How maths can save you rlife
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How maths can save you rlife Presentation Transcript

  • 1. HOW MATHS CAN SAVE YOUR LIFE AND SEE THE WORLD IN A DIFFERENT WAY Chris Budd
  • 2. SOME COMMON VIEWS OF MATHEMATICS
    • MATHS IS HARD
    • MATHS IS BORING
    • MATHS HAS NOTHING TO DO WITH REAL LIFE
    • ALL MATHEMATICIANS ARE MAD!
    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
    • Burglar robs a bank
    • Escapes in getaway car
    • Pursued by police
    • GOOD NEWS Police take a photo
    • BAD NEWS Photo is blurred
  • 4. Original Blurred
  • 5. SOLUTION Take the photo to a mathematician Original f(x) Blurring h(x) = f(x)*g(x)
    • Maths gives a formula for blurring convolution
    • By inverting the formula we can get rid of the blur
    g(x)
  • 6. Processed image : Image Processing
  • 7. MATHS AND PICTURES
    • PICTURES AND IMAGES ARE ALL AROUND US
    • TV
    • DVD
    • COMPUTER GRAPHICS
    • SPECIAL EFFECTS
    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!
    • Ultra sound : sound waves
    • MRI : magnetism
    • CAT scans: X rays
    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
    • 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
  • 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!