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Visualization of a closed three-dimensional surface using portal-based rendering
Abstract
The complexity and size of data is rapidly increasing in modern science, business and engineering. This has resulted in increasing demands for more sophisticated data analysis methods. Multidimensional scaling has been used to visualize large high-dimensional datasets in the form of a map. Such maps are very intuitive for us, as we are familiar with reading geographical maps. However, they typically result in a flat space (world), which presents undefined discontinuous edges at the end of the world. In order to provide a continuous space for visualiz- ing high-dimensional data, a closed three-dimensional (3D) surface, such as surfaces of a sphere and a torus, has been used as a target mapping space for multi- dimensional scaling. This paper proposes an applica- tion of a portal-based rendering technique to visualize such closed 3D surfaces. This 3D rendering technique allows users to see and navigate through the surface in a natural manner. This eliminates any discontin- uous boundaries usually introduced by a process of projection from 3D to two-dimensional space. Fur- thermore, as our initial investigations show, it is a general technique that is applicable to any closed 3D surface.
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Information visualization is an interdisciplinary research area in which cartographic efforts have mostly addressed the handling of geographic information. Some cartographers have recently become involved in attempts to extend geographic principles and cartographic techniques to the visualization of non-geographic information. This paper reports on current progress and future opportunities in this emerging research field commonly known as spatialization. The discussion is mainly devoted to the computational techniques that turn high-dimensional data into visualizations via processes of projection and transformation. It is argued that cartographically informed engagement of computationally intensive techniques can help to provide richer and less opaque information visu-alizations. The discussion of spatialization methods is linked to another priority area of cartographic involvement, the development of theory and principles for cognitively plausible spatialization. The paper distinguishes two equally important sets of challenges for cartographic success in spatialization research. One is the recognition that there are distinct advantages to applying a cartographic perspective in information visualization. This requires our community to more thoroughly understand the essence of cartographic activity and to explore the implications of its metaphoric transfer to non-geographic domains. Another challenge lies in cartographers becoming a more integral part of the information visualization community and actively engaging its constituent research fields.
Self-Organizing map (SOM) is a widely used tool to find clustering and also to visualize high dimensional data. Several spherical SOMs have been proposed to create a more accurate representation of the data by removing the "border effect". In this paper, we compare several spherical lattices for the purpose of implementation of a SOM. We then introduce a 2D rectangular grid data structure for representing the geodesic dome. This new approach improves the neighborhood searching process in the spherical gird. The new Geodesic SOM and its data structure are tested using socio-demographic data. In the experiments, we try to create a notion of direction in the Geodesic SOM. The direction facilitates more consistent visual comparison of different datasets as well as to assist viewers building their mental maps.
Multidimensional scaling can be considered as involving three basic steps. In the first step, a scale of comparative distances between all pairs of stimuli is obtained. This scale is analogous to the scale of stimuli obtained in the traditional paired comparisons methods. In this scale, however, instead of locating each stimulus-object on a given continuum, the distances between each pair of stimuli are located on a distance continuum. As in paired comparisons, the procedures for obtaining a scale of comparative distances leave the true zero point undetermined. Hence, a comparative distance is not a distance in the usual sense of the term, but is a distance minus an unknown constant. The second step involves estimating this unknown constant. When the unknown constant is obtained, the comparative distances can be converted into absolute distances. In the third step, the dimensionality of the psychological space necessary to account for these absolute distances is determined, and the projections of stimuli on axes of this space are obtained. A set of analytical procedures was developed for each of the three steps given above, including a least-squares solution for obtaining comparative distances by the complete method of triads, two practical methods for estimating the additive constant, and an extension of Young and Householder's Euclidean model to include procedures for obtaining the projections of stimuli on axes from fallible absolute distances.
In this paper, we identify a general paradigm for portal-based rendering and present an image-space algorithm for rendering complex portals. Our general paradigm is an abstraction of portal-based rendering that is independent of scene geometry. It provides a framework for flexible and dynamic scene composition by connecting cells with transformative portals. Our rendering algorithm maintains a visible volume in image-space and uses fragment culling to discard fragments outside of this volume. We discuss our implementation in OpenGL and present results that show it provides correct rendering of complex portals at interactive rates on current hardware. We believe that our work will be useful in many applications that require a means of creating dynamic and meaningful visual connections between different sets of data.
This paper introduces a method called relational perspective map (RPM) to visualize distance information in high-dimensional spaces. Like conventional multidimensional scaling, the RPM algorithm aims to produce proximity preserving 2-dimensional (2-D) maps. The main idea of the RPM algorithm is to simulate a multiparticle system on a closed surface: whereas the repulsive forces between the particles reflect the distance information, the closed surface holds the whole system in balance and prevents the resulting map from degeneracy. A special feature of RPM algorithm is its ability to partition a complex dataset into pieces and map them onto a 2-D space without overlapping. Compared to other multidimensional scaling methods, RPM is able to reveal more local details of complex datasets. This paper demonstrates the properties of RPM maps with four examples and provides extensive comparison to other multidimensional scaling methods, such as Sammon Mapping and Curvilinear Principle Analysis.
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- C B Jones