Abstract:
In recent work, various fractal image coding methods are reported, which adopt the self-similarity of images to compress the size of images. However, till now, no solutions for the security of fractal encoded images have been provided. In this paper, a secure fractal image coding scheme is proposed and evaluated, which encrypts some of the fractal parameters during fractal encoding, and thus, produces the encrypted and encoded image. The encrypted image can only be recovered by the correct key. To keep secure and efficient, only the suitable parameters are selected and encrypted through in-vestigating the properties of various fractal parameters, including parameter space, parameter distribu-tion and parameter sensitivity. The encryption process does not change the file format, keeps secure in perception, and costs little time or computational resources. These properties make it suitable for secure image encoding or transmission.

Abstract:
Intersection of a random fractal or self-affine set with a linear manifold or another fractal set is studied, assuming that one of the sets is in a translational motion with respect to the other. It is shown that the mass of such an intersection is a self-affine function of the relative position of the two sets. The corresponding Hurst exponent h is a function of the scaling exponents of the intersecting sets. A generic expression for h is provided, and its proof is offered for two cases --- intersection of a self-affine curve with a line, and of two fractal sets. The analytical results are tested using Monte-Carlo simulations.

Abstract:
The focus here is on connected fractal sets with topological dimension 1 and a lot of topological activity, and their connections with analysis.

Abstract:
In this paper are presented some results of implementation of digitalwatermarking methods into image coding based on fractal principles. Thepaper focuses on two possible approaches of embedding digitalwatermarks into fractal code of images - embedding digital watermarksinto parameters for position of similar blocks and coefficients ofblock similarity. Both algorithms were analyzed and verified on grayscale static images.

Abstract:
We consider the optimal covering of fractal sets in a two-dimensional space using ellipses which become increasingly anisotropic as their size is reduced. If the semi-minor axis is \epsilon and the semi-major axis is \delta, we set \delta=\epsilon^\alpha, where 0<\alpha<1 is an exponent characterising the anisotropy of the covers. For point set fractals, in most cases we find that the number of points N which can be covered by an ellipse centred on any given point has expectation value < N > ~ \epsilon^\beta, where \beta is a generalised dimension. We investigate the function \beta(\alpha) numerically for various sets, showing that it may be different for sets which have the same fractal dimension.

Abstract:
We discuss various questions which arise when one considers the central projection of three dimensional fractal sets (galaxy catalogs) onto the celestial globe. The issues are related to how fractal such projections look. First we show that the lacunarity in the projection can be arbitrarily small. Further characteristics of the projected set---in particular scaling---depend sensitively on how the apparent sizes of galaxies are taken into account. Finally, we discuss the influence of opacity of galaxies. Combining these ideas, seemingly contradictory statements about lacunarity and apparent projections can be reconciled.

Abstract:
This paper is a survey, with few proofs, of ideas and notions related to self-similarity of groups, semi-groups and their actions. It attempts to relate these concepts to more familiar ones, such as fractals, self-similar sets, and renormalizable dynamical systems. In particular, it presents a plausible definition of what a "fractal group" should be, and gives many examples of such groups. A particularly interesting class of examples, derived from monodromy groups of iterated branch coverings, or equivalently from Galois groups of iterated polynomials, is presented. This class contains interesting groups from an algebraic point of view (just-non-solvable groups, groups of intermediate growth, branch groups,...), and at the same time the geometry of the group is apparent in that a limit of the group identifies naturally with the Julia set of the covering map. In its survey, the paper discusses finite-state transducers, growth of groups and languages, limit spaces of groups, hyperbolic spaces and groups, dynamical systems, Hecke-type operators, C^*-algebras, random matrices, ergodic theorems and entropy of non-commuting transformations. Self-similar groups appear then as a natural weaving thread through these seemingly different topics.

Abstract:
In this paper is presented a new approach in fractal image codingbased on trigonometric approximation. The least square approximationmethod is used for approximation of blocks in standard fractal imagecompression algorithm. In the paper is shown that it is possible to usealso trigonometric approximation for describing of blocks in fractalimage coding. This approximation was implemented and analyzed frompoint of view of quality of reconstructed images. The experimentalresults of this method were tested on static grayscale images.

Abstract:
fractal codification of images is based on self-similar and self-affine sets. the codification process consists of construction of an operator which will represent the image to be encoded. if a complicated picture can be represented by an operator then it will be transmitted or stored very efficiently. clearly, this has many applications on data compression. the great disadvantage of the automatic form of fractal compression is its encoding time. most of the time spent in construction of such operator is due on finding the best match between parts of the image to be encoded. however, since the conception of automatic fractal image compression, researches on improvement of the compression time are widespread. this work aims to provide a new idea for decrease the encoding time: a classification of image parts based on their local fractal dimension. the idea is implemented on two steps. first, a preprocessing analysis of the image identify the complexity of each image block computing its dimension. then, only parts within the same range of complexity are used for testing the better self-affine pairs, reducing the compression time. the performance of this proposition, is compared with others fractal image compression methods. the points considered are image fidelity, encoding time and amount of compression on the image file.

Abstract:
Fractal codification of images is based on self-similar and self-affine sets. The codification process consists of construction of an operator which will represent the image to be encoded. If a complicated picture can be represented by an operator then it will be transmitted or stored very efficiently. Clearly, this has many applications on data compression. The great disadvantage of the automatic form of fractal compression is its encoding time. Most of the time spent in construction of such operator is due on finding the best match between parts of the image to be encoded. However, since the conception of automatic fractal image compression, researches on improvement of the compression time are widespread. This work aims to provide a new idea for decrease the encoding time: a classification of image parts based on their local fractal dimension. The idea is implemented on two steps. First, a preprocessing analysis of the image identify the complexity of each image block computing its dimension. Then, only parts within the same range of complexity are used for testing the better self-affine pairs, reducing the compression time. The performance of this proposition, is compared with others fractal image compression methods. The points considered are image fidelity, encoding time and amount of compression on the image file.