Directional Field Synthesis, Design, and Processing

Abstract

SummaryDirection 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. PrerequisitesThe 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. Intended AudienceThe 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. SourcesThese 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 Further ReadingBeing 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 RWTH Aachen University, GermanyDavid Bommes is an assistant professor in the Computer Science ...

Integer-only cross field computation

“…Without the unit-length constraint, this formulation leads to an unconstrained linear least squares problem that can be efficiently and globally solved [Knöppel et al 2013]. Our main interest is in the angle-based energy, as it has various advantages in applications; see [Vaxman et al 2016]. We show that for this energy our algorithm achieves lower energy values, with a smaller number of singularities, compared to competing approaches.…”

“…A recent review [Vaxman et al 2016] covers the latest developments, and we therefore focus our literature review on methods closest to our approach.…”

“…Then, in Section 3 we generalize TC, and show that the new formulation is equivalent to MIQ, yet simpler to optimize. We provide only a brief overview of the methods, and refer to specific sections of the survey [Vaxman et al 2016] and course ] for basic concepts. Bommes et al [2009] represented a cross field by an angle per face, θ ∈ R m , relative to a fixed local orthogonal frame.…”

“…They are used in many applications, from quadrangular remeshing to non-photorealistic rendering [Vaxman et al 2016]. Computing smooth cross fields on triangle meshes is challenging, as the problem formulation inherently depends on integer variables to encode the invariance of the crosses to rotations by integer multiples of π /2.…”

Autonomous reconstruction of unknown indoor scenes guided by time-varying tensor fields

“…Directional fields, including those which are orientation-dependent or orientation-free, have many applications in computer graphics [Vaxman et al 2016], ranging from surface parameterization and remeshing [Ray et al 2009], nonphotorealistic rendering [Hertzmann and Zarin 2000] to street and urban modeling [Chen et al 2008). We are particularly interested in symmetric tensor fields (also known as 2-direction fields or line fields).…”