Extending a shape-driven map to the interior of the input shape and to the surrounding volume is a difficult problem since it typically relies on the integration of shape-based and volumetric information, together with smoothness conditions, interpolating constraints, preservation of feature values at both a local and global level. This survey discusses the main volumetric approximation schemes for both 3D shapes and d-dimensional data, and provides a unified discussion on the integration of surface-based and volume-based shape information. Then, it describes the application of shape-based and volumetric techniques to shape modeling through volumetric parameterization and polycube splines; feature-driven approximation through kernels and radial basis functions. We also discuss the Hamilton's Ricci flow, which is a powerful tool to compute the conformal shape structure and to design Riemannian metrics of manifolds by prescribed curvatures. We conclude the presentation by discussing applications to shape analysis and medicine.

Level

Advanced

Prerequisites

Knowledge about differential geometry, mesh processing, function approximation.

Intended Audience

The target audience of this tutorial includes graduate students and researchers interested in Riemannian geometry, spectral geometry processing, and implicit modeling.

Presenter(s)

Giuseppe Patane, Consiglio Nazionale delle Ricerche, Instituto di Matematics Applicata e Tecnologie Informatiche (CNR-IMATI)
Xin Shane Li, Louisiana State University
David Xiangfeng Gu, State University of New York at Stony Brook
Ronald Lok Ming Lui, Department of Mathematics, The Chinese University of Hong Kong

Giuseppe Patane` (CNR-IMATI, Genova, Italy) is researcher at CNR-IMATI (2001-today). He received a Ph.D. in ”Mathematics and Applications” from the University of Genova (2005) and a Post Lauream Degree Master from the ”F. Severi National Institute for Advanced Mathematics” (2000). From 2001, his research activities have been focused on the definition of paradigms and algorithms for modeling and analyzing digital shapes and multidimensional data.

David Gu is an associated professor in Computer Science Department, Stony Brook University. He received a Ph.D. from Harvard university (2003), supervised by a Fields medalist, Prof. Shing-Tung Yau. His research focuses on computational conformal geometry, and its applications in graphics, vision, geometric modeling networks and medical imaging.

Xin Li is an assistant professor in School of Electrical Engineering and Computer Science, Louisiana State University. He received his Ph.D. in Computer Science from Stony Brook University (SUNY) in 2008. His research focus is on geometric modeling and computing, and their applications in graphics, vision, medical imaging, and computational forensics.

Ronald Lui is an assistant professor in Mathenatics Department, The Chinese University of Hong Kong. He received a Ph.D. from UCLA (2008), supervised by Prof. Tony F. Chan. After graduation, he worked as a postdoctoral scholar at Harvard University under the supervision of Prof. Shing-Tung Yau. His research focuses on computational quasi-conformal geometry, and its applications in graphics, vision and medical imaging. His research is partly supported by HKRGC GRF grant (CUHK Project ID: 404612).