Spectral Geometry of Shapes

Dr. Jing Hua is a Professor of Computer Science and the founding director of Computer Graphics and Imaging Lab (GIL) and Visualization Lab (VIS) at Computer Science at Wayne State University (WSU). He received his Ph.D. degree (2004) in Computer Science from the State University of New York at Stony Brook. He also received his M.S. degree (1999) in Pattern Recognition and Artificial Intelligence from the Institute of Automation, Chinese Academy of Sciences in Beijing, China and his B.S. degree (1996) in Electrical Engineering from the Huazhong University of Science & Technology in Wuhan, China. His research interests include Computer Graphics, Visualization, Image Analysis and Informatics, Computer Vision, etc. He received the Gaheon Award for the Best Paper of International Journal of CAD/CAM in 2009, the Best Paper Award at ACM Solid Modeling 2004, the WSU Faculty Research Award in 2005, the College of Liberal Arts and Sciences Excellence in Teaching Award in 2008, the K. C. Wong Research Award in 2010, and the Best Demo Awards at GENI Engineering Conference 21 (2014) and 23 (2015), respectively. Zichun Zhong is Assistant Professor of Computer Science at Wayne State University (WSU) since August 2015. He was a postdoctoral fellow in Department of Radiation Oncology at UT Southwestern Medical Center at Dallas (UTSW) from August 2014 to August 2015. He received Ph.D. degree in Computer Science at The University of Texas at Dallas (UTD) in Summer 2014, a B.S. degree in Computer Science and Technology (Software Engineering) and a M.S. degree in Computer Science in The University of Electronic Science and Technology of China (UESTC) in 2006 and 2009, respectively. Dr. Jiaxi Hu is a software engineer at Google LLC, Mountain View, California. He received his B.S. and M.S. from Huazhong University of Science and Technology, Wuhan, China, and his Ph.D. from Wayne State University, Detroit, Michigan before he joined Google LLC in 2014. After a decade of research of Computer Graphics in the university, Dr. Hu brought his expertise to the world through Google web services. He worked on Imagery Viewer which is a backend rendering engine to serve a broad range of Google products, such as Google Map, Street View, Local Search, etc., and third-party partners. After that, he found the new challenge in bringing Google Assistant to Next Billion Users.

• 1. Introduction2. Spectral Geometry Operation3. Spectral Geometric Features for Shapes4. Isometric Shape Analysis Using Spectral Geometry5. Near Isometric Shape Motion Analysis Using Spectral Geometry6. Non-Isometric Shape Motion Analysis by Variation of Shape Spectrum7. Machine Learning of Spectral Geometry8. Conclusions

"Beltrami spectral domain. An important property of the thus defined features is invariance or at least stability with respect to global isometries, scaling, isometric deformations, and changes in the triangulation. The next chapter is dedicated to the analysis of near-isometric poses of unregistered triangular meshes. The approach via spectral geometry allows the labelling of different semantic parts. An important application is skeleton extraction and animation of triangular meshes. Again, the algorithms are stable with respect to Euclidean transformations and triangulation. The chapter on nonisometric motion analysis is based on a chapter in the second author’s PhD thesis. The main theoretical result states that the shape spectrum (a family of eigenfunctions) is analytic and provides analytic expressions for its derivatives. These can be used for the analysis of nonisometrically deforming triangular meshes, and the authors present important applications in medicine. Registration of nonisometric surfaces can be accomplished by simultaneously matching eigenvectors and eigenvalues of the Laplace-Beltrami spectrum by optimizing a suitable energy functional [H. Hamidian et al., “Surface registration with eigenvalues and eigenvectors”, IEEE Trans. Vis. Comput. Gr., 1–1 (2019; doi:10.1109/tvcg.2019.2915567)]. This approach also allows to determine the point-to-point correspondence and compares favourably with other registration methods. The book concludes with a chapter on deep learning of spectral geometry. After a brief overview on the basics of deep learning, it discusses existing approaches based on multiple view, volumetric, point cloud or mesh representations of the geometry and, finally, adds ideas for deep learning in spectral domains." --ZBMath