovariate functions. The implementation of the decomposition proposed by Igelnik, and modified for image processing, is combined with a wavelet decomposition, 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, in which we combine KSTand multiresolution approach. Taking advantage of the KST decomposition scheme, we use a decomposition into simplified monovariate functions to compress the high frequencies. We detail our approach and the different methods used to simplify the monovariate functions. We present the reconstruction quality as a function of the quantity of pixels contained in monovariate functions, as well as the image reconstructions obtained with each simplification approach.
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Pierre-Emmanuel Leni, Yohan D. Fougerolle, and Frédéric Truchetet "Kolmogorov superposition theorem and wavelets for image compression", Proc. SPIE 7535, Wavelet Applications in Industrial Processing VII, 753502 (26 January 2010); https://doi.org/10.1117/12.838874
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