Linear geometric algebra rotor estimator for efficient mesh deformation

Abstract

The authors solve the problem of estimating the best rotation aligning two sets of corresponding vectors (also known as Wahba's problem or point cloud registration). The proposed method is among the fastest methods reported in recent literatures, moreover it is robust to noise, accurate and simpler than most other methods. It is based on solving the linear equations derived from the formulation of the problem in Euclidean Geometric Algebra. The authors show its efficiency in two applications: the as-rigid-as-possible (ARAP) surface modelling and the more smooth rotation enhanced ARAP mesh animation which is the only method capable of deforming surface modes with quality of tetrahedral models. Mesh deformation is a key technique in games, automated construction and robotics. The ARAP technique along with its improved variants, although have been extensively studied, can still not be achieved efficiently. Linear geometric algebra based rotor solution proposed in this study gives another perspective of the kernel problem. This, however, not only improves the real performance of the three-dimensional mesh deformation, but also provides a brand new computationally efficient solution to the Wahba's problem and point cloud registration, which has been closely related to the automation science and engineering.

Keywords

• computer animation
• mesh generation
• deformation
• computational geometry
• algebra
• rotors (mechanical)
• mechanical engineering computing
• solid modelling
• three-dimensional mesh deformation
• wahba's problem
• linear equations
• surface modelling
• arap mesh animation
• surface modes
• tetrahedral models
• rotor solution
• kernel problem
• euclidean geometric algebra
• point cloud registration
• as-rigid-as-possible surface modelling
• arap surface modelling
• linear geometric algebra
• rotor estimator