CAN MATHS HELP IN THE FIGHT AGAINST CRIME?

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 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. But what happened given the evidence? Inverse problems … what causes lead to what effects? For example, find the shape of an object only knowing its shadows Nasa

6. How to solve an inverse problem • Agree on a physical model of the event • Understand what causes lead to what effects • Given known effectsuse mathsto give possible causes. • Find the limitations and errors of the answer Different causes may give very similar effects (think of shadows!) Object Shadow

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

8. Evidence: sdistance of skid Cause: uspeed Other data: friction force Model links cause to effect Given the effect mathsgives thecause 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 Blurring function g Blurred image h Original image f • 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 Hawassreveals the probable cause of death …… National Geographic

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

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

16. Object Edge Edge Attenuation R(ρ, θ) Edge Edge

17. REMARKABLE FACT If we can measure R(ρ, θ) 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

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

19. 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

20. Find the trip wires in this picture

21. Digital picture of foliage is taken by camera on a long pole Effect: Image intensityf 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

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