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

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## A note about the "radius"

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).

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