[upbeat music] We live in a digital world.

A new study by the company Photo Tutorial estimates 1.9 trillion photos are taken every year, 5.3 billion photos taken every day, 61,000 photos taken per second.

The study also estimates we share a staggering 14 billion images every day through social media.

[upbeat music] We can create and share those digital images of our world, thanks to Duke University math professor, Ingrid Dobschi.

I'm exhilarated by human creativity.

It's wonderful.

I mean, we use so many applications all the time.

Dobschi is known as the godmother of the digital image, because of her pioneering work in signal processing.

It's all thanks to math, because a digital signal conveys information, just like a math equation.

That's the beauty of mathematics, that the patterns and formalisms you learn have applications in many places.

That's the reason we invent mathematics to begin with.

I mean, mathematics start with us realizing that things in different circumstances have patterns, have concepts in common.

[MUSIC] Dobschi's great mathematical discovery was the creation of what's known as the Dobschi wavelet.

Essentially, wavelets allow computers to function much like the human eye.

The image provides greater resolution and detail at the focal point and leaves the rest of the image a little more blurred.

Here is where I need to be really sharp, and here, and that can be at the edge of the image.

But on the whole image, it tells me here's where I need to be sharp, but it typically doesn't need to be sharp everywhere.

The algorithm Dobschi created allowed images to be compressed.

It was called JPEG 2000, the first version of what we know as a JPEG file.

She created it in 1987 when she was 33.

The image itself has been compressed.

It has been transformed into a version that will take less memory.

Here's how it works.

[upbeat music] Take this image of sailboats.

The image itself is made up of tens of thousands of picture elements.

We know them as pixels.

Pixels are the smallest single component in a digital image.

So these are the pixels then?

Pixels.

Yeah.

Okay.

Each represents a number.

This could be 81, and this could be 83, and so on.

Many of them will be very similar because something similar is going on in the image.

Dobschi's algorithm assigns a number corresponding to the grayscale for each pixel.

But look closely.

Many of those pixels are pretty similar to each other.

The water is about the same color.

So are the cliffs, the trees, and much of the sky.

83 and 85 that were here, if I replace them by 84, that's a gray level you wouldn't see a difference.

It's fine to think of each of these as 84.

And you can see that by the fact that when I compute their difference, that's just two.

It's very little.

When there's a dramatic color change in the image, there's a big change in the number assigned.

There may be here 147, and then 145, and then so on.

So this is something from dark to light.

You say this is a big change in the grayscale.

Yeah.

The wavelet's algorithm then averages out the differences, which highlight the action in an image.

But here, their difference is much more.

It's 62.

Not much is lost if all that ocean with a similar color is blurry.

The focus of the picture are the sailboats.

And so when I compute things like that, I've replaced every two numbers by just one.

And the red differences tell me where I don't have to care about the fact that I forgot information.

But they say, "Here, please, remember me, 64, 62.

Remember that I was a big difference.

You need me in order to keep track of that sudden transition."

And so you can compress by retaining only half the original amount of numbers and the differences where they're big.

Remember, it's all about mathematics.

And in math terms, a signal is something that conveys information.

Signal processing and the creation of wavelets deals with the geometry of information rather than the geometry of shapes, motions, and forces.

Mathematics brings it all together.

I realized that all this mathematics that I had learned for correspondence between quantum and classical mechanics was also useful for signal processing.

Which is a completely different thing.

But in both cases, you try to think in quantum mechanics of particles and waves, things that have both these properties.

In signal processing, you have things that are very localized in time, but you also think of frequency.

If you think of a music note, the notes on a sheet of paper on bars, it tells you what notes to play when.

Notes is how many oscillations per time unit.

Well, that is not something that happens right now.

It happens over a little time.

But on the other hand, you want to know when.

So it has a duality there, and it's that duality that is similar to what you have in quantum mechanics.

While technology has moved on from the original JPEG 2000, basic concepts of wavelets and signal processing survive and are now contributing to a variety of fields.

So I have worked with people in many different fields.

I mean, in neuroscience, in geophysics, in biology, in art history even, because I like learning things.

I love listening to people who are good at something that they do and that I don't know.

[upbeat music]