Known Motion Geometry

Suppose we know that there's a plane in our image, and so the motion model has to follow that of a plane.

1. Perspective

$$aX + bY + cZ + d = 0$$

The velocities: $$u(x,y) = a_1 + a_2x + a_3y +a_7x^2 + a_8xy \\ v(x,y) = a_4 + a_5x + a_6y +a_7xy + a_8y^2$$

2. Orthographic

If our plane lies at a sufficient distance from the camera, the distance between points on the plane is minuscule relative to that distance. So we can disregard the depth in this case. Thus we need 3 correspondence points to solve: $$u(x,y) = a_1 + a_2x + a_3y\\ v(x,y) = a_4 + a_5x + a_6y$$ This is an affine transformation.