Wróblewski and Jarosław Andrzejczak Wave propagation time optimization for geodesic distances calculation using the Heat Method

Abstract Finding the geodesic path defined as the shortest paths between two points on three-dimensional surface P is a well known problem in differential and computational geometry. Surfaces are not differentiable in a discrete way, hence known geometry algorithms can’t be used directly - they have to be discretized first. Classic algorithms for geodesic distance calculation such as Mitchell-Mount-Papadimitriou method (MMP) are precise but slow. Therefore modern solutions are developed for fast calculations. One of them is Heat Method which approximates such paths with some accuracy. In this paper we propose the extension of Heat Method to reduce the approximation error. A new formula for calculating value of the parameter t (wave propagation time step) which outperforms the original one in terms of mean and median error is presented. Also, correlation between mesh properties and best wave propagation time step as well as influence of variable node spacing on heat map based method were thoroughly analysed.

[1]  Piotr Napieralski,et al.  Physically Based Area Lighting Model for Real-Time Animation , 2016, ICCVG.

[2]  Jindong Chen,et al.  Shortest Paths on a Polyhedron , Part I : Computing Shortest Paths , 1990 .

[3]  Shi-Qing Xin,et al.  Improving Chen and Han's algorithm on the discrete geodesic problem , 2009, TOGS.

[4]  Hiromasa Suzuki,et al.  Approximate shortest path on a polyhedral surface based on selective refinement of the discrete graph and its applications , 2000, Proceedings Geometric Modeling and Processing 2000. Theory and Applications.

[5]  Leonidas J. Guibas,et al.  Earth mover's distances on discrete surfaces , 2014, ACM Trans. Graph..

[6]  Ligang Liu,et al.  Fast Wavefront Propagation (FWP) for Computing Exact Geodesic Distances on Meshes , 2015, IEEE Transactions on Visualization and Computer Graphics.

[7]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[8]  Yijie Han,et al.  Shortest paths on a polyhedron , 1990, SCG '90.

[9]  K. Guzek,et al.  Efficient rendering of caustics with streamed photon mapping , 2017 .

[10]  Ying He,et al.  Saddle vertex graph (SVG) , 2013, ACM Trans. Graph..

[11]  Joseph S. B. Mitchell,et al.  The Discrete Geodesic Problem , 1987, SIAM J. Comput..

[12]  Steven J. Gortler,et al.  Fast exact and approximate geodesics on meshes , 2005, ACM Trans. Graph..

[13]  Keenan Crane,et al.  Geodesics in heat: A new approach to computing distance based on heat flow , 2012, TOGS.

[14]  Sanjiv Kapoor,et al.  Efficient computation of geodesic shortest paths , 1999, STOC '99.

[15]  J A Sethian,et al.  Computing geodesic paths on manifolds. , 1998, Proceedings of the National Academy of Sciences of the United States of America.