HOW CAN MATHS HELP TO SOLVE CRIME

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

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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
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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?

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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
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Other examples of inverse problems
Nasa
Tumour images
Remote sensing the land
Medical imaging
Nasa
Nasa
Remote sensing the sea
Seismograph prospecting for oil
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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
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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!!!
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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
Pump
Junction
Pipe
Measurement
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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
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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.
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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

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Simulated Original
Blurred
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SOLUTION
Mathematical model of the blurring process
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

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Inversion formula
Deblurred image
Image processing .. Also used in TV, Medical
Imaging, Astronomy
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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
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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 to use his formula
• Machine inventor Cormack got a Nobel prize
• Radon got nothing!

Tutenkhamen died of a broken leg
University of St. Andrews
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Radons formula
Radon transform
Inverse
Many other applications Medical
imaging Monitoring Furnaces Remote Sensing
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
Same maths as for King Tutankhamen but in
reverse!!!
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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?!?