Particle tracking using IDL -- John C. Crocker and Eric R. Weeks

The radius is calculated in a funny way. The moment of inertia for an object is defined as

I = sum(mass_i * r_i^2) = (total mass)*(Rg)^2
The last term defines the radius of gyration Rg. The sum is over i, each pixel with a given "mass" (its brightness) at a given position r_i, measured from the center of mass of the particle. Rewriting, we get that
(Rg)^2 = sum(mass_i*r_i^2) / (total mass).
But we can do better than that... thanks to David Grier for thinking of this, and thanks to John Crocker for explaining to me (Eric) what is going on. The calculation that feature does is actually
(Rg)^2 = sum(mass_i*(r_i^2 + 1/6)) / (total mass)
The factor "1/6 pix^2 * mass" is the moment of inertia of a square prism with that mass and a square side length 1 pixel. So, we are not just thinking of the image as a set of point masses on a square grid. Rather, we are thinking of the image as a continuous function approximated as a bunch of square prisms.

Note athat what is returned is the square of the radius, but throughout the tutorial we always call this "radius". Note also that the same definition is used in the 3D code (feature3d.pro).