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Mathematics in Everyday Life Gilad Lerman Department of Mathematics University of Minnesota ighland park elementary (6 th graders)

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 Presentation

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Page 1: Mathematics in Everyday Life

Mathematics in Everyday Life

Gilad Lerman

Department of Mathematics

University of Minnesota

Highland park elementary (6th graders)

Page 2: Mathematics in Everyday Life

What do mathematicians do?What homework do I give my students?

• Example of a recent homework: Denoising

Page 3: Mathematics in Everyday Life

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

Page 4: Mathematics in Everyday Life

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)

Page 5: Mathematics in Everyday Life

Experiment with PhotoStitch

Experiment done by Rebecca Szarkowski

Input: 10 images along a bridge

Page 6: Mathematics in Everyday Life

Experiment continued…

Experiment done by Rebecca Szarkowski

Output: Panorama (PhotoStitch)

Output: Panorama (by a more careful mathematical algorithm)

Page 7: Mathematics in Everyday Life

What’s math got to do with it?

From visual images to numbers (or digital images)

New Topic: Relation of Imaging and Mathematics

Page 8: Mathematics in Everyday Life

Digital Image Acquisition

Page 9: Mathematics in Everyday Life

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

Page 10: Mathematics in Everyday Life

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

Page 11: Mathematics in Everyday Life

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

Page 12: Mathematics in Everyday Life

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)

Page 13: Mathematics in Everyday Life

Image to Numbers (Continued)We’ll zoom on middle of eye image (10*10)

Page 14: Mathematics in Everyday Life

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

Page 15: Mathematics in Everyday Life

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=

Page 16: Mathematics in Everyday Life

Example: Smoothing by averaging

Original image on top left It is then averaged with neighborsof distances 3, 5, 19, 15, 35, 45

Page 17: Mathematics in Everyday Life

Example: Smoothing by averaging

And removing wrinkles by both….

Page 18: Mathematics in Everyday Life

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

Page 19: Mathematics in Everyday Life

Moon sharpening (continued)

Page 20: Mathematics in Everyday Life

Real Life Applications

• Many…• From a Minnesota based company…

• Their main job: maintaining railroads• Main concern: Identify cracks in railroads,

before too late…

Page 21: Mathematics in Everyday Life

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

Page 22: Mathematics in Everyday Life

Millions of images then collected

Page 23: Mathematics in Everyday Life

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)

Page 24: Mathematics in Everyday Life

Summary

• Math is useful (beyond the grocery store)• Images are composed of numbers• Good math ideas good image processing