2D transformation
Isometric transformation
- rotation translation
- Preserve shape and area
- Usually describes the motion of rigid objects
similarity transformation
- Add scaling features based on isometric transformation
Projective transformation
- The intersection ratio of collinear and four-collinear points remains unchanged.
Affine transformation
- Area ratios, parallel relationships, etc. remain unchanged
- Affine transformation is a special projective transformation
Shadow cancellation point and shadow cancellation line
2D infinity point
- The intersection of two straight lines can be obtained by the cross product of the two straight lines, expressed as (x 1, x 2, z) (x_1, x_2, z)(x1,x2,z)。若 z = 0 z=0 z=0 , then the point is an infinite point (Euclidean coordinates are expressed as(x 1 z, x 2 z) (\frac{x_1}{z},\frac{x_2}{z})(zx1,zx2)）。
- A point at infinity becomes a point at infinite distance after projective transformation.
- An infinity point remains an infinity point after affine transformation.
2D infinity line
- The set of points at infinity lies on a line, which becomes the infinity line (can be expressed as linf = [ 0 0 1 ] l_{inf}=[0 \space 0 \space 1]linf=[0 0 1]）。
- A point at infinity becomes a point at infinite distance after projective transformation.
- An infinity point remains an infinity point after affine transformation.
line transformation
It is known that lx = 0 lx=0lx=0 , solvel ′ H x = 0 l'Hx=0l′Hx=0.
The derivation process is: known equation: l T x = 0 Add inverse matrix: l TH − 1 H x = 0 Ungroup: ( H − 1 l ) T ( H x ) = 0 Obtain: l ′ = H
− T l = 0 \begin{equation} \begin{split} Known equation: l^{T}x=0 \\ Add inverse matrix: l^{T}H^{-1}Hx=0 \\ Split Group: ({H^{-1}l})^T(Hx)=0 \\ Obtain: l'=H^{-T}l=0 \\ \end{split} \end{equation}Known equation: lTx=0Add the inverse matrix: lTH−1Hx=0Disassembly: ( H−1l)T(Hx)=0Available: l′=H−Tl=0
The infinity line is expressed as [ 0 0 1 ] \begin{bmatrix} 0\\ 0\\ 1 \end{bmatrix}
001
- Infinity line perspective (projection) transformation H = [ A tvb ] H=\begin{bmatrix} A & t\\ v &b \end{bmatrix}H=[Avtb] is no longer an infinity line.
- Infinity line affine transformation H = [ A t 0 b ] H=\begin{bmatrix} A & t\\ 0 &b \end{bmatrix}H=[A0tb] is the infinity line.
Points and areas in space
- 面： a x + b y + c z + d = 0 ax+by+cz+d=0 ax+by+cz+d=0
- 点： x ∞ = [ a b c 0 ] x_{\infty}=\begin{bmatrix} a\\ b\\ c\\ 0 \end{bmatrix} x∞= abc0
vanishing point
- The projection point p ∞ of the infinity point in the three-dimensional space on the two-dimensional pixel plane = [ abc ] p_{\infty}=\begin{bmatrix} a\\ b\\ c \end{bmatrix}p∞= abc 。
- Shadow extinction point = camera internal parameters * straight line direction.
shadow elimination line
- A collection of vanishing points.
- Identifying shadow cancellation lines helps reconstruct three-dimensional scenes.
Relationship with plane normal vector:
plane normal vector = camera internal parameter transpose matrix * shadow cancellation line
infinity plane
- Parallel planes at infinity are compared to a common line - the infinity line.
- The set of 2 or more infinite straight lines is defined as an infinite plane.
Single view reconstruction
step
- Calibrate camera internal parameters
- Restore information of 3D scene
- Reconstruction
Disadvantages: manual selection of shadow cancellation points and shadow cancellation lines; requires scene priori; the actual proportion of the scene cannot be restored.