Mathematics in Everyday Life

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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

• By rotating line camera
• Stitching together multiple images
• Your camera can do it this way
• E.g. PhotoStitch (Canon PowerShot SD600)

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Experiment with PhotoStitch
Input 10 images along a bridge
Experiment done by Rebecca Szarkowski
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Experiment continued
Output Panorama (PhotoStitch)
Output Panorama (by a more careful mathematical
algorithm)
Experiment done by Rebecca Szarkowski
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Whats math got to do with it?
New Topic Relation of Imaging and Mathematics
From visual images to numbers (or digital images)
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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 1black, 8white
• rest of colors are in between

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One more time

• Now well 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 1black, 128white
• rest of colors are in between

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Lets 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
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From an Image to Its Numbers

• We start with clown image
• It has 200320 numbers
• I cant show you all
• Lets zoom on eye (4050)

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Image to Numbers (Continued)

• Well zoom on middle of eye image (1010)

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The Numbers (Continued)

• The middle of eye image (1010)

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
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
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Example Smoothing by averaging
Original image on top left It is then averaged
with neighbors of 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 moon On 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