How maths can save you rlifePresentation Transcript
HOW MATHS CAN SAVE YOUR LIFE AND SEE THE WORLD IN A DIFFERENT WAY Chris Budd
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 !!!!!!!!!
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
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
Processed image : Image Processing
MATHS AND PICTURES
PICTURES AND IMAGES ARE ALL AROUND US
IMAGES ARE STORED AS NUMBERS USING THESE NUMBERS WE CAN PROCESS THE PICTURES BY USING MATHEMATICS
SOME APPLICATIONS PRODUCING THE PICTURES IN THE FIRST PLACE TRANSMITTING THE PICTURES WITHOUT MISTAKES Error Correcting Codes Galois
SOME MORE APPLICATIONS DEBLURRING ORIGINALS FINDING THINGS HIDDEN IN AN IMAGE Edges Brains Landmines
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!!
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
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
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
MOVE SOURCE AND DETECTOR AROUND GET SHADOWS OF THE OBJECT FROM MANY ANGLES AND MEASURE X-RAY INTENSITY
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
Object Attenuation R( ρ , θ ) θ ρ
Object Attenuation R( ρ , θ ) Edge Edge Edge Edge
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!