# Remarks on Correlation Tracking

The problem is to track the motion of part of a field of view. Let us assume that the scene is a two-dimensional picture in a plane perpendicular to the roll axis. (these simplifying assumptions, of course, are a main problem in estimating how the system works in real life). So we can think of the picture as a function f(x,y) in some plane. Now suppose that at time to the scene is fo(x,y) and at some time later it has moved, and is ft(x,y). Suppose also that the scene has not changed, but has only been moved rigidly in the plane. Then an elegant mathematical way to estimate this motion is to compute the cross-correlation of the original and current picture. First let us review a basic simple mathematical fact. Given any function f(x) and any displacement {triangle}, it is true that sf(x)f(x)>_sf(x)f(x+triangle).