This paper develops a new mathematical framework that enables nonparametric joint semantic and geometric representation of continuous functions using data. The joint embedding is modeled by representing the processes in a reproducing kernel Hilbert space. The functions can be defined on arbitrary smooth manifolds where the action of a Lie group aligns them. The continuous functions allow the registration to be independent of a specific signal resolution. The framework is fully analytical with a closed-form derivation of the Riemannian gradient and Hessian. We study a more specialized but widely used case where the Lie group acts on functions isometrically. We solve the problem by maximizing the inner product between two functions defined over data, while the continuous action of the rigid body motion Lie group is captured through the integration of the flow in the corresponding Lie algebra. Low-dimensional cases are derived with numerical examples to show the generality of the proposed framework. The high-dimensional derivation for the special Euclidean group acting on the Euclidean space showcases the point cloud registration and bird's-eye view map registration abilities. An implementation of this framework for RGB-D cameras outperforms the state-of-the-art robust visual odometry and performs well in texture and structure-scarce environments.

翻译：本文开发了一个新的数学框架, 使使用数据的连续函数能够进行非对称联合语义和几何表达。 联合嵌入通过在复制内核的Hilbert空间中代表过程进行模型化。 函数可以在任意的平滑的柱体上定义, 利伊小组的行动可以对之进行匹配。 连续的函数允许注册独立于特定的信号分辨率。 框架是完全分析的, 对里格曼梯度和赫西安的封闭式衍生。 我们研究了一个更加专业化但广泛使用的案例, 利伊小组对功能采取行动是测量性的。 我们通过在数据上定义的两个函数之间实现内部产品最大化来解决问题, 而僵硬体动作组的连续动作则通过在相应的利格布拉的流中进行整合来捕捉。 低维度案例与数字示例一起产生, 以显示拟议框架的一般性。 在 Euclidean 空间上采取行动的特别尤克利德小组的高维度衍生数据, 展示了点云层登记和鸟眼视的地图登记能力。 在 R- GB 光学 和光学摄影外的图像环境中, 执行这一框架。