The motion field

When an object moves in front of a camera, there is a corresponding change in the image. Thus, if a point p_o on a object moves with a velocity v_o, then the imaged point p_i can be assigned a vector v_i to indicate its movement on the image plane. The collection of all these vectors forms the motion field.

 

Figure: Object motion creates a motion field in the image.

If we are only dealing with rigid body translations and rotations, then the motion field will be continuous except at the silhouette boundaries of objects.

 

Figure: The motion field of a moving square.

In the case of pure camera translation, the direction of motion is along the projection ray through that image point from which (or towards which) all motion vectors radiate. The point of divergence (or convergence) of all motion field vectors is called the focus of expansion FOE (or focus of contraction FOC). Thus, in the case of divergence we have forward motion of the camera, and in the case of convergence, backwards motion.

If we take the axis of camera translation as the camera baseline in stereo, then every projection of a fixed scene point must translate along an epipolar line, and all such lines converge at the epipole, which is just the FOE.