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Nouvelles méthodes de traitement de signaux multidimensionnels par décomposition suivant le théorème de Superposition de Kolmogorov

Abstract : The processing of multidimensional signal remains difficult when using monodimensional-based methods. Therefore, it is either required to extend monodimensional methods to several dimensions, which is not always possible, or to convert the multidimensional signals into 1D signals. In this case, the priority is to preserve most of the properties of the original signal. In this context, the Kolmogorov Superposition Theorem offers a promising theoretical framework for multidimensional signal conversion. In 1957, Kolmogorov demonstrated that any multivariate function can be written as sums and compositions of monovariate functions.We have focused on the image decomposition according to the superposition theorem scheme, to study the possible applications of this decomposition to image processing. We have first studied the monovariate function constructions. Various studies have dealt with this problem, and recently, two algorithms have been proposed. Sprecher has proposed in [Sprecher, 1996; Sprecher, 1997] an algorithm in which the method to exactly build the monovariate functions is described, as well as fundamental notions for the understanding of the theorem. Igelnik and Parikh have proposed in [Igelnik and Parikh, 2003; Igelnik, 2009] an algorithm to approximate the monovariate functions by a Spline network. We have applied both algorithms to image decomposition. We have chosen to use Igelnik’s algorithm which is easier to modify and provides an analytic representation of the functions, to propose two novel applications for classical problems in image processing : for compression : we have studied the quality of a reconstructed image using a spline network built with only a fraction of the pixels of the original image. To improve this reconstruction, we have proposed to apply this decomposition on images of details obtained by wavelet transform. We have then combined this method with JPEG 2000, and we show that the JPEG 2000 compression scheme is improved, even at low bitrates. For progressive transmission : by modifying the spline network construction, the image can be decomposed into one monovariate function. This function can be progressively transmitted, which allows to reconstruct the image by progressively increasing its resolution. Moreover, we show that such a transmission is resilient to information lost.
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Pierre-Emmanuel Leni. Nouvelles méthodes de traitement de signaux multidimensionnels par décomposition suivant le théorème de Superposition de Kolmogorov. Autre. Université de Bourgogne, 2010. Français. ⟨NNT : 2010DIJOS034⟩. ⟨tel-00581756v2⟩

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