Forensic Maths Chris Budd

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

How to find out what happened.. forensic 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  Reconstruct what happened inverse problems  Store and interpret data probability, statistics  Transmit data in a secure way prime numbers 2,3,7,11 Using maths they can

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

W hat can we learn from the evidence? Forensics But what happened given the evidence? For example, find the shape of an object only knowing its shadows Nasa

How to solve a forensic problem  Agree on the science  Understand what causes lead to what evidence  Given known evidence use maths to give possible causes.  Find the errors of the answer Where has a bullet come from?

Not all forensic problems occur in crime fighting Oil prospecting Medical imaging Pollution Weather forecasting Sudoku Griddler Can you think of any more?

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

Evidence: s distance of skid Cause: u speed Other data: F brake force Science links cause to effect Given the effect maths gives the cause

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

GOOD NEWS Police take a photo BAD NEWS Photo is blurred

SOLUTION Work out the maths of the blurring process 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 Blurring function g

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

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

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

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

Object Attenuation R(ρ, θ) 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 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 University of St. Andrews Tutenkhamen died of a broken leg The murder mystery resolved …

Radon’s formula Also used in Medical imaging Tumour images

CASE STUDY 4: 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

Find the trip wires in this picture

Digital picture of foliage is taken by camera on a long pole Effect : Image intensity f Cause : Trip wires.. These 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

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