Subject and Goals

• MOTIVATION
• THEORETICAL RESEARCH SUPPORTING COPING WITH COMPLEXITY
• THEORETICAL RESEARCH SUPPORTING IMAGE PROCESSING
• SOLUTION OF PROBLEMS ARISING FROM PRACTICAL APPLICATIONS
• SPECIFICATION OF THE RESEARCH GOALS OF THE CENTRE

MOTIVATION

Almost unexpectedly, swift development of computer technology in the second half of the last century made it possible that models and tools that seemed just a field of study of unrealistic theoreticians only two decades ago, started to be used daily as a part of routine processes. Sharply increasing computer capacities open possibilities for employment of more and more sophisticated procedures of data processing and data-based decisions. If these procedures are to help human beings, they have to cope with uncertainty and vagueness, which is an inherent part of human data acquisition, communication and way of thinking. Furthermore, though psychologists claim that the human brain is not capable of combining more than seven factors, experts have developed approaches enabling them to take into account a vast number of conditions. For example, a medical expert, when diagnosing, usually has to take into account tens (and sometimes even hundreds) of symptoms, observations and results of laboratory and other special examinations. In addition to that, some of these features are of graphical character (RTG, ECG, EEG and so on), which often need to be translated into a multidimensional object for purposes of computer processing. Therefore, when intending to develop tools applicable in real life problems, one has to be prepared to cope with models of very high dimensionality, with models of extreme complexity.

Similarly, an operator of a rolling mill may influence several tens of variables, using about the same number of measurements updated in a fraction of second. Only an extreme skill, learned through several years' training period, allows him to cope with this problem. The desirable computer support can substitute him only when using high-dimensional dynamic models and the corresponding optimisation algorithms designing appropriate strategies of operations [37] . Dynamic optimisation of a highway or urban traffic is another problem exhibiting the same features as discussed above, see reference [100] . Here too high-dimensional probabilistic models should be implemented and optimisation of recommended strategies is required.

In addition to the complexity problems, one has to cope with the fact that some significant data are registered as linguistic expressions with the natural vagueness of their specifications (e.g. expert opinions, medical decisions, and many others). Therefore mathematical methods making possible to work and utilise information subjected to indeterminacy are to be exploited. The discussed methods fall into an area called soft computing.

THEORETICAL RESEARCH SUPPORTING COPING WITH COMPLEXITY

Mathematical procedures of optimal statistical decisions developed half-a-century ago for independent real valued observations were more recently, and still are, systematically extended to dependent data from more and more complex functional spaces. Typical observations for the presently studied procedures are various time-dependent stochastic signals and space-and/or-time-dependent random fields, or multivariate matrices of digits resulting, e.g., from digitisations of observations of the previous type.

Even more, development of techniques and tools for knowledge-processing and dynamic decision-making based on probabilistic models enables the users to represent and process multidimensional probability distributions, whose dimensions are hundreds and thousands rather than tens. As a result, using the modern methods based on recent theoretical achievements, we are able to deal with distributions in which the number of probabilities is much greater than the number of atoms in the globe. Even more, real data processed are often imperfect – vague, subjective or essentially uncertain, and yet they are expected to be processed on a sufficient level of accuracy[85].

On the other hand, the employed procedures are mathematically so sophisticated that their understanding and eventual algorithmisation require deeply educated specialised experts, which are rare even in academia and exceptional at applied research and development companies.

THEORETICAL RESEARCH SUPPORTING IMAGE PROCESSING

In many situations a decision is to be based on pictorial information. In classical image analysis theory, one often works with an ideal image function, which is considered to be "perfect" representation of a given scene. Usually that means all objects can be measured (both in space and colour domains) with infinite accuracy. This is, however, far from the reality. Since imaging sensors and other devices have their physical limits and the observational conditions tend to be imperfect, the acquired image represents only a degraded version of the original scene. Two main categories of degradations are recognised: colour (or brightness) degradations and geometric ones. The former degradations are caused by such factors as wrong focus, motion of the scene, media turbulence and limited spatial and spectral resolution of the sensor; they usually result in a blurring of the image. The latter degradations originate from the fact that each image is a 2-D projection of 3-D world. They cause deformation of object shapes and other spatial distortions of the image. An efficient way how to suppres the impact of these degradations is to integrate various images of the same scene together. This approach is called image fusion.

SOLUTION OF PROBLEMS ARISING FROM PRACTICAL APPLICATIONS

Modelling, sophisticated representation of high-dimensional models and optimisation do not exhaust possibilities for further progress. Due to the convergence of several strands of scientific and technological progress, we are witnessing the emergence of unprecedented opportunities to start solving new problems. Indeed, databases are accruing large amounts of complex multimedia documents. Sensors and storage are becoming cheap, networks allow fast and almost ubiquitous access to an abundance of resources and processors have the computational power to perform sophisticated and demanding algorithms. As a consequence, we are routinely amassing huge quantities of multimedia and multimodal data. These data have a rich and complex structure and potentially represent an extremely valuable source of information. They can be processed by both classical and advanced statistical methods. However, progress is hampered by the sheer amount and diversity of the available multidimensional data. Access can only be efficient if based directly on content and semantics, the extraction and understanding of which is only feasible if achieved automatically.

