can maths help in the fight against crime?
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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 - PowerPoint PPT PresentationTRANSCRIPT
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CAN MATHS HELP IN THE FIGHT AGAINST CRIME?
Chris Budd
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A crime has been committed
What challenges
Do they face?
The police arrive in force
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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
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
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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?
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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
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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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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!!!
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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
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)()()(
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An example of Image Processing
2// 222 dxgdexhdeef xixiyi
Inversion formula
h(x) f(x)
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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
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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
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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
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Object
Attenuation R(ρ, θ)
Edge Edge
Edge Edge
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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 …
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Radon transform
Inverse
Radon’s formula
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
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
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Mathematics finds the land mines!
Who says that maths isn’t relevant to real life?!?