Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6th graders) What homework I give What mathematicians do? my students? • Example of a recent homework: Denoising What projects I assign What mathematicians do? my students? • Example of a recent project: Recognizing Panoramas • Panorama: wide view of a physical space • How to obtain a panorama? How to obtain a panorama By “rotating line camera” Stitching together multiple images Your camera can 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 New Topic: Relation of Imaging What’s math got to with it? and Mathematics From visual images to numbers (or digital images) Digital Image Acquisition From Numbers to Images Let us type the following numbers 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 2 2 2 2 4 4 4 4 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 8 8 8 8 1 1 1 1 2 2 2 2 4 4 4 4 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 77 77 81 80 80 80 80 80 80 73 77 70 80 81 80 80 80 79 77 70 54 70 80 80 80 80 80 77 70 22 37 22 80 80 80 80 77 66 22 2 80 80 80 77 77 54 57 77 77 37 66 77 66 51 22 77 37 11 66 80 77 70 37 8 37 11 11 66 54 77 80 66 54 51 70 37 22 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 80 81 80 80 80 80 77 77 77 77 81 80 80 80 80 80 80 73 77 70 80 81 80 80 80 79 77 70 54 70 For example, has the neighborhood So replace by 70+22+57+22+2+2+37+1+6 = 24 80 80 80 80 80 77 70 22 37 22 80 80 80 80 77 66 22 2 80 80 80 77 77 54 57 77 77 37 66 77 66 51 22 70 22 37 22 57 77 37 11 66 80 77 70 37 8 37 11 11 66 54 77 80 66 54 51 70 37 22 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 Moon sharpening (continued) 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 Millions of images then collected 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 [...]... of the first Moon sharpening (continued) 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 Millions of... 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 Moon sharpening (continued)... 37 8 8 37 11 9 6 2 11 66 54 77 80 66 54 51 70 37 22 2 6 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 80 81 80 80 80 80 77 77 77 77 81 80 80 80 80 80 80 73 77 70 80 81 80 80 80 79 77 70 54 70 For example, 2 has the neighborhood... Millions of images then collected 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 ... 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 77 77 81 80 80 80 80 80 80 73 77 70 80 81 80 80 80 79 77 70 54 70 80 80 80 80 80 77 70 22 37 22 80 80 80