| Abstract [eng] |
The research object of this work is multidimensional data visualization algorithms and methods that preserve the local structure of multidimensional data, as well as the evaluation criteria of multidimensional data projections in a low-dimensional space. There is none activity field of mankind that would not collect and analyse multidimensional data. A typical example of multidimensional data is related with image processing. Frequently data are comprised of photos of the same object, obtained by gradually turning the object at a certain angle or by taking its photos at different moments. Each photo is digitized, i.e., the coordinates of a data point consist of colour parameters of photo points and therefore the number of the coordinates of this point is very large. Dimensionality reduction methods (projection methods) are developed rather intensively. By transforming multidimensional data into a two- or three-dimensional space and after visualizing them, it is much easier to conceive the structure of data and connections among them. However, while transforming data into a lower dimensional space, data projection distortions are inevitable. That is why evaluation of the projection quality obtained remains a topical problem. We often work with datasets that are constantly supplemented with new data. It is of great importance to immediately map the new data points without loss of a high accuracy. Therefore, the mapping of new points, their insertion among the earlier mapped points is one of the problems considered in the thesis. Projection methods of multidimensional data come across two main problems. The first one is to find multidimensional data projections in a space of lower dimensionality (two or three-dimensional space) with a view to preserve the proximity (similarities or dissimilarities) of objects of the analysed set as exactly as possible. The second one is to map multidimensional data in a low-dimensional space so that their projections did not overlap. This problem is also one of the problems solved in the thesis. Frequently in practical problems multidimensional data are accumulated and the points corresponding to them are considered in a high-dimensional space, while in fact they are either points of a manifold of some lower dimensionality or the points close to that manifold. Thus, one of the major problems of the thesis is to discover a low-dimensional nonlinear manifold in a high-dimensional data space and then transfer the data points that lie on or near to this manifold, into this low-dimensional space. An important point of a manifold is its topology. There are a lot of different measures of topology preservation in the literature. Another important problem solved in the thesis is to find and explore those measures that would be suitable to analyse the manifold topology preservation after its transformation into a low-dimensional space. |