Demand for a higher efficiency and reliability calls for systematic, theoretically and algorithmically supported solutions. Brute-force solutions soon fail due to the dimensionality problems. For instance, the industrial partners of the Centre claim that they need to treat problems with 103-105 variables. Thus, there is an urgent need for theoretically well-grounded tools whose background reflects real requirements of problems met in practice.

Profit-driven, short term horizons of companies, daily problems of public services and lack of knowledge of scientists on “ordinary” application problems as well as scientific expertise on the applied side imply that a broad-scale systematic transfer of these new information technologies from the academic circles to the companies is hardly possible on the basis of individual contracts. Its systematic support obviously requires an institutional basis such as the proposed Centre. It also determines the Centre's orientation. The research in the proposed Centre will aim at further development of theoretical tools and simultaneously at design of necessary effective algorithms and procedures, enabling the users to apply the advanced models to support human activities via dynamic decision support systems, search engines, data mining tools etc.

CHARACTERIZATION OF THE RESEARCH ACTIVITIES

The proposed Centre aims to contribute systematically to development and transfer of tools supporting important human activities in complex uncertain situations via algorithmic realisation of high-dimensional models, optimisation of feasible strategies as well as via statistically, context- and semantics-based extraction of knowledge from large databases, extensive sets of sensors, and subjective or vague sources of information. Such sources may cover also information provided imprecisely, possibly only in natural languages.

This global aim will be reached through the following interrelated activities:

(1) The proposed Centre will put together experts dealing with problems dominated by aspects of uncertainty and complexity within various traditional research fields. The application partners of the Centre will bring the motivating real-life problems of this type as well as the expertise necessary for their solutions. This merging and focus on selected sample applications will build and/or strengthen mutual connections both across research branches leading to improved and widely applicable techniques coping with probabilistic models of high dimensions, and their applications in decision-making.

(2) The Centre will focus on advanced education of future experts both in theoretic and applied domains. Learning by doing will be the predominantly used technique, complemented by specialised courses and supported by internet-based education. Systematic development and deepening of the existing accredited courses, their extension and propagation to other universities, systemisation of topics for student research projects, diploma and PhD theses, and wider exploitation of international networks of excellence are the main tools to be used.

(3) The Centre will carry out systematic dissemination of acquired knowledge across field boundaries as well as boundaries of academic and applied worlds. Classical forms, such as research reports, regular seminars, specialised workshops and conferences will be complemented by a wider exploitation of the Internet as an advanced communication channel. It will allow the Centre members and a wider technical audience to exploit the fact that the involved researchers will systematically supervise the new international achievements in the areas of pure and applied mathematics and engineering dealing with broad topics of the Centre, as an extension of their own professional interests and contributions in these areas. If a potential user is interested in one of these methods more thoroughly, the Centre Board will help to create an adequate group for a deeper research, algorithmisation and/or testing of the method.

Symmetrically, the discussed communication channels will also provide a ground for the flow of information in the opposite direction, from the companies to the academic circles. In this sense the real-life problems encountered by the applicators will stimulate the research of the academic members of the Centre.

SPECIFICATION OF THE RESEARCH GOALS OF THE CENTRE

Global aim of the Centre is to contribute substantially to development of a unified theoretical, algorithmic and software background for solution of real life problems, above all decision-making in complex situations based on different sources of knowledge and data.

To achieve this, the following are the main partial goals from several theoretical fields that have to be tackled:

Knowledge acquisition and representation:

• modelling and learning with mixed dynamic graphical models;
• combinations of low dimensional models into higher dimensional ones when input data are not fully compatible;
• compound Markov random fields, their parameter estimation, optimal model selection and synthesis;
• dependence structure representation and its exploitation to graphical Markov model verification;
• algorithms for learning Markov models;

Decision-making:

• feasible optimisation of decision-making strategies when both model and aims are described by mixed graphical models;
• advanced statistical data analysis;
• description of the numerical relations between the information-theoretic divergences and the corresponding Bayes risk and its exploitation to optimisation of various digitalisations of observations (with respect to the Bayes risk);

Multidimensional signal processing and pattern recognition:

• methods for automatic fusion of digital images, which are incomplete, noisy, and/or degraded in an a priori unknown way;
• feature selection;
• numerical evaluation of the loss of information due to quantisation of observations on random processes and random fields;
• contextual classification and segmentation;
• applied medical image fusion, reconstruction, restoration and analysis.

Soft Computing and Fuzzy Modelling:

• development of methods and algorithms based on the theory of fuzzy logic with the stress to approximate reasoning and fuzzy approximation;
• improvement of existing methods for advanced fuzzy modelling, methods developed on the basis of results in fuzzy approximation theory and methods combining stochastic models and fuzzy logic;
• development of software tools for fuzzy modelling of complex processes.

Let us stress at this point that based on theoretical results, most of the problems will require design of specific algorithms necessary for the development of the required software tools. Recall that, namely, size of problems of practice requires sophisticated optimization of the algorithms regarding their time and space complexity.