由平行平面的投影确定无穷远平面的单应矩阵

Determination of the Infinite Homography Through Scene Parallel Planes

在三维计算机视觉中,无穷远平面的单应矩阵扮演了极其重要的角色,可使众多视觉问题的求解得到简化.主要讨论如何利用平行平面的投影来求解两个视点间的无穷远平面的单应矩阵,用代数方法构造性地证明了下述结论:(1) 如果场景中含有一组平行平面,则可以通过求解一个一元4次方程来确定两个视点间的无穷远平面对应的单应矩阵;(2) 如果场景中含有两组平行平面,则可以线性地确定两个视点间的无穷远平面对应的单应矩阵.并对上述结果给出了相应的几何解释和具体算法.所给出的结果在三维计算机视觉,特别是摄像机自标定中具有一定的理论意义和应用价值.

The homography induced by the plane at infinity between two images, namely the infinite homography, plays a very important role in 3D computer vision since many vision problems could be substantially simplified by knowing it. Unlike homographies induced by ordinary planes which can usually be determined by correspondences of image points, the infinite homography must be determined indirectly since no real physical points lie on the plane at infinity. In this paper, how to determine the infinite homography through scene parallel planes is studied, and the following two conclusions are proved: (1) If only a pair of parallel planes is present in the scene, the infinite homography can be obtained by solving a 4th order polynomial, and at maximum, four possible solutions exist. (2) If at least two pairs of parallel planes exist in the scene, and if planes in different pairs are not parallel, then the infinite homograpgy can be linearly and uniquely determined. In addition, a geometric interpretation to the above results, and some practical algorithms are also provided. The proposed results in the paper are of interests in camera self-calibration and image based 3D reconstruction under both theoretical and practical standpoints.