Mathematics in everyday life

Mathematics in Everyday Life

Gilad Lerman
Department of Mathematics
University of Minnesota
Highland park elementary (6th graders)


What
homework
do
I
give
What do mathematicians do?
my students?
• Example of a recent homework: Denoising


What
projects
do
I
assign
What do mathematicians do?
my students?
• Example of a recent project:
Recognizing Panoramas
• Panorama: wide view of a physical space

• How to obtain a panorama?

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
Input: 10 images along a bridge

Experiment done by Rebecca Szarkowski


Experiment continued…
Output: Panorama (PhotoStitch)

Output: Panorama (by a more careful mathematical algorithm)

Experiment done by Rebecca Szarkowski


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


Digital Image Acquisition

From Numbers to Images
Let us type the following numbers
1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3

4
5
6
7
8

1
2
3
4
5
6
7
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 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
1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7

8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1
2
3
4
5
6
7
8

1

2
3
4
5
6
7
8

1
2
3
4
5
6
7
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
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)
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
77
77

81
80
80
80
80
80
80
73
77

70

80
81
80
80
80
79
77
70
54
70

80
80
80
80
80
77
70
22
37
22

80
80
80
80
77
66

22
2
1
2

80
80
80
77
77
54
57
2
6
2

77
77
37
66
77
66
51
22
2
6

77
37
11

66
80
77
70
37
8
8

37 11
9 6
2 11
66 54
77 80
66 54
51 70
37 22
2 6
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

80
81
80
80
80
80
77
77
77
77

81
80
80
80
80
80
80
73
77
70

80
81
80
80
80
79
77
70

54
70

For example, 2 has the neighborhood
So replace 2 by

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

80
80
80
80
80
77
70
22
37
22

80
80
80
80
77
66
22

2
1
2

80
80
80
77
77
54
57
2
6
2

77
77
37
66
77
66
51
22
2
6

70
22
37

22
2
1

57
2
6

77
37
11
66
80
77
70
37
8
8

37 11
9 6
2 11
66 54
77 80
66 54
51 70
37 22
2 6
8 6

Example: Smoothing by averaging

Original image on top left
It is then averaged with neighbors
of 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 moon
On 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