Howmathscansaveyourlife
- 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
- 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
- 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!