DOI: 10.1111/j.1467-8659.2012.03231.x COMPUTER GRAPHICS forum Volume 32 (2013), number 1 pp. 49–74 Visualization and Analysis of Second-Order Tensors: Moving Beyond the Symmetric Positive-Definite Case A. Kratz 1 , C. Auer 1 , M. Stommel 2 and I. Hotz 1 1 Zuse Institute, Berlin, Germany {kratz, auer, hotz}@zib.de 2 Institute of Polymer Materials, Saarland University, Germany m.stommel@mx.unisaarland.de Abstract Tensors provide a powerful language to describe physical phenomena. Consequently, they have a long tradition in physics and appear in various application areas, either as the final result of simulations or as intermediate product. Due to their complexity, tensors are hard to interpret. This motivates the development of well-conceived visualization methods. As a sub-branch of scientific visualization, tensor field visualization has been especially pushed forward by diffusion tensor imaging. In this review, we focus on second-order tensors that are not diffusion tensors. Until now, these tensors, which might be neither positive-definite nor symmetric, are under-represented in visualization and existing visualization tools are often not appropriate for these tensors. Hence, we discuss the strengths and limitations of existing methods when dealing with such tensors as well as challenges introduced by them. The goal of this paper is to reveal the importance of the field and to encourage the development of new visualization methods for tensors from various application fields. Keywords: tensors, second-order tensors, tensor visualization, scientific visualization ACM CCS: I.3.3 [Computer Graphics]: Picture/Image Generation—Line and curve generation 1. Introduction Tensor mathematics is a beautiful, simple, and useful language for the description of natural phenomena. [Dan97] Visualization is fundamental to understanding models of complex phenomena [ ... ]. [MJM*06] Using tensors and the underlying mathematics, physical phenom- ena can be formulated and analysed. Tensors describe entities that scalars and vectors cannot describe sufficiently, for example, the stress at a point in a continuous medium under load. Consequently, tensors play an important role in many application areas such as medicine, geology, astrophysics, continuum mechanics and many more. Current visualization research, however, concentrates more on scalar and vector fields. Compared to their significance and exclud- ing diffusion tensor imaging (DTI), tensors are an under-represented topic in visualization. Looking at tensors from a visualization point of view, the extrac- tion of relevant information from the literature is a non-trivial task. Several workshops have been held with the goal to bring researchers from different disciplines together and to foster new collaborations and research. As a result, several books [WH05, LW08, AFdL- GTL09] have been published that give insight into the multiple re- lated research areas. However, a clear focus on DTI (diffusion tensor as output of simulations) and image processing (structure tensor as feature descriptor during data analysis) can be observed. Recently, a tutorial on tensors in visualization was presented [KST*10] that also covers tensor topology, asymmetric tensors and higher-order tensors for DTI. A compact review that summarizes work focusing on tensors that are not symmetric positive definite, however, is still missing. This paper surveys analysis and visualization methods of second- order tensors. The focus are tensors that are not symmetric positive definite. Such tensors are, for example, generated as result of simu- lations in engineering applications but they are also used as feature descriptors in flow analysis. In the field of visualization, appropri- ate tools for dealing with these types of tensors are still missing. Domain experts often analyse only derived scalar values although there clearly is an interest to understand and analyse the whole tensor as one entity, or at least to keep the tensor data as long as possible before extracting relevant features. Well-conceived visu- alization methods can support such a process and help to develop c  2012 The Authors Computer Graphics Forum c  2012 The Eurographics Association and Blackwell Publishing Ltd. Published by Blackwell Publishing, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA. 49