東京大学 情報理工学系研究科 創造情報学専攻 2009年8月実施 筆記試験 第2問

标签：

• Tokyo-University
• Projection-and-Reconstruction

Author

itsuitsuki

Description

Answer the following questions.

(1) As Figure 1 shows, an orthogonal coordinate frame $\Sigma_C$ of a camera with the lens axis $CZ$ and the projection plane $S$ is placed at the point $C$. The plane $S$ is orthogonal to the lens axis $CZ$ and has the distance $f$ from $C$. The point $Q$ is projected to the point $P$ on the plane $S$ with the coordinates $\mathbf{P}_C=(P_X,P_Y,f)^t$ in $\Sigma_C$. The coordinates of three orientation vectors $CX,CY,CZ$ are described as $\mathbf{X}_\mathbf{W}=(X_X,X_Y,X_Z)^t$, $\mathbf{Y}_\mathbf{W}=(Y_X,Y_Y,Y_Z)^t$ and $\mathbf{Z}_\mathbf{W}=(Z_X,Z_Y,Z_Z)^t$, and the position vector of $C$ is $\mathbf{C}_\mathbf{W}=(C_X,C_Y,C_Z)^t$ in the coordinate frame $\Sigma_W$. The superscript $t$ indicates transpose.

Assume the distance from $C$ to $Q$ is $d$, show the vector $\mathbf{Q}_C$ from the point $C$ to the point $Q$ with $\mathbf{P}_C$ and $d$. When the vector $\mathbf{Q}_\mathbf{W}$ is the position vector of $Q$ and the rotation matrix of $\Sigma_C$ is $R_C$ in $\Sigma_W$, we have $\mathbf{Q}_\mathbf{W}=R_C\mathbf{Q}_C+\mathbf{C}_\mathbf{W}$. Show the elements of the rotation matrix $R_C$.

(2) When we observe the point $Q$ from the camera placed at a point $A$, the projection point is $\mathbf{P}_A=(a_X,a_Y,f)^t$ in the camera coordinate frame $\Sigma_A$. Then, we translate the camera with the distance $\ell$ along the axis $X$ to a point $B$ and rotate it around the axis $Y$ of the translated coordinate frame with the angle $\alpha$. The rotated camera coordinate frame is $\Sigma_B$. The projection point becomes $\mathbf{P}_B=(b_X,b_Y,f)^t$ in $\Sigma_B$. Show the method to get the distance $d_A$ from $A$ to $Q$ and the distance $d_B$ from $B$ to $Q$, when $\mathbf{P}_A=\mathbf{P}_B$ is obtained. Assume there is no error in the translation and rotation, and the $XZ$ planes of $\Sigma_A$ and $\Sigma_B$ are aligned in the same plane.

(3) Two cameras are placed at the points $M$ and $N$, respectively. Let the position vectors of $M$ and $N$ be $\mathbf{M}_\mathbf{W}$ and $\mathbf{N}_\mathbf{W}$ and the rotation matrices be $R_M$ and $R_N$. The projection points of $Q$ on these two cameras become $\mathbf{P}_M$ and $\mathbf{P}_N$. As the position vectors $\mathbf{Q}_\mathbf{M}$ and $\mathbf{Q}_\mathbf{N}$ of the point $Q$ are the same in the coordinate frame $\Sigma_W$. Denote the condition which the projection points $\mathbf{P}_M$ and $\mathbf{P}_N$ should satisfy.

(4) Assume the projection points are described in an array and the condition in (3) is not satisfied. Let the evaluation function be $J=|(R_M\mathbf{Q}_M+\mathbf{M}_\mathbf{W})-(R_N\mathbf{Q}_N+\mathbf{N}_\mathbf{W})|^2$, and consider minimizing $J$ to get $\mathbf{Q}_\mathbf{W}$. Let $d_M$ and $d_N$ be the distances from $M$ and $N$ to $Q$, respectively, when $J$ is minimized. Denote $d_M$ and $d_N$. Then explain the method to get $\mathbf{Q}_\mathbf{W}$ in $\Sigma_W$ with $d_M$, $d_N$.

(5) Explain the best arrangement to minimize errors when we measure a three dimensional position by two cameras such as (3).

题目描述

回答以下相机投影与三维重建问题。

1. 如图 1，在点 $C$ 建立相机正交坐标系 $\Sigma_C$，镜头轴为 $CZ$，投影平面 $S$ 与 $CZ$ 正交且距 $C$ 为 $f$。空间点 $Q$ 投影到 $S$ 上的点 $P$，其相机坐标为 $\mathbf P_C=(P_X,P_Y,f)^t$。在世界坐标系 $\Sigma_W$ 中，相机三轴方向向量分别为

   $$\mathbf X_W=(X_X,X_Y,X_Z)^t,\quad \mathbf Y_W=(Y_X,Y_Y,Y_Z)^t,\quad \mathbf Z_W=(Z_X,Z_Y,Z_Z)^t,$$

   光心位置为 $\mathbf C_W=(C_X,C_Y,C_Z)^t$，上标 $t$ 表示转置。若 $C$ 到 $Q$ 的距离为 $d$，用 $\mathbf P_C,d$ 表示从 $C$ 指向 $Q$ 的向量 $\mathbf Q_C$。又已知

   $$\mathbf Q_W=R_C\mathbf Q_C+\mathbf C_W,$$

   写出旋转矩阵 $R_C$ 的各元素。

2. 相机位于 $A$ 时，$Q$ 的投影为 $\mathbf P_A=(a_X,a_Y,f)^t$。随后相机沿 $X$ 轴平移距离 $\ell$ 到 $B$，再绕平移后坐标系的 $Y$ 轴旋转角 $\alpha$，得到坐标系 $\Sigma_B$，投影为 $\mathbf P_B=(b_X,b_Y,f)^t$。在观测到 $\mathbf P_A=\mathbf P_B$ 时，说明如何求 $A,Q$ 间距离 $d_A$ 和 $B,Q$ 间距离 $d_B$。假设平移、旋转无误差，且 $\Sigma_A,\Sigma_B$ 的 $XZ$ 平面共面。

3. 两台相机分别位于 $M,N$，世界坐标位置为 $\mathbf M_W,\mathbf N_W$，旋转矩阵为 $R_M,R_N$，对 $Q$ 的投影为 $\mathbf P_M,\mathbf P_N$。因为由两台相机得到的 $Q$ 的世界坐标应相同，写出 $\mathbf P_M,\mathbf P_N$ 必须满足的条件。

4. 若投影点以数组给出且不严格满足第 3 问条件，定义

   $$J=\left|(R_M\mathbf Q_M+\mathbf M_W)-(R_N\mathbf Q_N+\mathbf N_W)\right|^2.$$

   通过最小化 $J$ 求 $\mathbf Q_W$。设最小时从 $M,N$ 到 $Q$ 的距离分别为 $d_M,d_N$，求 $d_M,d_N$，并说明如何由二者得到世界坐标系中的 $\mathbf Q_W$。

5. 说明用第 3 问所述双目相机测量三维位置时，为减小误差应如何布置两台相机。