Mathematics in Everyday Life

Gilad LermanDepartment of Mathematics

University of Minnesota

Highland park elementary (6th graders)

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

• Example of a recent homework: Denoising

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

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

Experiment done by Rebecca Szarkowski

Input: 10 images along a bridge

Experiment continued…

Experiment done by Rebecca Szarkowski

Output: Panorama (PhotoStitch)

Output: Panorama (by a more careful mathematical algorithm)

What’s math got to do with it?

From visual images to numbers (or digital images)

New Topic: Relation of Imaging and Mathematics

Digital Image Acquisition

From Numbers to ImagesLet 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…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

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

From an Image to Its NumbersWe start with clown imageIt has 200*320 numbersI 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 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

Averaging numbers smoothing imagesIdea of averaging: take an imageReplace 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 1249 3

=

Example: Smoothing by averaging

Original image on top left It is then averaged with neighborsof 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 moonOn 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