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- 1. HOW MATHS CAN SAVE YOUR LIFE AND SEE THE WORLD IN A DIFFERENT WAY Chris Budd
- 2. SOME COMMON VIEWS OF MATHEMATICS <ul><li>MATHS IS HARD </li></ul><ul><li>MATHS IS BORING </li></ul><ul><li>MATHS HAS NOTHING TO DO WITH REAL LIFE </li></ul><ul><li>ALL MATHEMATICIANS ARE MAD! </li></ul>BUT I CAN SHOW YOU THAT MATHS IS IMPORTANT IN CRIME DETECTION MEDICINE FINDING LANDMINES AND EVEN DISNEYLAND !!!!!!!!!
- 3. MATHS AND CRIME A short mathematical story <ul><li>Burglar robs a bank </li></ul><ul><li>Escapes in getaway car </li></ul><ul><li>Pursued by police </li></ul><ul><li>GOOD NEWS Police take a photo </li></ul><ul><li>BAD NEWS Photo is blurred </li></ul>
- 4. Original Blurred
- 5. SOLUTION Take the photo to a mathematician Original f(x) Blurring h(x) = f(x)*g(x) <ul><li>Maths gives a formula for blurring convolution </li></ul><ul><li>By inverting the formula we can get rid of the blur </li></ul>g(x)
- 6. Processed image : Image Processing
- 7. MATHS AND PICTURES <ul><li>PICTURES AND IMAGES ARE ALL AROUND US </li></ul><ul><li>TV </li></ul><ul><li>DVD </li></ul><ul><li>COMPUTER GRAPHICS </li></ul><ul><li>SPECIAL EFFECTS </li></ul>IMAGES ARE STORED AS NUMBERS USING THESE NUMBERS WE CAN PROCESS THE PICTURES BY USING MATHEMATICS
- 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
- 10. MATHS AND MEDICINE Modern medicine has been transformed by methods of seeing Inside you without cutting you open! <ul><li>Ultra sound : sound waves </li></ul><ul><li>MRI : magnetism </li></ul><ul><li>CAT scans: X rays </li></ul>ALL USE MATHS TO WORK!!
- 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
- 18. REMARKABLE FACT If we can measure R( ρ , θ ) accurately we can calculate The density f(x,y) of the object at any point <ul><li>Mathematical formula discovered by Radon (1917) </li></ul><ul><li>Took 60 years before computers and machines were developed to use his formula </li></ul><ul><li>Machine inventor Cormack got a Nobel prize </li></ul><ul><li>Radon got nothing! </li></ul><ul><li>Process is called Back Projection </li></ul>
- 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!

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