mathematics in everyday life
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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?. - PowerPoint PPT PresentationTRANSCRIPT
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
1. By “rotating line camera”
2. Stitching together multiple images
Your camera can do it this way…
E.g. PhotoStitch (Canon PowerShot SD600)
Experiment with PhotoStitch
Experiment done by Rebecca Szarkowski
Input: 10 images along a bridge
Experiment continued…
Experiment done by Rebecca Szarkowski
Output: Panorama (PhotoStitch)
Output: Panorama (by a more careful mathematical algorithm)
What’s math got to do with it?
From visual images to numbers (or digital images)
New Topic: Relation of Imaging and Mathematics
Digital Image Acquisition
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 64128 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 64128 128 128 128 128 128 128 128
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)
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
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=
Example: Smoothing by averaging
Original image on top left It is then averaged with neighborsof 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 moonOn 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