Search results for "Superposition theorem"
showing 7 items of 7 documents
Kolmogorov Superposition Theorem and Wavelet Decomposition for Image Compression
2009
International audience; 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 decomp…
The Kolmogorov Spline Network for Image Processing
2011
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 &…
New Representations for Multidimensional Functions Based on Kolmogorov Superposition Theorem. Applications on Image Processing
2012
Mastering the sorting of the data in signal (nD) can lead to multiple applications like new compression, transmission, watermarking, encryption methods and even new processing methods for image. Some authors in the past decades have proposed to use these approaches for image compression, indexing, median filtering, mathematical morphology, encryption. A mathematical rigorous way for doing such a study has been introduced by Andrei Nikolaievitch Kolmogorov (1903-1987) in 1957 and recent results have provided constructive ways and practical algorithms for implementing the Kolmogorov theorem. We propose in this paper to present those algorithms and some preliminary results obtained by our team…
A new Adaptive and Progressive Image Transmission Approach using Function Superpositions
2010
International audience; We present a novel approach to adaptive and progressive image transmission, based on the decomposition of an image into compositions and superpositions of monovariate functions. The monovariate functions are iteratively constructed and transmitted, one after the other, to progressively reconstruct the original image: the progressive transmission is performed directly in the 1D space of the monovariate functions and independently of any statistical properties of the image. Each monovariate function contains only a fraction of the pixels of the image. Each new transmitted monovariate function adds data to the previously transmitted monovariate functions. After each tra…
Kolmogorov superposition theorem for image compression
2012
International audience; The authors present a novel approach for image compression based on an unconventional representation of images. The proposed approach is different from most of the existing techniques in the literature because the compression is not directly performed on the image pixels, but is rather applied to an equivalent monovariate representation of the wavelet-transformed image. More precisely, the authors have considered an adaptation of Kolmogorov superposition theorem proposed by Igelnik and known as the Kolmogorov spline network (KSN), in which the image is approximated by sums and compositions of specific monovariate functions. Using this representation, the authors trad…
Kolmogorov Superposition Theorem and Its Application to Multivariate Function Decompositions and Image Representation
2008
International audience; In this paper, we present the problem of multivariate function decompositions into sums and compositions of monovariate functions. We recall that such a decomposition exists in the Kolmogorov's superposition theorem, and we present two of the most recent constructive algorithms of these monovariate functions. We first present the algorithm proposed by Sprecher, then the algorithm proposed by Igelnik, and we present several results of decomposition for gray level images. Our goal is to adapt and apply the superposition theorem to image processing, i.e. to decompose an image into simpler functions using Kolmogorov superpositions. We synthetise our observations, before …
Progressive transmission of secured images with authentication using decompositions into monovariate functions
2014
International audience; We propose a progressive transmission approach of an image authenticated using an overlapping subimage that can be removed to restore the original image. Our approach is different from most visible water- marking approaches that allow one to later remove the watermark, because the mark is not directly introduced in the two-dimensional image space. Instead, it is rather applied to an equivalent monovariate representation of the image. Precisely, the approach is based on our progressive transmission approach that relies on a modified Kolmogorov spline network, and therefore inherits its advantages: resilience to packet losses during transmis- sion and support of hetero…