從影像決定基本二次曲面物的定位及外形參數

Abstract

本論文探討電腦視覺中二次曲面物基本物件的幾何特性及定位和重建的方法。我們提出一些新的觀念及方法，並提出定理加以證明。根據文獻資料，85%的工業零件可以用少樣的線、平面、圓、球面、圓柱、及角錐來表示。我們以線為基礎，首先指出消失點存在有一個對偶的特性及對偶的消失線等幾何特性。。 接著我們探討這些基本物件的定位及重建問題。目標是求得簡易的正解而不用近似解。這些物體上沒有特殊的點或線。因此文獻上使用點、線對應的方法不能使用。有些方法使用shape from shading、grid encoding、ranging finding、標示特徵等方法及相機安設限制，在實際使用時並不實用。在透視投影下，由於透視的失真導致定位及重建問題的困難且解為非線性。我們提出了利用重投影(reprojection)的方法簡化問題，並從原來的影像得到一新的影像。因而使得物體的定位及重建解簡易方便。 對平面物圓的定位而言，我們提出一個新幾何的特性--最大張角(the largest apex angle (LAA))來求得圓的位置及方向參數。這個方法可以處理退化(degenerated)的問題。又在橢圓太扁的情形下可以免除套合方法的誤差。它是一個通解，不論圓的投影形狀都可以得到正解。 將問題擴展到3D的圓柱體，其3D的定位參數也可以用類似的方法求得。我們的方法使用線性的向量運算及幾何的對稱性得解而更方便簡易。對於重建的問題，我們利用一平面鏡的對稱性，可以從單一影像，得到四個3D的切面而得到外形及定位的解。我們也考慮多個圓柱體在重建時對應及遮蔽的問題。相同的原理可以用在球及角錐的定位及重建。 論文也提出實驗，結果說明我們的方法是有效的。我們也藉由雜訊來分析我們方法的靈敏度，結果顯示我們的方法足夠穩健。

In this dissertation we perform pose determination and shape reconstruction to obtain 3D parameters of primitive quadric objects from a single image. To begin with we present the relevant geometric concepts and theorems which are bases of pose determination and shape reconstruction. This dissertation consists of five parts. In the first part we infer the 3D information of parallel lines from an image. In this manner, the dual property of the vanishing point, named the dual vanishing point property, is investigated and we obtain several properties in perspective geometry. In the same manner, the dual vanishing line can also be obtained. In the second part we deal with the pose parameters of a circle. In a circle feature, a general and geometric method is proposed for the pose estimation of a circle. The circle pose parameters are obtained directly through the determination of the largest apex angle (LAA) at the apex of the viewing cone. We then present theorems on the viewing geometry which are essential to the pose determination. We obtain the closed form and exact solution(s). Besides, it has a clear geometric interpretation. In the third part the same concept can also be used to provide a better approach to solving the pose determination problem for a cylinder. The same difficulties exist in the pose determination of cylinders. Also, extracting useful feature points from a right cylinder image is also relatively difficult. We also can obtain the exact and analytical solution. We use a special vector for the reprojection transformation and obtain a new image (the canonical image). The points on the symmetrical axis of the pattern are adequate for pose determination of a cylinder. In the fourth part we discuss the reconstruction of a cylinder from an image. We use a planar mirror to generate the stereo effect. From the double (or compound) image we can use four 3D tangent planes to the real cylinder and minimal least square error for the cylinder radius. The location, orientation of the axis, and the height of the cylinder can then be computed. The correspondences between real/reflected cylinder in this setup will be studied. Moreover, multiple cylinders with occlusion are considered in reconstruction. In the fifth part we extend our method to the pose determination and shape reconstruction of spheres and cones by using the same principles such as the LAA, the reprojection transformation, and four tangent planes. Finally, the experiments are included to illustrate the validness and effectiveness of our methods. We also analyze the noise sensitivity of our method. The results show that our method is robust.