Courses Information

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 trade-offs 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 finite-dimensional 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.

Mathieu Desbrun, California Institute of Technology
Fernando de Goes, PIxar Animation Studios
Yiying Tong, Michigan State University

Fernando de Goes, Research Scientist at Pixar Animation Studios. Fernando received his PhD from Caltech in 2014. His research centers on numerical methods for geometry processing and computational physics.

Mathieu Desbrun, John & Herberta Miles Professor at Caltech. His
research takes a geometric standpoint to develop differential, yet
readily-discretizable computational foundations, with applications
to modeling and simulation.

Yiying Tong, Associate Professor at Michigan State University. His
research focuses on a differential geometric approach to a number
of computational applications such as face recognition, fingerprint
analysis, and molecular surface computation.

Intended-Audience:
This course is intended for graduate students, researchers, and developers interested in geometry processing. Practitioners will find concrete data structures and numerical tools for the manipulation of vector fields on triangle surfaces, while scholars will learn how to preserve geometrical and topological continuous properties in the finite-dimensional realm.