Raw Data Image Data Processed Data Focus Data Geometric Data Data Analysis Filtering Mapping Rendering

Mathematical concepts and tools have shaped the field of visualization in fundamental ways and played a key role in the development of a large variety of visualization techniques. In this chapter, we sample the visualization literature to provide a taxonomy of the usage of mathematics in visualization, and to identify a fundamental set of mathematics that should be taught to students as part of an introduction to contemporary visualization research. Within the scope of this chapter, we are unable to provide a full review of all mathematical foundations of visualization; rather, we identify a number of concepts that are useful in visualization, explain their significance, and provide references for further reading. We assume the reader has basic knowledge of linear algebra [90], multivariate calculus [89], statistics, combinatorics, and stochastics [39]. Other topics not covered in this chapter, such as image analysis [88], computer graphics [86], signal processing [41], computational geometry [18], geometric modeling, mesh generation, computer aided geometric design [35, 106], and numerics [76], can be found in wellestablished textbooks. 1 Data and basic terminology You can have data without information, but you cannot have information without data. Daniel Keys Moran, programmer and science fiction writer Data are at the center of every visualization task and every step of the visualization pipeline, see Fig. 1. Raw Data Image Data Processed Data Focus Data Geometric Data Data Analysis Filtering Mapping Rendering Figure 1: A visualization pipeline. All steps in the pipeline involve the use of mathematical concepts and tools. We cover various aspects of data analysis, filtering, and mapping. The input to the visualization pipeline, the raw data, can be any collection of information in any form. In this chapter, we define a data set as a triplet D = (S ,A , f ) consisting of a set of structured items S , a set of attributes A , and a function that assigns attributes to the items. S consists of a set of items, continuous or discrete, together with a structure (such as a metric for a continuous domain or neighborhood relations for networks), see Fig. 2 for an example. ∗ingrid.hotz@liu.se, Linköping University, Sweden †bujack@lanl.gov, Los Alamos National Laboratory, USA ‡garth@cs.uni-kl.de, Technische Universität Kaiserslautern, Germany §beiwang@sci.utah. edu, University of Utah, USA 1 ar X iv :1 90 9. 04 83 5v 1 [ cs . H C ] 1 1 Se p 20 19 Structured Items set of points with neigborhood relation Attributes Temperature, a scalar value in [t0,t1] t1 t0 Assignmemt Assigning temperature value to points Figure 2: An example of a data set: S consists of a set of points with a neighborhood relation. Attributes in A are elements of the interval [t0, t1]. f assigns temperature values to the points. The tools used for the analysis and visualization of data sets depend on the nature of S and A . The most important distinctions are continuous vs. discrete structures, and quantitative vs. categorical attributes, see Table 1. In this section, we emphasize continuous structures and quantitative attributes. A more detailed classification of data sets concerning types, structures, and organizations can be found in Munzner [68]. An introduction to data representations from a scientific visualization perspective can be found in Telea [92].