In computer science permutations are used, e.g., in the tasks of pattern searching, duplicate documents detection and data compression [1], [2]. For this reason the reduction of redundancy leading to succinct representation of permutations is of great importance. In this paper, we introduce a novel method for succinct representation of permutations where the average number of bits per element required to encode permutations is $\log_{2}n-1.269$, which is close to the theoretic limit. Furthermore, it is possible to formulate precise expressions for the average value, lower, and upper bounds to the number of bits required by the method. Let n be an integer power of 2. Then the proposed method can be described as follows: (i) the method follows the ‘‘divide-and-conquer’’ strategy and at each stage a considered permutation is divided into two equal halves (bins), (ii) binary encoding is used to describe elements-to-bins assignment (’ 0’-first, ‘l’-second bin), (iii) depending on a permutation some bits can be omitted, which leads to succinct representation. For instance, let $\pi_{2}=(0,2,1,3,7,6,4,5)$. We start with the identity permutation $\pi_{1}=(0,1,2,3,4,5,6,7)$. At the first stage $\pi_{1}$ is split between two bins in relation to $\pi_{2}$ as $\pi_{1}=(0,1,2,3|4,5,6,7)$ which is encoded with bits ‘0000’. At the second stage we repeat the same operations leading to $\pi_{1}=(0,2|1,3|6,7|4,5)$, and formulate the coding bits ‘01011’ Finally, at the last stage, we get $\pi_{1}=\pi_{2}=(0|2|1|3|7|6|4|5)$ encoded as ‘0010’. The concatenated bits give the unique code $C=0000010110010$ for $\pi_{2}$. The lower and upper bounds for the length of codes $\displaystyle \frac{1}{n}|C|$ are $G^{\min}(n)=\displaystyle \frac{1}{2}\log_{2}n$ and $G^{\max}\left(n\right)=\displaystyle \log_{2}n-\left(1-\frac{1}{n}\right)$. The average number of bits per element required to encode permutations can be calculated as:

The Karhunen-Loève transform (KLT), as a component of block scalar quanti-zation, is optimal among linear orthonormal transforms and allows to obtain the smallest value of mean squared error (MSE) for a given rate of data representation. In this paper we propose a novel and robust optimization scheme designed for arti-ficial neural networks that implies possibly minimal constraints and allows to obtain the KLT up to the permutation of basis vectors. The proposed scheme involves two optimization criteria: (i) minimization of the MSE of signal reconstruction and (ii) minimization of the entropy related criterion, see Fig. 1(a).

In this paper the authors propose a novel structure of convolutional neural network for lossy compression of images intended to be used as an extension of JPEG image compression standard. The convolutional network is trained on the set of images randomly selected from the database of high-quality images representing human faces and its effectiveness is verified experimentally using both human faces images as well as standard test images. The performance of the proposed network expressed in terms of its compression capabilities and image reconstruction quality is compared to other approaches utilizing the standard Discrete Cosine Transform, Lapped Orthogonal Transform, Modulated Lapped Transform and Karhunen-Loeve Transform, also incorporated into JPEG image compression standard. The obtained experimental results indicate that the proposed approach not only performs significantly better than the remaining approaches in terms of objective image quality measures, but also enables significant reduction of the blocking defects, which was verified by visual examination, when compared to the remaining tested transforms.

In this paper we present a new joint encryption and compression coding scheme of natural images which is intended for the use in conjunction with a popular JPEG image compression standard. The encryption is performed prior to compression step and is carried out using fast, parametrized with a private key, linear transformations which do not alter statistical characteristics of the input image data, what enables JPEG algorithm to maintain its full compression capabilities. The work also includes a mathematical model of the proposed scheme which allows for theoretical analysis of the impact of the image encryption step on the compression process. The presented experimental results indicate that the reconstructed images' qualities at a given compression ratios are comparable to those obtained for the JPEG standard without the encryption step.