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Mathematics in Everyday Life

Mathematics in Everyday Life. Gilad Lerman Department of Mathematics University of Minnesota. Highland park elementary (6 th graders). What do mathematicians do?. What homework do I give my students?. Example of a recent homework: Denoising. What do mathematicians do?.

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Mathematics in Everyday Life

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  1. Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6th graders)

  2. What do mathematicians do? What homework do I give my students? • Example of a recent homework: Denoising

  3. What do mathematicians do? What projects do I assign my students? • Example of a recent project: Recognizing Panoramas • Panorama: • How to obtain a panorama? wide view of a physical space

  4. How to obtain a panorama • By “rotating line camera” • Stitching together multiple images Your camera can do it this way… E.g. PhotoStitch (Canon PowerShot SD600)

  5. Experiment with PhotoStitch Input: 10 images along a bridge Experiment done by Rebecca Szarkowski

  6. Experiment continued… Output: Panorama (PhotoStitch) Output: Panorama (by a more careful mathematical algorithm) Experiment done by Rebecca Szarkowski

  7. What’s math got to do with it? New Topic: Relation of Imaging and Mathematics From visual images to numbers (or digital images)

  8. Digital Image Acquisition

  9. From Numbers to Images Let us type the following numbers 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 We then color them so 1=black, 8=white rest of colors are in between

  10. One more time… Now we’ll try the following numbers 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 16 16 16 16 16 16 16 16 32 32 32 32 32 32 32 32 64 64 64 64 64 64 64 64 128 128 128 128 128 128 128 128 We then color them so 1=black, 128=white rest of colors are in between

  11. Let’s compare 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 6 7 7 7 7 7 7 7 7 8 8 8 8 8 8 8 8 1 1 1 1 1 1 1 1 2 2 2 2 2 2 2 2 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8 16 16 16 16 16 16 16 16 32 32 32 32 32 32 32 32 64 64 64 64 64 64 64 64 128 128 128 128 128 128 128 128

  12. From an Image to Its Numbers We start with clown image It has 200*320 numbers I can’t show you all… Let’s zoom on eye (~40*50)

  13. Image to Numbers (Continued) We’ll zoom on middle of eye image (10*10)

  14. The Numbers (Continued) The middle of eye image (10*10) 80 81 80 80 80 80 77 77 37 11 81 80 81 80 80 80 77 37 9 6 80 80 80 80 80 80 37 11 2 11 80 80 80 80 80 77 66 66 66 54 80 80 80 80 77 77 77 80 77 80 80 80 79 77 66 54 66 77 66 54 77 80 77 70 22 57 51 70 51 70 77 73 70 22 2 2 22 37 37 22 77 77 54 37 1 6 2 8 2 6 77 70 70 22 2 2 6 8 8 6 Note the rule: Bright colors – high numbers Dark colors - low numbers

  15. More Relation of Imaging and Math Averaging numbers  smoothing images Idea of averaging: take an image Replace each point by average with its neighbors For example, 2 has the neighborhood So replace 2 by 80 81 80 80 80 80 77 77 37 11 81 80 81 80 80 80 77 37 9 6 80 80 80 80 80 80 37 11 2 11 80 80 80 80 80 77 66 66 66 54 80 80 80 80 77 77 77 80 77 80 80 80 79 77 66 54 66 77 66 54 77 80 77 70 22 57 51 70 51 70 77 73 70 22 2 2 22 37 37 22 77 77 54 37 1 6 2 8 2 6 77 70 70 22 2 2 6 8 8 6 80 81 80 80 80 80 77 77 37 11 81 80 81 80 80 80 77 37 9 6 80 80 80 80 80 80 37 11 2 11 80 80 80 80 80 77 66 66 66 54 80 80 80 80 77 77 77 80 77 80 80 80 79 77 66 54 66 77 66 54 77 80 77 70 22 57 51 70 51 70 77 73 70 22 2 2 22 37 37 22 77 77 54 37 1 6 2 8 2 6 77 70 70 22 2 2 6 8 8 6 70 22 57 22 2 2 37 1 6

  16. Example: Smoothing by averaging Original image on top left It is then averaged with neighbors of distances 3, 5, 19, 15, 35, 45

  17. Example: Smoothing by averaging And removing wrinkles by both….

  18. More Relation of Imaging and Math Differences of numbers  sharpening images On left image of moon On right its edges (obtained by differences) We can add the two to get a sharpened version of the first

  19. Moon sharpening (continued)

  20. Real Life Applications • Many… • From a Minnesota based company… • Their main job: maintaining railroads • Main concern: Identify cracks in railroads, before too late…

  21. How to detect damaged rails? • Traditionally… drive along the rail (very long) and inspect • Very easy to miss defects (falling asleep…) • New technology: getting pictures of rails

  22. Millions of images then collected

  23. How to detect Cracks? • Human observation… • Train a computer… • Recall that differences detect edges… Work done by Kyle Heuton (high school student at Saint Paul)

  24. Summary • Math is useful (beyond the grocery store) • Images are composed of numbers • Good math ideas  good image processing

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