CAN MATHS HELP IN THE FIGHT AGAINST CRIME

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CAN MATHS HELP IN THE FIGHT AGAINST CRIME?
Chris Budd
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A crime has been committed
The police arrive in force
What challenges Do they face?
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• 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

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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
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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
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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
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Case study 1 Catching a speeding motorist
.. Was the car speeding?
Evidence collision damage, witness
statements, tyre skid marks
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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!!!
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Case study 2 Deblurring a number plate
A short crime story

• Burglar robs a bank
• Escapes in a getaway car
• Pursued by police

Nasa
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GOOD NEWS Police take a photo BAD NEWS Photo
is blurred
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SOLUTION
Find a model of the blurring process
Blurring function g
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

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Inversion formula
h(x)
f(x)
An example of Image
Processing
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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
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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
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Source
X-Ray
Detector
Object
? Distance from the object centre ? Angle
of the X-Ray
Measure attenuation of X-Ray R(?, ?)
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Object
Edge
Edge
Attenuation
R(?, ?)
Edge
Edge
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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
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Radons formula
Radon transform
Inverse
Also used in Medical imaging
Tumour images
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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
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Find the trip wires in this picture
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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(?,?)
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
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Mathematics finds the land mines!
Who says that maths isnt relevant to
real life?!?