1. Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6 th 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 1.By “rotating line camera” 2.Stitching together multiple images Your camera can do it this way… E.g. PhotoStitch (Canon PowerShot SD600)

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

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

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

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 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

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 70 22 57 22 2 2 37 1 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

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