can maths help in the fight against crime?

CAN MATHS HELP IN THE FIGHT AGAINST CRIME?

Chris Budd

A crime has been committed

What challenges

Do they face?

The police arrive in force

• 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

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

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 informationAnd retrieve it using wavelets

How likely was it to have come from a suspect?

Inverse problems … what causes lead to what effects?

But what happened given the evidence?

For example, find the shape of an object only knowing its shadows

Nasa

How 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.

Find the limitations and errors of the answer

Different causes may give very similar effects (think of shadows!)

Object

Shadow

Case study 1: Catching a speeding motorist

..

Was the car speeding?

Evidence: collision damage, witness statements,

tyre skid marks

su 2

2

2us

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!!!

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

Find a model 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

ydygyxfxh 2)()()(

An example of Image Processing

2// 222 dxgdexhdeef xixiyi

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 Edge

Edge 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 transform

Inverse

Radon’s formula

Also used in

Medical imaging

Tumour images

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

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?!?