HOW MATHS CAN SAVE YOUR LIFE AND SEE

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

Original Blurred

SOLUTION Take the photo to a mathematician g(x) 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 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

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 scanners work by casting many shadows with X-rays and using maths to assemble these into a picture

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

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

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

Object Attenuation R(ρ, θ) ρ θ

Object Edge Attenuation R(ρ, θ) 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’s formula Radon transform Back Projection 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 R(ρ, θ) f(x, y) Radon transform y • • x Points of high intensity in R correspond to trip wires Isolate points and transform back to find the wires θ ρ

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