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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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**Mathematics in Everyday Life**Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6th graders)**What do mathematicians do?**What homework do I give my students? • Example of a recent homework: Denoising**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**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)**Experiment with PhotoStitch**Input: 10 images along a bridge Experiment done by Rebecca Szarkowski**Experiment continued…**Output: Panorama (PhotoStitch) Output: Panorama (by a more careful mathematical algorithm) Experiment done by Rebecca Szarkowski**What’s math got to do with it?**New Topic: Relation of Imaging and Mathematics From visual images to numbers (or digital images)**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**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**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**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)**Image to Numbers (Continued)**We’ll zoom on middle of eye image (10*10)**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**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**Example: Smoothing by averaging**Original image on top left It is then averaged with neighbors of distances 3, 5, 19, 15, 35, 45**Example: Smoothing by averaging**And removing wrinkles by both….**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**Real Life Applications**• Many… • From a Minnesota based company… • Their main job: maintaining railroads • Main concern: Identify cracks in railroads, before too late…**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**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)**Summary**• Math is useful (beyond the grocery store) • Images are composed of numbers • Good math ideas good image processing

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