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The Kolmogorov Spline Network for Image Processing
Abstract
In 1900, Hilbert stated that high order equations cannot be solved by sums and compositions of bivariate functions. In 1957, Kolmogorov proved this hypothesis wrong and presented his superposition theorem (KST) that allowed for writing every multivariate functions as sums and compositions of univariate functions. Sprecher has proposed in (Sprecher, 1996) and (Sprecher, 1997) an algorithm for exact univariate function reconstruction. Sprecher explicitly describes construction methods for univariate functions and introduces fundamental notions for the theorem comprehension (such as tilage). Köppen has presented applications of this algorithm to image processing in (Köppen, 2002) and (Köppen & Yoshida, 2005). The lack of flexibility of this scheme has been pointed out and another solution which approximates the univariate functions has been considered. More specifically, it has led us to consider Igelnik and Parikh’s approach, known as the KSN which offers several perspectives of modification of the univariate functions as well as their construction. This chapter will focus on the presentation of Igelnik and Parikh’s Kolmogorov Spline Network (KSN) for image processing and detail two applications: image compression and progressive transmission. Precisely, the developments presented in this chapter include: (1)Compression: the authors study the reconstruction quality using univariate functions containing only a fraction of the original image pixels. To improve the reconstruction quality, they apply this decomposition on images of details obtained by wavelet decomposition. The authors combine this approach into the JPEG 2000 encoder, and show that the obtained results improve JPEG 2000 compression scheme, even at low bitrates. (2)Progressive Transmission: the authors propose to modify the generation of the KSN. The image is decomposed into univariate functions that can be transmitted one after the other to add new data to the previously transmitted functions, which allows to progressively and exactly reconstruct the original image. They evaluate the transmission robustness and provide the results of the simulation of a transmission over packet-loss channels.
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