Discrete multiscale vector field decomposition
ACM SIGGRAPH 2003 Papers
Discrete exterior calculus
Rotational symmetry field design on surfaces
ACM SIGGRAPH 2007 papers
Design of tangent vector fields
ACM SIGGRAPH 2007 papers
Geometry-aware direction field processing
ACM Transactions on Graphics (TOG)
Discrete Lie Advection of Differential Forms
Foundations of Computational Mathematics
Interactive Visualization of Rotational Symmetry Fields on Surfaces
IEEE Transactions on Visualization and Computer Graphics
Functional maps: a flexible representation of maps between shapes
ACM Transactions on Graphics (TOG) - SIGGRAPH 2012 Conference Proceedings
ACM Transactions on Graphics (TOG) - SIGGRAPH 2012 Conference Proceedings
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In this paper, we introduce a novel coordinate-free method for manipulating and analyzing vector fields on discrete surfaces. Unlike the commonly used representations of a vector field as an assignment of vectors to the faces of the mesh, or as real values on edges, we argue that vector fields can also be naturally viewed as operators whose domain and range are functions defined on the mesh. Although this point of view is common in differential geometry it has so far not been adopted in geometry processing applications. We recall the theoretical properties of vector fields represented as operators, and show that composition of vector fields with other functional operators is natural in this setup. This leads to the characterization of vector field properties through commutativity with other operators such as the Laplace-Beltrami and symmetry operators, as well as to a straight-forward definition of differential properties such as the Lie derivative. Finally, we demonstrate a range of applications, such as Killing vector field design, symmetric vector field estimation and joint design on multiple surfaces.