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Forensic Maths

Forensic Maths. Chris Budd. A crime has been committed. The police arrive in force. What challenges Do they face?. How to find out what happened .. forensic How to interpret confusing data How to store a mass of data and mine it for information

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Forensic Maths

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  1. Forensic Maths Chris Budd

  2. A crime has been committed The police arrive in force What challenges Do they face?

  3. How to find out what happened .. forensic • 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 probability, statistics • Transmit data in a secure way prime numbers 2,3,7,11

  4. 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

  5. But what happened given the evidence? What can we learn from the evidence? Forensics For example, find the shape of an object only knowing its shadows Nasa

  6. How to solve a forensic problem Where has a bullet come from? • Agree on the science • Understand what causes lead to what evidence • Given known evidenceuse mathsto give possible causes. • Find theerrors of the answer

  7. Not all forensic problems occur in crime fighting • Oil prospecting • Medical imaging • Pollution • Weather forecasting • Sudoku • Griddler Can you think of any more?

  8. Case study 1: Catching a speeding motorist .. Was the car speeding? Forensic evidence: collision damage, witness statements, skid marks

  9. Evidence: sdistance of skid Cause: uspeed Other data: F brake force Science links cause to effect Given the effect mathsgives thecause

  10. Case study 2: Deblurring a number plate A short crime story • Burglar robs a bank • Escapes in a getaway car • Pursued by police Nasa

  11. GOOD NEWS Police take a photo BAD NEWS Photo is blurred

  12. SOLUTION Work out the maths of the blurring process Blurring function g Blurred image h Original image f • Blurring formula • Inverting the formula we can get rid the blur • BUT need to know the blurring function g

  13. Inversion formula h(x) f(x) An example of Image Processing

  14. 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 Hawassreveals the probable cause of death …… National Geographic

  15. 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

  16. Source X-Ray Detector Object ρ : Distance from the object centre θ : Angle of the X-Ray Measure attenuation of X-Ray R(ρ, θ)

  17. Object Edge Edge Attenuation R(ρ, θ) Edge Edge

  18. 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 The murder mystery resolved … Tutenkhamen died of a broken leg University of St. Andrews

  19. Radon’s formula Also used in Medical imaging Tumour images

  20. CASE STUDY 4: 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

  21. Find the trip wires in this picture

  22. Digital picture of foliage is taken by camera on a long pole Effect: Image intensityf Cause: Trip wires .. These are like X-Rays R(ρ,θ) f(x,y) • Radon transform • • y ρ • x θ Points of high intensity in R correspond to trip wires Isolate points and transform back to find the wires

  23. Mathematics finds the land mines! Who says that maths isn’t relevant to real life?!?

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