This document discusses how mathematics can be useful in various fields like crime detection, medicine, and finding landmines. It provides examples of how math is used in image processing and crime scene analysis using blurred photos. It also explains how CAT scans and MRI use math concepts like the Radon transform to reconstruct medical images from X-rays and detect densities inside the body without invasive procedures. Mathematics allows detection of hidden trip wires in photos that can identify locations of landmines.

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

4. Original
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

16. Object
Attenuation
R(ρ, θ)
θ
ρ

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!