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

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)

Processed image : Image Processing

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 PROCESSTHE PICTURES BY USING MATHEMATICS

SOME APPLICATIONS

PRODUCING THE PICTURES IN THE FIRST PLACE

TRANSMITTING THE PICTURES WITHOUT MISTAKES

Error Correcting CodesGalois

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

AttenuationR(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!