This document discusses the relationship between mathematics and imaging/image processing. It provides examples of how 6th grade students used math concepts like denoising and panorama recognition in their homework. The document then explains how digital images are composed of numbers and how basic math operations like averaging and differences can be used to smooth and sharpen images. It concludes by discussing a real-world application of using image processing to detect cracks in railroad tracks from pictures.

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

2. WWhaht adto h momatehweomrka tdicoia In gsi vdeo ?
my students?
• Example of a recent homework: Denoising

3. WWhaht adto p mroajethctesm daot iIc iaasnssig dno ?
my students?
• Example of a recent project:
Recognizing Panoramas
• Panorama:
wide view of a physical space
• How to obtain a panorama?

4. 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)

5. Experiment with PhotoStitch
Input: 10 images along a bridge
Experiment done by Rebecca Szarkowski

6. Experiment continued…
Output: Panorama (PhotoStitch)
Output: Panorama (by a more careful mathematical algorithm)
Experiment done by Rebecca Szarkowski

7. New Topic: Relation of Imaging
What’s math got to do with it?
and Mathematics
From visual images to numbers (or digital images)

8. Digital Image Acquisition

9. 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

10. 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

11. 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

12. 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)

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

14. 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

15. 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
70+22+57+22+2+2+37+1+6 24 1
=
9 3

16. Example: Smoothing by averaging
Original image on top left
It is then averaged with neighbors
of distances 3, 5, 19, 15, 35, 45

17. Example: Smoothing by averaging
And removing wrinkles by both….

18. 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

19. Moon sharpening (continued)

20. Real Life Applications
• Many…
• From a Minnesota based company…
• Their main job: maintaining railroads
• Main concern: Identify cracks in railroads,
before too late…

21. 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

22. Millions of images then collected

23. 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)

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