CAPTURE: A Communications Architecture for Progressive Transmission via Underwater Relays With Eavesdropping

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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The Kolmogorov Spline Network for Authentication Data Embedding in Images

Efficiency and Scalability Methods for Computational Intellect ◽

10.4018/978-1-4666-3942-3.ch005 ◽

2013 ◽

pp. 96-114

Author(s):

Pierre-Emmanuel Leni ◽

Yohan D. Fougerolle ◽

Frédéric Truchetet

Keyword(s):

Continuous Functions ◽

Data Embedding ◽

Progressive Transmission ◽

Secret Key ◽

Transmission Scheme ◽

Black And White ◽

Novel Approach ◽

Univariate Functions ◽

Univariate Representation ◽

Superposition Theorem

In 1900, Hilbert declared that high order polynomial equations could not be solved by sums and compositions of continuous functions of less than three variables. This statement was proven wrong by the superposition theorem, demonstrated by Arnol’d and Kolmogorov in 1957, which allows for writing all multivariate functions as sums and compositions of univariate functions. Amongst recent computable forms of the theorem, Igelnik and Parikh’s approach, known as the Kolmogorov Spline Network (KSN), offers several alternatives for the univariate functions as well as their construction. A novel approach is presented for the embedding of authentication data (black and white logo, translucent or opaque image) in images. This approach offers similar functionalities than watermarking approaches, but relies on a totally different theory: the mark is not embedded in the 2D image space, but it is rather applied to an equivalent univariate representation of the transformed image. Using the progressive transmission scheme previously proposed (Leni, 2011), the pixels are re-arranged without any neighborhood consideration. Taking advantage of this naturally encrypted representation, it is proposed to embed the watermark in these univariate functions. The watermarked image can be accessed at any intermediate resolution, and fully recovered (by removing the embedded mark) without loss using a secret key. Moreover, the key can be different for every resolution, and both the watermark and the image can be globally restored in case of data losses during the transmission. These contributions lie in proposing a robust embedding of authentication data (represented by a watermark) into an image using the 1D space of univariate functions based on the Kolmogorov superposition theorem. Lastly, using a key, the watermark can be removed to restore the original image.

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