Transcript HOW MATHS CAN SAVE YOUR LIFE AND SEE THE WORLD IN A DIFFERENT WAY Chris Budd.

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
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
scanner
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
Edge
R(ρ, θ)
Attenuation
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’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