Differential methods for multi-dimensional visual data analysis

Abstract

Images in scientific visualization are the end product of data processing. Starting from higher-dimensional data sets such as scalar, vector, and tensor fields given on 2D, 3D, and 4D domains, the objective is to reduce this complexity to two-dimensional images comprehensible to the human visual system. Various mathematical fields such as in particular differential geometry, topology (theory of discretized manifolds), differential topology, linear algebra, Geometric Algebra, vector field and tensor analysis, and partial differential equations contribute to the data filtering and transformation algorithms used in scientific visualization. The application of differential methods is core to all these fields. The following chapter will provide examples from current research on the application of these mathematical domains to scientific visualization. Ultimately the use of these methods allows for a systematic approach for image generation resulting from the analysis of multidimensional datasets.