IMAGES ARE STORED AS NUMBERS USING THESE NUMBERS WE CAN PROCESS THE PICTURES BY USING MATHEMATICS
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SOME APPLICATIONS PRODUCING THE PICTURES IN THE FIRST PLACE TRANSMITTING THE PICTURES WITHOUT MISTAKES Error Correcting Codes Galois
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SOME MORE APPLICATIONS DEBLURRING ORIGINALS FINDING THINGS HIDDEN IN AN IMAGE Edges Brains Landmines
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MATHS AND MEDICINE Modern medicine has been transformed by methods of seeing Inside you without cutting you open!
Ultra sound : sound waves
MRI : magnetism
CAT scans: X rays
ALL USE MATHS TO WORK!!
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WHAT IS A CAT SCAN?? CAT = Computerised axial tomography Based on X-Rays discovered by Roengten X-Rays cast a shadow GOOD for looking at bones BAD for looking at soft tissue
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USING MODERN MATHS WE CAN DO A LOT BETTER Modern CAT scanner CAT scanners work by casting many shadows with X-rays and using maths to assemble these into a picture
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X-Ray source Object Detector X Intensity of X-ray at detector depends on width of object We can find the thickness … can we find the shape? Intensity X
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MOVE SOURCE AND DETECTOR AROUND GET SHADOWS OF THE OBJECT FROM MANY ANGLES AND MEASURE X-RAY INTENSITY
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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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REMARKABLE FACT If we can measure R( ρ , θ ) accurately we can calculate The density f(x,y) of the object at any point
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!
Process is called Back Projection
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Radon transform Back Projection Radon’s formula Many other applications X-raying Mummies Monitoring Furnaces Remote Sensing
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USING THE RADON TRANSFORM TO FIND ANTI-PERSONEL LAND MINES Land mines are hidden in foliage and triggered by trip wires Trip wires are well hidden – can they be quickly and safely detected
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Digital picture of foliage is taken by camera on a long pole Image intensity f(x,y) • • • • Trip wires 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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Used by the Canadian Peace keeping forces Mathematics finds the land mines!
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