1. HOW CAN MATHS HELP TO SOLVE CRIME?
CAN MATHS TELL WHAT HAPPENED?
Chris Budd

2. SOME COMMON VIEWS ABOUT MATHS
Maths is boring
Maths is exact
Maths is irrelevant
Maths is scary
Apart from the last point ………
Maths seems very different from the confused, unpredictable and
Highly relevant business of fighting crime
But I will show that maths is very relevant to ….
Accident and number-plate reconstruction, finding a poison, an ancient murder mystery and detecting landmines

3. SOME CHALLENGES FACING THE POLICE
How to find out what happened at the scene of a crime or accident
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
Mathematics gives a way to
Reduce the ambiguity in reconstructing what happened inverse problems
Store and interpret data, wavelets, probability, statistics
Transmit data in a secure way, prime numbers

4. An example: finger prints
Can we say how likely it was to have come from a particular suspect?
These can be clear
Or blurred
Using maths we 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. What happened at a crime or accident given the evidence?
Inverse problems … mathematical detective problems …what causes lead to what effects?
Eg. trying to find the shape of an object only knowing its shadows
Nasa
Can we do this at all?
What sort of errors are we likely to make?
How much extra information is needed?

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

7. Other examples of inverse problems
Nasa
Tumour images
Remote sensing the land
Medical imaging
Nasa
Nasa
Remote sensing the sea
Seismograph … prospecting for oil

8. Case study 1: Mechanics applied to crime fighting
Several applications of mechanics in crime fighting
Finding the trajectories of bullets given angle and depth of impact
Reconstructing what happened in an accident ..
Was a car speeding?
Evidence: tyre skid marks, collision damage, radar, witnesses
istockphoto
Causes of marks: speed of the car, braking force, friction with the road, impacts with other vehicles

9. Evidence: s distance of skid Cause: u speed
Other data: coefficient of friction times braking efficiency, g: acceleration due to gravity
Model links cause to effect
Given the effect maths gives the cause
(usually a lower bound)
BUT Need to know accurately!!!

10. Case Study 2: Where is the poison?
A contaminant is illegally released into a water network.
Can we find where it was released some time later?
Reservoir
Nasa B
pipe
junction A
Pump
Junction
Pipe
Measurement

11. The model
Contaminant C is convected and diluted by pipe water flow Q
Pipe C
Convection and dilution
It also gets mixed at the junctions
Mixing

12. The solution Find the pipe flow rates
Measure the present contaminant concentrations at the junctions.
Guess the initial contaminant concentrations, flow forward in time and compare with what we measure.
Adjust the initial concentrations until the measured and predicted concentrations agree. Nonlinear optimisation
Single possible cause for simple pipe networks
But several possible causes if there are circulation paths.

13. Case study 3: Deblurring a number plate
A short mathematical story
Burglar robs a bank
Escapes in getaway car
Pursued by police
Nasa
GOOD NEWS Police take a photo
BAD NEWS Photo is blurred

14. Simulated
Original
Blurred

15. SOLUTION Mathematical model of the blurring process Original image
Blurring function g
Original image f
Blurred image h
Formula for blurring
By inverting the formula we can get rid of some of the blur
BUT need to know the blurring function g

16. Image processing .. Also used in TV, Medical Imaging, Astronomy
Inversion formula
Deblurred image
Image processing .. Also used in TV, Medical Imaging, Astronomy

17. Case study 4: Who or what killed Tutankhamen?
Another application of image processing sheds light on an ancient ‘murder mystery’
Bible images
X-ray CAT scan of the mummy of Tutankhamen reveals the probable cause of death ……
Research by National Geographic, Zahi Hawass, Brando Quilici
BBC News

18. 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

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

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

21. Tutenkhamen died of a broken leg
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 to use his formula
Machine inventor Cormack got a Nobel prize
Radon got nothing!
Tutenkhamen died of a broken leg
University of St. Andrews

22. Many other applications Medical imaging Monitoring Furnaces
Radon’s formula
Radon transform
Inverse
Many other applications
Medical imaging
Monitoring Furnaces
Remote Sensing
Tumour images

23. 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
Same maths as for King Tutankhamen … but in reverse!!!

24. Find the trip wires in this picture

25. Effect: Image intensity f
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

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