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

近年来，三维图像和三维计算机图形学在虚拟环境、多媒体通信和数字内容等方面发挥着重要的作用。因此，为了有效地表示、编码、识别和合成三维物体，需要一个功能强大、高效的三维自由曲面模型.本文提出了一种利用切向和法向李代数表示三维物体在欧氏或仿射运动下不变的曲面模型的全局方法.作为李群的整体形状完全由这些李代数中的纯局部信息来描述。特别是，我们专注于线性李代数和哈密顿李代数，它可以表示代数形状和更广泛的一类非代数形状以及。我们得到了在欧氏运动下唯一确定和再现物体的不变量的完整集合，并提出了通过求解线性方程组从表面上的局部数据中简单而鲁棒地提取这些不变量的算法。我们还提出了一种新的自由曲面模型--纤维束模型，它将任意曲面表示为基曲线和纤维曲线的局部乘积。特别地，该模型使用纤维作为线性李代数或汉密尔顿李代数的1-参数群是非常有效的，因为在线性李代数的情况下，表面可以由基曲线和6个不变量或15个参数表示。该曲面可快速合成，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.