Motions, depends on the Motion models

For an image with width $W$ and height $H$, the equations mapping integer pixel coordinates $\bar X=(\bar x,\bar y)$ to normalized device coordinates $x=(x,y)$ are

where $S=max(W,H)$

Basic set

2D motions

• 2D translations : $x′=x+t$ or $x′=[IT] \tilde x$ , where $\tilde x=(x,y,1)$: the homogeneous or projective 2D coordinate
• 2D rotation+translation : $x′=Rx+t$ (rigid body morion or 2D Euclidean transformation), Euclidean distances are preserved

• Scaled rotation : $x′=sRx+t$ (similarity transform), angles (between lines) are preserved
• Affine : $x′=Ax$
• Projective : operates on homogeneous coordinate x,x', i.e., $x′∼Hx$ (perspective transform or homography), straight line are preserved

3D transformations

• The process of central maps 3D coordinates p = (X,Y,Z) to 2D coordinates x = (x,y,1) through a pinhole at the camera origin onto a 2D projection plane a distance f along the z axis,

  $x=f \frac XZ,y=f \frac YZ$