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- Title
Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills.
- Authors
Mejia-Parra, Daniel; Sánchez, Jairo R.; Posada, Jorge; Ruiz-Salguero, Oscar; Cadavid, Carlos
- Abstract
In the context of CAD, CAM, CAE, and reverse engineering, the problem of mesh parameterization is a central process. Mesh parameterization implies the computation of a bijective map ϕ from the original mesh M ∈ R 3 to the planar domain ϕ (M) ∈ R 2 . The mapping may preserve angles, areas, or distances. Distance-preserving parameterizations (i.e., isometries) are obviously attractive. However, geodesic-based isometries present limitations when the mesh has concave or disconnected boundary (i.e., holes). Recent advances in computing geodesic maps using the heat equation in 2-manifolds motivate us to revisit mesh parameterization with geodesic maps. We devise a Poisson surface underlying, extending, and filling the holes of the mesh M. We compute a near-isometric mapping for quasi-developable meshes by using geodesic maps based on heat propagation. Our method: (1) Precomputes a set of temperature maps (heat kernels) on the mesh; (2) estimates the geodesic distances along the piecewise linear surface by using the temperature maps; and (3) uses multidimensional scaling (MDS) to acquire the 2D coordinates that minimize the difference between geodesic distances on M and Euclidean distances on R 2 . This novel heat-geodesic parameterization is successfully tested with several concave and/or punctured surfaces, obtaining bijective low-distortion parameterizations. Failures are registered in nonsegmented, highly nondevelopable meshes (such as seam meshes). These cases are the goal of future endeavors.
- Subjects
PARAMETERIZATION; GEODESICS; GEODESIC distance; MULTIDIMENSIONAL scaling; HEAT equation; EUCLIDEAN distance; ISOMETRICS (Mathematics)
- Publication
Mathematics (2227-7390), 2019, Vol 7, Issue 8, p753
- ISSN
2227-7390
- Publication type
Article
- DOI
10.3390/math7080753