Mbius-invariant curve and surface energies and their applications

作     者：YOSHIZAWA Shin BELYAEV Alexander 

作者机构：Image Processing Research Team RIKEN Institute of Sensors Signals & Systems Heriot-Watt University 

出 版 物：《Science China(Information Sciences)》 (中国科学:信息科学(英文版))

年 卷 期：2013年第56卷第9期

页      面：18-27页

学科分类：1305[艺术学-设计学（可授艺术学、工学学位）] 13[艺术学] 08[工学] 080203[工学-机械设计及理论] 081304[工学-建筑技术科学] 0802[工学-机械工程] 0813[工学-建筑学] 080201[工学-机械制造及其自动化] 

基　　金：supported in part by Grants-in-Aid for Scientific Research of Japan (24700182  20113007) 

主　　题：Willmore energy minimum variation surfaces Dupin’s cyclides Mbius/conformal invariance 

摘      要：Curvature-based surface energies are frequently used in mathematics, physics, thin plate and shell engineering, and membrane chemistry and biology studies. Invariance under rotations and shifts makes curvature-based energies very attractive for modeling various phenomena. In computer-aided geometric design, the Willmore surfaces and the so-called minimum variation surfaces (MVS) are widely used for shape modeling purposes. The Willmore surfaces are invariant w.r.t conformal transformations (Mbius or conformal invariance), and studied thoroughly in differential geometry and related disciplines. In contrast, the minimum variation surfaces are not conformal invariant. In this paper, we suggest a simple modification of the minimum variation energy and demonstrate that the resulting modified MVS enjoy Mbius invariance (so we call them conformal-invariant MVS or, shortly, CI-MVS). We also study connections of CI-MVS with the cyclides of Dupin. In addition, we consider several other conformal-invariant curve and surface energies involving curvatures and curvature derivatives. In particular, we show how filtering with a conformal-invariant curve energy can be used for detecting salient subsets of the principal curvature extremum curves used by Hosaka and co-workers for shape quality inspection purposes.