Efficient Computation of Smoothed Exponential Maps. (14th March 2019)

• Record Type: Journal Article
• Authors: Herholz, Philipp; Alexa, Marc
• Is Part Of: Computer graphics forum. Volume 38:Number 6(2019)
• Page Start: 79
• Page End: 90
• Publication Date: 2019-03-14
• Subjects: computational geometry -- modelling -- digital geometry processing -- Computational Geometry and Object Modelling Curve -- surface -- solid -- and object representations; Computer graphics -- Periodicals; 006.605
• DOI: 10.1111/cgf.13607
• Languages: English
• ISSNs: 0167-7055

Abstract

Abstract: Many applications in geometry processing require the computation of local parameterizations on a surface mesh at interactive rates. A popular approach is to compute local exponential maps, i.e. parameterizations that preserve distance and angle to the origin of the map. We extend the computation of geodesic distance by heat diffusion to also determine angular information for the geodesic curves. This approach has two important benefits compared to fast approximate as well as exact forward tracing of the distance function: First, it allows generating smoother maps, avoiding discontinuities. Second, exploiting the factorization of the global Laplace–Beltrami operator of the mesh and using recent localized solution techniques, the computation is more efficient even compared to fast approximate solutions based on Dijkstra's algorithm. Abstract : Many applications in geometry processing require the computation of local parameterizations on a surface mesh at interactive rates. A popular approach is to compute local exponential maps, i.e. parameterizations that preserve distance and angle to the origin of the map. We extend the computation of geodesic distance by heat diffusion to also determine angular information for the geodesic curves. This approach has two important benefits compared to fast approximate as well as exact forward tracing of the distance function: First, it allows generating smoother maps, avoiding discontinuities. Second, exploiting the factorization of the global Laplace–Beltrami operator of the mesh and using recent localized solution techniques, the computation is more efficient even compared to fast approximate solutions based on Dijkstra's algorithm.