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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

Mathematics in Everyday Life

Gilad Lerman

Department of Mathematics

University of Minnesota

Highland park elementary (6th graders)

what do mathematicians do
What do mathematicians do?

What homework do I give my students?

  • Example of a recent homework: Denoising
what do mathematicians do3
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
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
Experiment with PhotoStitch

Input: 10 images along a bridge

Experiment done by Rebecca Szarkowski

experiment continued
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
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
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
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
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
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
Image to Numbers (Continued)

We’ll zoom on middle of eye image (10*10)

the numbers continued
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
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

slide16

Example: Smoothing by averaging

Original image on top left

It is then averaged with neighbors

of distances 3, 5, 19, 15, 35, 45

slide17

Example: Smoothing by averaging

And removing wrinkles by both….

more relation of imaging and math18
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
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
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
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
Summary
  • Math is useful (beyond the grocery store)
  • Images are composed of numbers
  • Good math ideas  good image processing