Vector field processing on triangle meshes

Abstract

While scalar fields on surfaces have been staples of geometry processing, the use of tangent vector fields has steadily grown in geometry processing over the last two decades: they are crucial to encoding directions and sizing on surfaces as commonly required in tasks such as texture synthesis, non-photorealistic rendering, digital grooming, and meshing. There are, however, a variety of discrete representations of tangent vector fields on triangle meshes, and each approach offers different tradeoffs among simplicity, efficiency, and accuracy depending on the targeted application.This course reviews the three main families of discretizations used to design computational tools for vector field processing on triangle meshes: face-based, edge-based, and vertex-based representations. In the process of reviewing the computational tools offered by these representations, we go over a large body of recent developments in vector field processing in the area of discrete differential geometry. We also discuss the theoretical and practical limitations of each type of discretization, and cover increasingly-common extensions such as n-direction and n-vector fields.While the course will focus on explaining the key approaches to practical encoding (including data structures) and manipulation (including discrete operators) of finitedimensional vector fields, important differential geometric notions will also be covered: as often in Discrete Differential Geometry, the discrete picture will be used to illustrate deep continuous concepts such as covariant derivatives, metric connections, or Bochner Laplacians. PrefaceT hese course notes provide a review of the various representations of tangent vector fields on triangulated surfaces. Over the past decades, a number of approaches to express, design, and analyze vector fields on arbitrary meshes have been proposed in geometry processing, targeting a series of applications varying from rendering to animation. Some describe vector fields through values on vertices, some on edges, and some on faces; some treat vector fields as piecewise constant, some as piecewise linear, and some even involve non-linear finite elements. This lack of a unified approach to vector field representation and manipulation can be confusing for practitioners, and rare are the references that discuss the pros and cons of these various representations. The chapters in this document were written with this fact in mind, as a way to offer an introduction to the motivations, benefits, and shortcomings of the most common discretizations of vector fields in graphics-along with their associated discrete differential operators (curl, divergence, Laplacian, etc). Emphasis is placed on the tradeoffs between simplicity, efficiency, and accuracy of these geometry processing tools for applications ranging from texture synthesis to meshing.Course rationale. In many ways, this course is a continuation of ACM SIGGRAPH courses on Discrete Differential Geometry (SIGGRAPH '05 and '06, and SIGGRAPH Asia '09) and Discrete Exte...

A Vector Field Design Approach to Animated Transitions

“…A vector field refers to a function where each point in a space is assigned a vector -often, the space is a surface and the vectors are tangent to the surface. A vector field is frequently required as input in many applications including texture synthesis, nonphotorealistic rendering, anisotropic shading and fluid simulation, to control surface appearance [40]. Due to these applications in computer graphics, vector field design, or methods to create vector fields that satisfy user constraints, has been an important area of research [41], [42].…”

Directional Field Synthesis, Design, and Processing

“…Thus, the directional information is solely contained in the traceless deviatoric part . Consequently, the eigenvalues λ′ i of D encode a 2‐vector field ± λ′ i u i [dGDT15] with singularities at points where D vanishes. A common approach to 2‐tensor fields is to choose local coordinate systems (e.g.…”

“…A recent course on vector field processing [dGDT15] provides an additional introduction to the topic, with a focus on the aspects of discretization (vertex vs. edge vs. face based) of vector fields.…”

Spectral Processing of Tangential Vector Fields

“…They are used for controlling anisotropic shading of surfaces [MRMH12, RGB*14], non‐photorealistic rendering [HZ00, YCLJ12, CYZL14], texture generation [WL01, KCPS15], simulation of fluid and liquids on surfaces [AWO*14, AVW*15], surface segmentation [SBCBG11, ZZCJ14] and surface construction [IBB15, PLS*15]. For a recent surveys on direction field synthesis, design, and processing, we refer to [dGDT15, VCD*16]. Methods that use tangential vector fields and more general direction fields for surface meshing have received much attention in recent years.…”

“…We want to remark that after the submission of this paper we found that concurrently to our work the discrete Hodge–Laplace operator for piecewise constant vector fields was also introduced in [dGDT15] and a scheme for computing the eigenfields using eigenfunctions was introduced in [LMH*15], also concurrent to this work.…”