REMARKABLE FACT If we can measure R( ρ , θ ) accurately we can calculate The density f(x,y) of the object at any point
Mathematical formula discovered by Radon (1917)
Took 60 years before computers and machines were developed to use his formula
Machine inventor Cormack got a Nobel prize
Radon got nothing!
Process is called Back Projection
Radon transform Back Projection Radon’s formula Many other applications X-raying Mummies Monitoring Furnaces Remote Sensing
USING THE RADON TRANSFORM TO FIND ANTI-PERSONEL LAND MINES Land mines are hidden in foliage and triggered by trip wires Trip wires are well hidden – can they be quickly and safely detected
Find the trip wires in this picture
Digital picture of foliage is taken by camera on a long pole Image intensity f(x,y) • • • • Trip wires are like X-Rays Radon transform x y f(x,y) R( ρ , θ ) Points of high intensity in R correspond to trip wires θ ρ Isolate points and transform back to find the wires
Used by the Canadian Peace keeping forces Mathematics finds the land mines!