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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 information And retrieve it using wavelets How likely was it to have come from a suspect?

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W hat can we learn from the evidence? Inverse problem 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 evidence Given known evidence use maths to give possible causes. Find the limitations and errors of the answer Where has a bullet come from?

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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 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

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An example of Image Processing 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 AttenuationR(ρ, θ) 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 Radons 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 isnt relevant to real life?!?

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