arXiv 1801.070992018-01-14math.CVmath.AP

关于Multiply-connected Domains的基于散度的距离函数

On Divergence-based Distance Functions for Multiply-connected Domains

Renjie Chen, Craig Gotsman, Kai Hormann

备注：: 7 pages

摘要：

给定有限连接的有界平面域$Ω$，可以定义从$x\inΩ$到$y\inΩ$的{ \it发散距离} $D(x,y)$，这考虑了域的复杂几何形状。该距离函数基于$f$ -发散的概念，该距离度量传统上用于测量两个概率分布之间的差异。在我们的例子中，相关的概率分布是$x$和$y$域的泊松核。证明了对于$χ^2$ -发散距离，$D$的$x$梯度与具有极点的格林函数$G(x,y)$的$x$梯度方向相反.由于$G$是调和的，这意味着$D$，像$G$一样，在$Ω$中具有单个极值，即在$D$为零的$y$处。因此，$D$可用于跟踪从$x$到$y$的~ $Ω$内的梯度下降路径，通过遵循$\nabla_x D(x,y)$，这与跟踪$G$的梯度相比具有显著的计算优势。该结果可用于复杂几何环境下的机器人路径规划。

英文摘要：

Given a finitely-connected bounded planar domain $Ω$, it is possible to define a {\it divergence distance} $D(x,y)$ from $x\inΩ$ to $y\inΩ$, which takes into account the complex geometry of the domain. This distance function is based on the concept of $f$-divergence, a distance measure traditionally used to measure the difference between two probability distributions. The relevant probability distributions in our case are the Poisson kernels of the domain at $x$ and at $y$. We prove that for the $χ^2$-divergence distance, the gradient by $x$ of $D$ is opposite in direction to the gradient by $x$ of $G(x,y)$, the Green's function with pole $y$. Since $G$ is harmonic, this implies that $D$, like $G$, has a single extremum in $Ω$, namely at $y$ where $D$ vanishes. Thus $D$ can be used to trace a gradient-descent path within~$Ω$ from $x$ to $y$ by following $\nabla_x D(x,y)$, which has significant computational advantages over tracing the gradient of $G$. This result can be used for robotic path-planning in complex geometric environments.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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