Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills

Daniel Mejia-Parra, Jairo R. Sánchez, Jorge Posada, Oscar Ruiz-Salguero and Carlos Cadavid
Additional contact information
Daniel Mejia-Parra: Laboratory of CAD CAM CAE, Universidad EAFIT, Cra 49 no 7-sur-50, 050022 Medellín, Colombia
Jairo R. Sánchez: Vicomtech, Paseo Mikeletegi 57, Parque Científico y Tecnológico de Gipuzcoa, 20009 Donostia/San Sebastián, Spain
Jorge Posada: Vicomtech, Paseo Mikeletegi 57, Parque Científico y Tecnológico de Gipuzcoa, 20009 Donostia/San Sebastián, Spain
Oscar Ruiz-Salguero: Laboratory of CAD CAM CAE, Universidad EAFIT, Cra 49 no 7-sur-50, 050022 Medellín, Colombia
Carlos Cadavid: Matemáticas y Aplicaciones, Departamento de Ciencias Matemáticas, Universidad EAFIT, Cra 49 no 7-sur-50, 050022 Medellín, Colombia

Mathematics, 2019, vol. 7, issue 8, 1-17

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 ? from the original mesh M ∈ R 3 to the planar domain ? ( 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.

Keywords: mesh parameterization; geodesic maps; heat transfer analysis; Poisson fills (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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