This paper presents a Riemannian approach for free-space extraction and path planning using color catadioptric vision. The problem is formulated considering color catadioptric images as Riemannian manifolds and solved using the Riemannian Eikonal equation with an anisotropic fast marching numerical scheme. This formulation allows the integration of adapted color and spatial metrics in an incremental process. First, the traversable ground (namely free-space) is delimited using a color structure tensor built on the multi-dimensional components of the catadioptric image. Then, the Eikonal equation is solved in the image plane incorporating a generic metric tensor for central catadioptric systems. This built Riemannian metric copes with the geometric distortions in the catadioptric image plane introduced by the curved mirror in order to compute the geodesic distance map and the shortest path between image points. We present comparative results using Euclidean and Riemannian distance transforms and show the effectiveness of the Riemannian approach to produce safest path planning.

本文提出了一种利用彩色折反射视觉进行自由空间提取和路径规划的黎曼方法。该问题以彩色折反射图像作为黎曼流形而拟定，并使用具有各向异性快速行进数值格式的黎曼Eikonal方程解决。这种表述允许在增量过程中整合适应的颜色和空间度量。首先是可遍历的地面（即自由空间）使用建立在折反射图像多维分量上的颜色结构张量来定界。然后，在像平面中求解Eikonal方程，并为中央折反射系统合并通用度量张量。这种建立的黎曼度量可以解决曲面镜引入的折反射像平面中的几何畸变，从而计算出测地距离图和像点之间的最短路径。我们提出了使用欧几里得距离和黎曼距离变换的比较结果，并显示了黎曼方法产生最安全路径规划的有效性。