论文标题
从经典理论的角度来看，3D几何瞬间不变性
3D geometric moment invariants from the point of view of the classical invariant theory
论文作者
论文摘要
本文的目的是清除3D几何矩之间的联系与不变理论之间的联系，考虑到描述3D几何矩不变的问题是经典不变理论的问题。使用一个非凡的事实，即$ so（3）$和$ sl（2）$是本地同构的，我们将推导3D几何矩不变性的问题减少到了经典不变理论的众所周知的问题。我们提供了3D几何不变力矩计算的精确陈述，其中引入了同时3D几何矩不变的代数的概念，并证明它们是$ SL（2）$ - 几种binary形式的关节$ SL（2）$的代数。为了简化不变性的计算，我们从Lie Group $ so（3）$的动作进行了Lie代数$ \ Mathfrak {SL} _2 _2 $的动作。作者希望结果对图像分析和模式识别领域的研究人员有用。
The aim of this paper is to clear up the problem of the connection between the 3D geometric moments invariants and the invariant theory, considering a problem of describing of the 3D geometric moments invariants as a problem of the classical invariant theory. Using the remarkable fact that the groups $SO(3)$ and $SL(2)$ are locally isomorphic, we reduced the problem of deriving 3D geometric moments invariants to the well-known problem of the classical invariant theory. We give a precise statement of the 3D geometric invariant moments computation, introducing the notions of the algebras of simultaneous 3D geometric moment invariants, and prove that they are isomorphic to the algebras of joint $SL(2)$-invariants of several binary forms. To simplify the calculating of the invariants we proceed from an action of Lie group $SO(3)$ to an action of its Lie algebra $\mathfrak{sl}_2$. The author hopes that the results will be useful to the researchers in the fields of image analysis and pattern recognition.
