Representation, recognition and synthesis of 3D images using Lie algebra surace model

近年来，3D图像和3D计算机图形学在虚拟环境、多媒体通信和数字内容中发挥着重要的作用。因此，为了高效地表示、编码、识别和合成三维物体，迫切需要一种强大而高效的三维自由曲面模型。该研究提出了一种利用三维物体的切向李代数和法向李代数来表示欧几里得或仿射运动下三维物体表面模型的全局方法。这些李代数中的纯局部信息完全描述了作为李群的全局形状。特别是，我们重点讨论了线性李代数和哈密顿李代数，它们既可以表示代数形状，也可以表示更广泛的非代数形状。我们得到了欧氏运动下唯一、确定和再现物体的不变量的完备集，并提出了通过求解一个线性方程组来从曲面上的局部数据中提取这些不变量的算法。我们还提出了一种新的曲面模型，称为自由曲面的纤维束模型，该模型将任意曲面表示为基础曲线和纤维曲线之间的局部积。特别是，这种使用纤维作为线性李代数或哈密顿李代数的单参数群的模型是非常有效的，因为在线性李代数的情况下，曲面可以用一条基本曲线和六个不变量或15个参数来表示。曲面合成速度快，无数值误差。

Recently, 3D images and 3D computer graphics are playing an important rule in virtual environment, multimedia communications and digital contents. Therefore, it is highly desirable to have powerful and efficient model for 3D free surfaces for efficient representation, coding, recognition and synthesis of 3D objects.This research presents a global method to represent surface models of 3D objects invariantly under Euclidean or Affine motions using their tangential and normal Lie algebras. The global shapes as Lie groups are completely described by purelylocal information in these Lie algebra. Particularly, we focus on linear Lie algebras and Hamiltonian Lie algebras, which can represent algebraic shapes and a much wider class of non-algebraic shapes as well. We obtain the complete sets of invariants under Euclidean motions which uniquely, determines and reproduces the objects.Algorithms are also proposed to extract these invariants easily and robustly from local data on the surfaces by solving a system of linear equations. These invariants can then be used in segmentation and recognition of the objects.We also propose a novel surface model called fibre-bundle model for free surfaces, which represents an arbitrary surface as a local product between a base curve and a fibre curve. In particular, this model using fibres as 1-parameter group of linear Lie algebra or Hamilton Lie algebra is very efficient in the sense that the surface can be represented by a base curve and six invariants or 15 parameters, in the linear Lie algebraic case. The surface can be synthesised fastly without numerical error.