Mathematics in everyday life

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

Mathematics in Everyday Life

Gilad Lerman Department 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 Images

Let us type the following numbers

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

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)

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

For example, 2 has the neighborhood

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)

• Math is useful (beyond the grocery store)

• Images are composed of numbers

• Good math ideas  good image processing