Secondary Laplace operator and generalized Giaquinta–Hildebrandt operator with applications on surface segmentation and smoothing |
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Affiliation: | 1. Department of Mechanical Engineering, Carnegie Mellon University, USA;2. Institute of Computational Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, China;1. Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str. bl. 8, 1113, Sofia, Bulgaria;2. “L. Karavelov” Civil Engineering Higher School, 175 Suhodolska Str., 1373 Sofia, Bulgaria;3. Istanbul Technical University, Faculty of Science and Letters, Department of Mathematics, 34469 Maslak, Istanbul, Turkey;1. Université Catholique de Louvain, Avenue Georges Lemaitre, 4, B-1348 Louvain-la-Neuve, Belgium;2. CEA, DAM, DIF, F-91297, Arpajon, France;3. Université Pierre et Marie Curie, 4 place Jussieu, 75005, Paris, France;1. College of Mathematics and Statistics, Hengyang Normal University, Hengyang, Hunan 421008, People''s Republic of China;2. Department of Mathematics, Shantou University, Shantou, Guangdong 515063, People''s Republic of China;3. School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, People''s Republic of China |
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Abstract: | Various geometric operators have been playing an important role in surface processing. For example, many shape analysis algorithms have been developed based on eigenfunctions of the ?Laplace–Beltrami operator (LBO), which is defined based on the first fundamental form of the surface. In this paper, we introduce two new geometric operators based on the second fundamental form of the surface, namely the secondary Laplace operator (SLO) and generalized Giaquinta–Hildebrandt operator (GGHO). Surface features such as concave creases/regions and convex ridges can be captured by eigenfunctions of the SLO, which can be used in surface segmentation with concave and convex features detected. Moreover, a new geometric flow method is developed based on the GGHO, providing an effective tool for sharp feature-preserving surface smoothing. |
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Keywords: | Secondary Laplace operator Generalized Giaquinta–Hildebrandt operator Eigenfunction Concave and convex feature Surface segmentation Geometric flow |
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