mathematics in everyday life gilad lerman department of mathematics university of minnesota highland...
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![Page 1: Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota Highland park elementary (6 th graders)](https://reader034.vdocument.in/reader034/viewer/2022050805/56649d4e5503460f94a2d5df/html5/thumbnails/1.jpg)
Mathematics in Everyday Life
Gilad Lerman
Department of Mathematics
University of Minnesota
Highland park elementary (6th graders)
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What do mathematicians do?What homework do I give my students?
• Example of a recent homework: Denoising
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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
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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)
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Experiment with PhotoStitch
Experiment done by Rebecca Szarkowski
Input: 10 images along a bridge
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Experiment continued…
Experiment done by Rebecca Szarkowski
Output: Panorama (PhotoStitch)
Output: Panorama (by a more careful mathematical algorithm)
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What’s math got to do with it?
From visual images to numbers (or digital images)
New Topic: Relation of Imaging and Mathematics
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Digital Image Acquisition
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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
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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 64128 128 128 128 128 128 128 128
We then color them so 1=black, 128=white rest of colors are in between
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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 64128 128 128 128 128 128 128 128
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From an Image to Its NumbersWe start with clown image
It has 200*320 numbers
I can’t show you all…
Let’s zoom on eye (~40*50)
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Image to Numbers (Continued)We’ll zoom on middle of eye image (10*10)
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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
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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
70+22+57+22+2+2+37+1+6 124
9 3=
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Example: Smoothing by averaging
Original image on top left It is then averaged with neighborsof distances 3, 5, 19, 15, 35, 45
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Example: Smoothing by averaging
And removing wrinkles by both….
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More Relation of Imaging and Math
Differences of numbers sharpening images
On left image of moonOn right its edges (obtained by differences)We can add the two to get a sharpened version of the first
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Moon sharpening (continued)
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Real Life Applications
• Many…• From a Minnesota based company…
• Their main job: maintaining railroads• Main concern: Identify cracks in railroads,
before too late…
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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
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Millions of images then collected
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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)
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Summary
• Math is useful (beyond the grocery store)• Images are composed of numbers• Good math ideas good image processing