"Computer Vision 3D Reconstruction" Note 3-Single Vision Geometry

Notes for the course "3D Reconstruction of Computer Vision (Sfm and SLAM Core Algorithms)" taught by Beiyou teacher Lu Peng

3. Single View Geometry

3.1 2D

straight line

Use l = ( a , b , c ) T l=(a,b,c)^T for a straight line on the plane$l=(a,b,c{)}^T$ to represent, point$x=(x,y,1{)}^T$ , the point on the line:
$$l^{T}x=$$The normal vector of the$$0$$
$$\vec{n}=(b,-a)$$ The direction vector of the straight line:
$$\vec{d}=(a,b)$$ Two straight lines$l,l^{\prime }$ intersection point:
$$x=l\times l^{\prime }$$

point line at infinity

Two parallel lines intersect at the point at infinity, the straight line ll can be obtainedInfinity point of$l$
$$x_{\infty }=(b,-a,0{)}^T$$

$x_{\infty }=l\times l^{\prime }=(a,b,c_1)\times (a,b,c_2)=(b{c}_2-b{c}_1,-(a{c}_2-a{c}_1),ab-ab)=(b,-a,0)$

All points at infinity have the shape $(x,y,0{)}^T$ , the resulting infinite line is:
$$l_{\infty }=(0,0,1{)}^T$$

3.2 3D

flat

Plane Π = ( a , b , c , d ) T \Pi=(a,b,c,d)^T in 3-dimensional space$Pi=(a,b,c,d{)}^T$，点$x=(x,y,z,1{)}^T$ , a point on the plane:
$${Pi}^{T}x=$$The normal vector of the$$0$$
$$n=(a,b,c{)}^T$$

straight line

A straight line in three-dimensional space can be obtained by intersecting two planes. This representation is a bit cumbersome. Here we only care about the direction of the straight line. Use the direction of the straight line $d$ to represent straight line$l$：
$$d=(a,b,c{)}^T$$

infinity

straight line llThe infinity point of$l$
$$l_{\infty }=(a,b,c,0{)}^T$$

hePoints on$l$$x=(a\lambda ,bl,cl,1{)}^T$，即$x=(a,b,c,\frac{1}{l}{)}^T$
$x_{\infty }={\lim }_{\lambda \to \infty }(a,b,c,\frac{1}{l}{)}^T=(a,b,c,0{)}^T$

The infinite surface formed by all infinite points:
$${Pi}_{\infty }=(0,0,0,1{)}^T$$

3.3 Vanishing points and vanishing lines

In the camera coordinate system, the line llThe direction of$l$$d=(a,b,c{)}^T$ , whose infinity point is$x_{\infty }=(a,b,c,0{)}^T$ , the point projected on the image becomesthevanishing point$v$：
$$v=Kd$$ $$d=\frac{K^{-1}}{\Vert K^{-1}v\Vert }$$

$v=M{x}_{\infty }=K[I,0]\left[\begin{array}{c}a\\ b\\ c\\ 0\end{array}\right]=K\left[\begin{array}{c}a\\ b\\ c\end{array}\right]=Kd$

Plane $The\; normal\; vector\; of\; \Pi$ is$n$ , its infinity line$l_{\infty }$The projection on the image is the shadow line $l_h$：
$$n=K^T{l}_h$$The vanishing line is composed of vanishing points.

$l_{\infty }$Point on $P=(a,b,c,0{)}^{}$The vanishing point vv${}^{of\; T}$ on the image$v$，$l_h^{T}v=0$
所以$l_h^{T}Kd=0$
planeΠ \PiDirection ddof a line on$\Pi$$d$ are both perpendicular to$n$，有$n^{T}d=0$
$\Rightarrow n=K^T{l}_h\square$

3.4 Single View Reconstruction

Two straight lines $d_1,d_2$The included angle is $\theta$ , shadow vanishing points$v_1,v_2$，令$oh=(K{K}^T{)}^{-1}$：
$$\cos i=\frac{v_1^T\omega v_2}{\sqrt{v_1^T\omega v_1}\sqrt{v_2^T\omega v_2}}$$

$$\cos i=\frac{d_1^T{d}_2}{\Vert d_1^T{d}_2\Vert }=\frac{(K^{-1}{v}_1{)}^T}{\Vert K^{-1}{v}_1\Vert }\frac{K^{-1}{v}_2}{\Vert K^{-1}{v}_2\Vert }=\frac{v_1{K}^{-T}{K}^{-1}{v}_2}{\sqrt{v_1{K}^{-T}{K}^{-1}{v}_1}\sqrt{v_2{K}^{-T}{K}^{-1}{v}_2}}=\frac{v_1^T\omega v_2}{\sqrt{v_1^T\omega v_1}\sqrt{v_2^T\omega v_2}}$$

When the line is vertical, $\cos i=0\Rightarrow$
$$v_1^T\omega v_2=0$$

single vision calibration

Find the image of three mutually perpendicular planes in space (general houses have this property), and find the shadow vanishing points $v_1,v_2,v_3$，则得到方程：
$$\{\begin{array}{l}{v}_1^T\omega v_2=0\\ v_2^T\omega v_3=0\\ v_3^T\omega v_1=0\end{array}$$If the camera has zero tilt and square pixels, then $K$ has three degrees of freedom, so$\omega$ has three degrees of freedom, then the equations can be solved to obtain the internal parameters of the camera.

single vision reconstruction

Get $After\; K$ , you can find a planar shadow line$l_h$, by $n=K^T{l}_h$The three-dimensional information of the plane can be recovered.
Single-view reconstruction needs to know some prior information, such as parallel lines and vertical lines, and these need to be manually marked on the image, which is suitable for scenes with a large number of parallel lines.