We present a survey of the recent applications of continuous domains for providing simple computational models for classical spaces in mathematics including the real line, countably based locally compact spaces, complete separable metric spaces, separable Banach spaces and spaces of probability distributions. It is shown how these models have a logical and effective presentation and how they are used to give a computational framework in several areas in mathematics and physics. These include fractal geometry, where new results on existence and uniqueness of attractors and invariant distributions have been obtained, measure and integration theory, where a generalization of the Riemann theory of integration has been developed, and real arithmetic, where a feasible setting for exact computer arithmetic has been formulated. We give a number of algorithms for computation in the theory of iterated function systems with applications in statistical physics and in period doubling route to chao...

: In fractal image compression the encoding step is computationally expensive. A large number of sequential searches through a list of domains (portions of the image) are carried out while trying to find a best match for another image portion. Our theory developed here shows that this basic procedure of fractal image compression is equivalent to multi-dimensional nearest neighbor search. This result is useful for accelerating the encoding procedure in fractal image compression. The traditional sequential search takes linear time whereas the nearest neighbor search can be organized to require only logarithmic time. The fast search has been integrated into an existing state-of-the-art classification method thereby accelerating the searches carried out in the individual domain classes. In this case we record acceleration factors from 1.3 up to 11.5 depending on image and domain pool size with negligible or minor degradation in both image quality and compression ratio. Furthermore, as comp...

We introduce the notion of weakly hyperbolic iterated function system (IFS) on a compact metric space, which generalises that of hyperbolic IFS. Based on a domain-theoretic model, which uses the Plotkin power domain and the probabilistic power domain respectively, we prove the existence and uniqueness of the attractor of a weakly hyperbolic IFS and the invariant measure of a weakly hyperbolic IFS with probabilities, extending the classic results of Hutchinson for hyperbolic IFSs in this more general setting. We also present finite algorithms to obtain discrete and digitised approximations to the attractor and the invariant measure, extending the corresponding algorithms for hyperbolic IFSs. We then prove the existence and uniqueness of the invariant distribution of a weakly hyperbolic recurrent IFS and obtain an algorithm to generate the invariant distribution on the digitised screen. The generalised Riemann integral is used to provide a formula for the expected value of almost everywh...

: In fractal image compression the encoding step is computationally expensive. A large number of sequential searches through a list of domains (portions of the image) are carried out while trying to find a best match for another image portion. We show that this step can be replaced by multi-dimensional nearest neighbor search which runs in logarithmic time instead of linear time required for the common sequential search. Keywords: Image compression, multi-dimensional search, nearest neighbors, fractals. 1 Introduction With the ever increasing demand for images, sound, video sequences, computer animations and volume visualization, data compression remains a critical issue regarding the cost of data storage and transmission times. While JPEG currently provides the industry standard for still image compression there is ongoing research in alternative methods. Fractal image compression is one of them. JPEG can be termed symmetric in the sense that the encoding and decoding phases require ...

Content creation, editing, and searching are extremely time consuming tasks that often require substantial training and experience, especially when high-quality audio and video are involved. "New media" represents a new paradigm for multimedia information representation and processing, in which the emphasis is placed on the actual content. It thus brings the tasks of content creation and searching much closer to actual users and enables them to be active producers of audiovisual information rather than passive recipients. We discuss the state-of-the-art and present next-generation techniques for content representation, searching, creation, and editing. We discuss our experiences in developing a Web-based distributed compressed video editing and searching system (WebClip), a media representation language (Flavor) and an object-based video authoring system (Zest) based on it, and large image/video search engines for the World-Wide Web (WebSEEk and VideoQ). We also present a case study of new media applications based on specific planned multimedia education experiments with the above systems in several K-12 schools in Manhattan.

Fractal image compression provides an innovative approach to lossy image encoding, with a potential for very high compression ratios. Because of prohibitive compression times, however, the procedure has proved feasible in only a limited range of commercial applications. In this paper we demonstrate that, due to the independent nature of fractal transform encoding of individual image segments, fractal image compression performs well in a coarsegrain distributed processing system. A sequential fractal compression algorithm is optimized and parallelized to execute across distributed workstations and an SP2 parallel processor using the Parallel Virtual Machine (PVM) software. The system utilizes both static and dynamic load allocation to obtain substantial compression time speedup over the original, sequential encoding implementation. Considerations such as workload granularity and compression time versus number of processors and RMS tolerance values are also presented. Keywords: Image com...

In fractal image compression the encoding step is computationally expensive. A large number of sequential searches through a list of domains (portions of the image) are carried out while trying to find best matches for other image portions called ranges. Our theory developed here shows that this basic procedure of fractal image compression is equivalent to multi-dimensional nearest neighbor search in a space of feature vectors. This result is useful for accelerating the encoding procedure in fractal image compression. The traditional sequential search takes linear time whereas the nearest neighbor search can be organized to require only logarithmic time. The fast search has been integrated into an existing state-of-the-art classification method thereby accelerating the searches carried out in the individual domain classes. In this case we record acceleration factors up to about 50 depending on image and domain pool size with negligible or minor degradation in both image quality and com...

This research is the first application of fractal compression to volumetric data. The various components of the fractal image compression method extend simply and directly to the volumetric case. However, the additional dimension increases the already high time complexity of the fractal technique, requiring the application of sophisticated volumetric block classification and search schemes to operate at a feasible rate. Numerous experiments over the many parameters of fractal volume compression show it to perform competitively against other volume compression methods, surpassing vector quantization and approaching the discrete cosine transform. Keywords: Data compression, fractal, iterated function system, volume visualization. 1 Introduction The problem of managing extremely large data sets often arises in applications employing volumetric data. This has prompted research in new techniques for economizing both storage space and processing time. Data compression techniques reduce th...

. Since the conception of fractal image compression by Michael F. Barnsley around 1987 the research literature on this topic has experienced a rapid growth, which is hard to keep track of. This paper contains a brief description of the major advances in the field. The corresponding relevant papers are presented in the form of the original (slightly edited) abstracts. Also included is a comprehensive bibliography, the largest published on this topic to this date. 1 Introduction While JPEG is becoming the industry standard for image compression technology there is ongoing research in alternative methods. Currently there are at least two exciting new developments: wavelet based methods and fractal image compression. This article is intended to provide the reader with an overview and a resource of the research on the latter. We attempt to put the work into a historical perspective and aim at providing the most comprehensive and up-to-date list of references in the field, which has truly b...

. This paper copes with the issue of adaptive partitioning in the context of Iterated Functions Systems (IFS) image coding. The main concern is to provide some insights on the optimum degree of partition adaptivity. The key point is to point out the proper balance of information between the partition representation and the transformation parameters of the system. Several systems involving different degrees of adaptivity will be compared. To that end a polygonal-based segmentation will be presented. This will allow to compare performances of partitioning using respectively square, rectangular and polygonal segments. 1. Introduction The concept of fractal has been introduced by Mandelbrot in the 1960's as an alternative to the classical Euclidean geometry mainly for describing shapes generated by Nature. Since then, this theory attracts the interest of many researchers in fields ranging from biology to physics through computer imaging and image processing. Iterated contractive function...