1. HOW MATHS CAN SAVE YOUR LIFE AND SEE THE WORLD IN A DIFFERENT WAY Chris Budd

4. Original Blurred

6. Processed image : Image Processing

8. SOME APPLICATIONS PRODUCING THE PICTURES IN THE FIRST PLACE TRANSMITTING THE PICTURES WITHOUT MISTAKES Error Correcting Codes Galois

9. SOME MORE APPLICATIONS DEBLURRING ORIGINALS FINDING THINGS HIDDEN IN AN IMAGE Edges Brains Landmines

11. WHAT IS A CAT SCAN?? CAT = Computerised axial tomography Based on X-Rays discovered by Roengten X-Rays cast a shadow GOOD for looking at bones BAD for looking at soft tissue

12. USING MODERN MATHS WE CAN DO A LOT BETTER Modern CAT scanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

13. X-Ray source Object Detector X Intensity of X-ray at detector depends on width of object We can find the thickness … can we find the shape? Intensity X

14. MOVE SOURCE AND DETECTOR AROUND GET SHADOWS OF THE OBJECT FROM MANY ANGLES AND MEASURE X-RAY INTENSITY

15. X-Ray Object ρ : Distance from the object centre θ : Angle of the X-Ray Measure attenuation of X-Ray R( ρ , θ ) Source Detector

16. Object Attenuation R( ρ , θ ) θ ρ

17. Object Attenuation R( ρ , θ ) Edge Edge Edge Edge

19. Radon transform Back Projection Radon’s formula Many other applications X-raying Mummies Monitoring Furnaces Remote Sensing

20. 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

21. Find the trip wires in this picture

22. 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

23. Used by the Canadian Peace keeping forces Mathematics finds the land mines!