Derivatives and Eigensystems for Volume-Data Analysis and Visualization. Jiri Hladuvka jiri@cg.tuwien.ac.at Abstract Volume data refer to sampled three-dimensional spatial signals. The tools which handle them can broadly be divided into two categories: visual tools which aim at an output interpretable by a human user and analytic tools which prepare the data for further machine processing. Although it would be natural that the two related disciplines, i.e., volume visualization and volume processing closely collaborate, they are still rather separated. In my talk I will concentrate on the recent research results which contribute to bridge the gap in between. I will address classification of volume samples based on observations of how the scalar values vary in their vicinity. I will investigate the first three terms of a Taylor series expansion of the corresponding scalar field at the inspected points. An important issue arising with such an analysis in higher dimensions are the directions to be examined. In order to find an answer to this problem I will discuss the eigensystems of algebraic structures composed of the first- and second-order partial derivatives. After a survey on derivative-based classification in volume visualization and processing I will talk about my own results which apply to three distinct problems: specification of transfer functions, content-based retrieval of volume-data features, and shape-based interpolation.