1. CAN MATHS HELP IN THE FIGHT AGAINST CRIME?
Chris Budd

2. A crime has been committed
The police arrive in force
What challenges
Do they face?

3. How to interpret confusing data
How to find out what happened
How to interpret confusing data
How to store a mass of data and mine it for information
How to guard against fraud and keep things secure
Using maths they can
Reconstruct what happened inverse problems
Store and interpret data wavelets, probability, statistics
Transmit data in a secure way prime numbers 2,3,7,113,511

4. For example, you find some fingerprints
How likely was it to have come from a suspect?
These can be clear
Or blurred
Maths can reduce the amount of blurring
And contain lots of information
Maths gives a way of storing
Only the relevant information
And retrieve it using wavelets

5. How to be a mathematical detective?
What can we learn from the evidence?
For example, find the shape of an object only knowing its shadows
Nasa

6. Being a mathematical detective
Where has a bullet come from?
Agree on a physical model of the event
Understand what causes lead to what evidence
Given known evidence use maths to give possible causes.
Find the limitations and errors of the answer

7. Case study 1: Catching a speeding motorist..
Was the car speeding?
Evidence: collision damage, witness statements,
tyre skid marks

8. Evidence: s distance of skid
Cause: u speed
Other data: friction force
Model links cause to effect
Given the effect maths gives the cause
BUT Need to know accurately!!!

9. Case study 2: Deblurring a number plate
A short crime story
Burglar robs a bank
Escapes in a getaway car
Pursued by police
Nasa

10. GOOD NEWS
Police take a photo
BAD NEWS
Photo is blurred

11. SOLUTION Find a model of the blurring process Original image
Blurring function g
Original image f
Blurred image h
Blurring formula
Inverting the formula we can get rid the blur
BUT need to know the blurring function g

12. Inversion formula
h(x)
f(x)
An example of Image Processing

13. Case study 3: Who or what killed Tutankhamen?
Image processing solves an ancient ‘murder mystery’
Bible images
X-ray CAT scan of the mummy of Tutankhamen by Zahi Hawass reveals the probable cause of death ……
National Geographic

14. Object eg. King Tutenkhamen
Detector
X-Ray source X
Intensity R of X-ray at detector depends on width and density of object
Intensity X
Now look at LOTS of X-rays and find R for each one

15. The murder mystery resolved …
REMARKABLE FACT
If we can measure R accurately we can calculate the density f(x,y) of the object at any point
Knowing f tells us the structure of the object
Mathematical formula discovered by Radon (1917)
Took 60 years before computers and machines were developed by Cormack to use his formula
The murder mystery resolved …
Tutenkhamen died of a broken leg
University of St. Andrews

16. Radon’s formula Also used in Medical imaging Radon transform Inverse
Tumour images

17. Land mines are hidden in foliage and triggered by trip wires
CASE STUDY 5: A CRIME AGAINST HUMANITY
ANTI-PERSONEL LAND MINES
Land mines are hidden in foliage and triggered by trip wires
Land mines are well hidden .. we can use maths to find them

18. Find the trip wires in this picture

19. Digital picture of foliage is taken by camera on a long pole
Effect: Image intensity f
Cause: Trip wires .. These are like X-Rays
R(ρ,θ)
f(x,y) •
Radon transform • • y ρ • x θ
Points of high intensity in R correspond to trip wires
Isolate points and transform back to find the wires

20. Mathematics finds the land mines!
Who says that maths isn’t relevant to real life?!?