Multiple View Geometry(多视图几何)学习笔记（20）―摄像机模型之无穷远摄像机

无穷远摄像机是中心在无穷远平面上的摄像机。无穷远摄像机可以分为仿射摄像机和非仿射摄像机。

P$P$的最后一行P3T$P^{3T}$形如（0,0,1）$（0,0,1）$的摄像机。

P0=KP[I|C~]=Kr1Tr2Tr3Tr1TC~r2TC~r3TC~$$P_0=KP[I|\tilde{C}]=K\left(\begin{array}{cc}{r}^{1T}& r^{1T}\tilde{C}\\ r^{2T}& r^{2T}\tilde{C}\\ r^{3T}& r^{3T}\tilde{C}\end{array}\right)$$

riT$r^{iT}$是旋转矩阵的第i$i$行。而数值的d0=r3TC~$d_0=r^{3T}\tilde{C}$是世界原点到摄像机中心在主射线方向上的距离。
(3，4)$(3，4)$元素用世界原点到摄像机中心的深度dt$d_t$替代。
k=dt/d0$k=d_t/d_0$,在时间t$t$所产生的摄像机矩阵是:

Pt=Kdt/d0dt/d01r1Tr2Tr3Tr1TC~r2TC~dt=dtd0Kr1Tr2Tr3Td0/dtr1TC~r2TC~d0$$P_t=K\left(\begin{array}{ccc}{d}_t/d_0& & \\ & d_t/d_0& \\ & & 1\end{array}\right)\left(\begin{array}{cc}{r}^{1T}& r^{1T}\tilde{C}\\ r^{2T}& r^{2T}\tilde{C}\\ r^{3T}& d_t\end{array}\right)=\frac{d_t}{d_0}K\left(\begin{array}{cc}{r}^{1T}& r^{1T}\tilde{C}\\ r^{2T}& r^{2T}\tilde{C}\\ r^{3T}{d}_0/d_t& d_0\end{array}\right)$$

dt$d_t$ 趋向无穷大时:

P∞=Kr1Tr2T0Tr1TC~r2TC~d0$$P_{\mathrm{\infty }}=K\left(\begin{array}{cc}{r}^{1T}& r^{1T}\tilde{C}\\ r^{2T}& r^{2T}\tilde{C}\\ 0^T& d_0\end{array}\right)$$

X=(αr1+βr21)$$X=\left(\begin{array}{c}\alpha r^1+\beta r^2\\ 1\end{array}\right)$$

P0X=PtX=P∞X$P_{0}X=P_{t}X=P_{\mathrm{\infty }}X$
X$X$离该平面的垂直距离是Δ$\mathrm{\Delta }$:

X=(αr1+βr2+Δr31)$$X=\left(\begin{array}{c}\alpha r^1+\beta r^2+\mathrm{\Delta }{r}^3\\ 1\end{array}\right)$$

P0$P_0$和P∞$P_{\mathrm{\infty }}$影像到：

Xproj=P0X=Kx~y~d0+Δ,Xaffine=P∞X=Kx~y~d0$$X_{proj}=P_{0}X=K\left(\begin{array}{c}\tilde{x}\\ \tilde{y}\\ d_0+\mathrm{\Delta }\end{array}\right),X_{affine}=P_{\mathrm{\infty }}X=K\left(\begin{array}{c}\tilde{x}\\ \tilde{y}\\ d_0\end{array}\right)$$

x~=αr1TC~,y~=βr2TC~$\tilde{x}=\alpha r^{1T}\tilde{C},\tilde{y}=\beta r^{2T}\tilde{C}$。记标定矩阵为：

K=(K2×20~TX~01)$$K=\left(\begin{array}{cc}{K}_{2\times 2}& {\tilde{X}}_0\\ {\tilde{0}}^T& 1\end{array}\right)$$

X~affineX~0=d0+Δd0(X~projX~0)$${\tilde{X}}_{affine}{\tilde{X}}_0=\frac{d_0+\mathrm{\Delta }}{d_0}({\tilde{X}}_{proj}{\tilde{X}}_0)$$

P∞$P_{\mathrm{\infty }}$仿射近似真实摄像机矩阵P0$P_0$的效应是把点X$X$的图像径向地靠近或离开主点x~0${\tilde{x}}_0$，其加权因子是(d0+Δ)/d0=1+Δ/d0$(d_0+\mathrm{\Delta })/d_0=1+\mathrm{\Delta }/d_0$

仿射影像条件

X~affineX~proj=Δd0(X~projX~0)$${\tilde{X}}_{affine}{\tilde{X}}_{proj}=\frac{\mathrm{\Delta }}{d_0}({\tilde{X}}_{proj}{\tilde{X}}_0)$$

• 深度的起伏( Δ$\mathrm{\Delta }$)相对平均深度(d0$d_0$) 较小
• 点离主轴的距离较小

P∞=(K2×20^T0^1)(R^0Tt^1)=(K2×20^TX~01)(R^0T0^1)$$P_{\mathrm{\infty }}=\left(\begin{array}{cc}{K}_{2\times 2}& \hat{0}\\ {\hat{0}}^T& 1\end{array}\right)\left(\begin{array}{cc}\hat{R}& \hat{t}\\ 0^T& 1\end{array}\right)=\left(\begin{array}{cc}{K}_{2\times 2}& {\tilde{X}}_0\\ {\hat{0}}^T& 1\end{array}\right)\left(\begin{array}{cc}\hat{R}& \hat{0}\\ 0^T& 1\end{array}\right)$$

平行投影

1.平行投影矩阵

100010000001$$\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)$$

代替了有限摄像机的规范射影矩阵[I|0]$[I|0]$

2.标定矩阵

(K2×20^T0^1)$$\left(\begin{array}{cc}{K}_{2\times 2}& \hat{0}\\ {\hat{0}}^T& 1\end{array}\right)$$

代替了有限摄像机的K$K$。

3.主点无定义

正投影

P=r1Tr2T0Tt1t21$$P=\left(\begin{array}{cc}{r}^{1T}& t_1\\ r^{2T}& t_2\\ 0^T& 1\end{array}\right)$$

R$R$的三个参数，加上两个偏置参数t1$t_1$和t2$t_2$

缩放正投影

P=r1Tr2T0Tt