Transcript CAN MATHS HELP IN THE FIGHT AGAINST CRIME? Chris Budd

CAN MATHS HELP IN THE
FIGHT AGAINST CRIME?
Chris Budd
A crime has been committed
The police arrive in force
What challenges
Do they face?
• 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
For example, you find some fingerprints
These can be clear
Or blurred
And contain lots of
information
How likely was it to have
come from a suspect?
Maths can reduce the
amount of blurring
Maths gives a way of storing
Only the relevant information
And retrieve it using wavelets
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
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
Case study 1: Catching a speeding motorist
..
Was the car speeding?
Evidence:
collision damage, witness statements,
tyre skid marks
Evidence:
s
distance of skid
Cause:
u
speed
Other data: 
friction force
2
Model links cause to effect
Given the effect maths
gives the cause
BUT Need to know 
u
s
2
u  2s
accurately!!!
Case study 2: Deblurring a number plate
A short crime story
• Burglar robs a bank
• Escapes in a getaway car
Nasa
• Pursued by police
GOOD NEWS
Police take a photo
BAD NEWS
Photo is blurred
SOLUTION
Find a model of the blurring process
Blurring function g
Blurred image
Original image
h
f
• Blurring formula
h( x )   f ( x  y ) g ( y ) d 2 y
• Inverting the formula we can get rid the blur
• BUT need to know the blurring function g
f   eiy 
 ix
2
 ix
2
2

e
hd
x
/
e
gd
x
d
 / 2


Inversion formula
h(x)
An example of Image Processing
f(x)
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
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
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
Radon’s formula
Radon transform
Inverse
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
R(ρ,θ)
f(x,y)
Radon
transform
y
•
•
•
•
x
Points of high intensity in R correspond to trip wires
Isolate points and transform back to find the wires
θ
ρ
Mathematics finds the land mines!
Who says that maths isn’t relevant to real life?!?