Direction fields and vector fields play an increasingly important role in computer graphics and geometry processing. The synthesis of directional fields on surfaces, or other spatial domains, is a fundamental step in numerous applications, such as mesh generation, deformation, texture mapping, and many more. The wide range of applications resulted in definitions for many types of directional fields: from vector and tensor fields, over line and cross fields, to frame and vector-set fields. Depending on the application at hand, researchers have used various notions of objectives and constraints to synthesize such fields. These notions are defined in terms of fairness, feature alignment, symmetry, or field topology, to mention just a few. To facilitate these objectives, various representations, discretizations, and optimization strategies have been developed. These choices come with varying strengths and weaknesses. This course provides a systematic overview of directional field synthesis for graphics applications, the challenges it poses, and the methods developed in recent years to address these challenges. The audience should have some prior experience with triangle mesh representation of geometric models, and a working knowledge of vector calculus, linear algebra, and general computer graphics fundamentals. Some familiarity with the basics of differential geometry and numerical optimization are helpful, but not required. The course targets researchers and developers who seek to understand the concepts and technologies used in direction field and vector field synthesis, learn about the most recent developments, and discern how this powerful tool, which has had impact in a variety of research and application areas, might benefit their area of work. Participants will get a broad overview, and obtain the knowledge on how to choose the proper combination of techniques for many relevant tasks. These notes are largely based on the following state-of-the-art report by the lecturers. It has been extended to include updates on the most recent developments. • The course was subsequently given at SIGGRAPH Asia 2016, including demos and real-time coding sessions. The entire course, including the notes, the presentation slides, and the demos, is provided in the following open-source GitHub repository: https://github.com/avaxman/DirectionalFieldSynthesis Being a relatively young and developing topic, no textbooks covering the various aspects of directional field synthesis in the context of computer graphics and geometry processing are available. The notes of a recent course on vector field processing offer another perspective on parts of the topic, with a focus on the discrete differential geometry aspects: • F. Her current interests are in geometry processing and modeling, specifically on vector field design, surface parametrizations, and inter-surface mappings. David Bommes is an assistant professor in the Computer Science department at RWTH Aachen University (Germany). His PhD thesis from 2012 (RWTH Aachen University) is on Quadrilateral Surface Mesh Generation for Animation and Simulation and was awarded the EUROGRAPHICS Best PhD Thesis Award. Furthermore, David was awarded the EUROGRAPHICS Young Researcher Award in 2016. Currently he is leading the Mesh Generation and Optimization group, which scientifically contributes to the areas of geometry processing, in particular direction fields, parametrization and quad/hex mesh generation, and nonlinear/mixed-integer optimization. Klaus Hildebrandt is an assistant professor in the Computer Graphics and Visualization group at Delft University of Technology, The Netherlands. He was a Senior Researcher at the Max Planck Institute for Informatics in Saarbrücken, Germany, where he headed the Applied Geometry group. He received his PhD from Freie University Berlin. His research interests are in geometric modeling and geometry processing, computational and discrete differential geometry and physically-based computer animation. Mirela Ben-Chen is an assistant professor in the Computer Graphics and Geometric Computing group at the Computer Science department at the Technion, Israel. She was a Fullbright postdoctoral scholar at Stanford University, and received her PhD from the Technion. Her research interests are computer graphics, geometry processing, discrete differential geometry, conformal geometry, and shape analysis and understanding.  The course will start with an overview over the broad variety of types of direction fields and vector fields that have found use in applications. We establish a concise taxonomy for these directional fields, and briefly introduce the necessary differential geometry background. After discussing the concepts of discretization and their specific pitfalls in the context of directional fields, we focus on the most important representations used for these fields in geometry processing. It will become clear that every choice of discretization and representation comes with different advantages and shortcomings; we provide a guide to making the proper choice, depending on the concrete requirements of the application. We illustrate the importance of these choices by comparing results from various field synthesis and optimization methods and demonstrate the algorithms to clarify the crucial differences. After introducing the most common synthesis objectives and constraints, such as smoothness and alignment, we also discuss advanced and specialized ones, pertaining to symmetry, conjugacy, holonomy, or based on differential operators. While directional fields are used as an abstract mathematical tool in many applications, it can be extremely helpful to actually visualize them-for purposes of understanding, debugging, and assessment. We explain efficient means of visualization suited for specific types of fields. The course will conclude with a demonstration of how directional field synthesis provides benefits in important application scenarios, and a discussion of the major open problems in this area of research.
Cross fields, and their application to quad meshing, have been studied extensively in geometry processing (cf. REF ).
Directional field synthesis, design, and processing
SIGGRAPH '17
Computer Science