In the 1920's, the German mathematician David Hilbert devised a famous thought experiment to show us just how hard it is to wrap our minds around the concept of infinity.
Imagine a hotel with an infinite number of rooms and a very hardworking night manager.
One night, the Infinite Hotel is completely full, totally booked up with an infinite number of guests.
A man walks into the hotel and asks for a room.
Rather than turn him down, the night manager decides to make room for him.
How?
Easy, he asks the guest in room number 1 to move to room 2, the guest in room 2 to move to room 3, and so on.
Every guest moves from room number "n" to room number "n+1".
Since there are an infinite number of rooms, there is a new room for each existing guest.
This leaves room 1 open for the new customer.
The process can be repeated for any finite number of new guests.
If, say, a tour bus unloads 40 new people looking for rooms, then every existing guest just moves from room number "n" to room number "n+40", thus, opening up the first 40 rooms.
But now an infinitely large bus with a countably infinite number of passengers pulls up to rent rooms.
countably infinite is the key.
Now, the infinite bus of infinite passengers perplexes the night manager at first, but he realizes there's a way to place each new person.
He asks the guest in room 1 to move to room 2.
He then asks the guest in room 2 to move to room 4, the guest in room 3 to move to room 6, and so on.
Each current guest moves from room number "n" to room number "2n" -- filling up only the infinite even-numbered rooms.
By doing this, he has now emptied all of the infinitely many odd-numbered rooms, which are then taken by the people filing off the infinite bus.
Everyone's happy and the hotel's business is booming more than ever.
Well, actually, it is booming exactly the same amount as ever, banking an infinite number of dollars a night.
Word spreads about this incredible hotel.
People pour in from far and wide.
One night, the unthinkable happens.
The night manager looks outside and sees an infinite line of infinitely large buses, each with a countably infinite number of passengers.
What can he do?
If he cannot find rooms for them, the hotel will lose out on an infinite amount of money, and he will surely lose his job.
Luckily, he remembers that around the year 300 B.C.E., Euclid proved that there is an infinite quantity of prime numbers.
So, to accomplish this seemingly impossible task of finding infinite beds for infinite buses of infinite weary travelers, the night manager assigns every current guest to the first prime number, 2, raised to the power of their current room number.
So, the current occupant of room number 7 goes to room number 2^7, which is room 128.
The night manager then takes the people on the first of the infinite buses and assigns them to the room number of the next prime, 3, raised to the power of their seat number on the bus.
So, the person in seat number 7 on the first bus goes to room number 3^7 or room number 2,187.
This continues for all of the first bus.
The passengers on the second bus are assigned powers of the next prime, 5.
The following bus, powers of 7.
Each bus follows: powers of 11, powers of 13, powers of 17, etc.
Since each of these numbers only has 1 and the natural number powers of their prime number base as factors, there are no overlapping room numbers.
All the buses' passengers fan out into rooms using unique room-assignment schemes based on unique prime numbers.
In this way, the night manager can accommodate every passenger on every bus.
Although, there will be many rooms that go unfilled, like room 6, since 6 is not a power of any prime number.
Luckily, his bosses weren't very good in math, so his job is safe.
The night manager's strategies are only possible because while the Infinite Hotel is certainly a logistical nightmare, it only deals with the lowest level of infinity, mainly, the countable infinity of the natural numbers, 1, 2, 3, 4, and so on.
Georg Cantor called this level of infinity aleph-zero.
We use natural numbers for the room numbers as well as the seat numbers on the buses.
If we were dealing with higher orders of infinity, such as that of the real numbers, these structured strategies would no longer be possible as we have no way to systematically include every number.
The Real Number Infinite Hotel has negative number rooms in the basement, fractional rooms, so the guy in room 1/2 always suspects he has less room than the guy in room 1.
Square root rooms, like room radical 2, and room pi, where the guests expect free dessert.
What self-respecting night manager would ever want to work there even for an infinite salary?
But over at Hilbert's Infinite Hotel, where there's never any vacancy and always room for more, the scenarios faced by the ever-diligent and maybe too hospitable night manager serve to remind us of just how hard it is for our relatively finite minds to grasp a concept as large as infinity.
Maybe you can help tackle these problems after a good night's sleep.
But honestly, we might need you to change rooms at 2 a.m.
One of the biggest challenges in computer graphics has been being able to create a photo-real, digital human face.
And one of the reasons it is so difficult is that, unlike aliens and dinosaurs, we look at human faces every day.
They are very important to how we communicate with each other.
As a result, we're tuned in to the subtlest things that could possibly be wrong with a computer rendering, in order to believe whether these things are realistic.
And what I'm going to do in the next five minutes is take you through a process where we tried to create a reasonably photo-realistic computer-generated face, using some computer graphics technology we've developed, and also some collaborators at a company called Image Metrics.
And we're going to try to do a photo-real face of an actress named Emily O'Brien, who is right there.
And that's actually a completely computer-generated rendering of her face.
By the end of the talk, we're going to see it move.
The way that we did this is we tried to start with Emily herself, who was gracious enough to come to our laboratory in Marina Del Rey, and sit for a session in Light Stage 5.
This is a face-scanning sphere, with 156 white LEDs all around that allow us to photograph her in a series of very controlled illumination conditions.
And the lighting that we use these days looks something like this.
We shoot all of these photographs in about three seconds.
And we basically capture enough information with video projector patterns that drape over the contours of her face, and different principle directions of light from the light stage, to figure out both the coarse-scale and the fine-scale detail of her face.
If we zoom in on this photograph right here, we can see it's a really nice photograph to have of her, because she is lit from absolutely everywhere at the same time to get a nice image of her facial texture.
And in addition, we've actually used polarizers on all the lights -- just like polarized sunglasses can block the glare off of the road, polarizers can block the shine off of the skin, so we don't get all those specular reflections to take this map.
Now, if we turn the polarizers around just a little bit, we can actually bring that specular reflection of the skin back in, and you can see she looks kind of shiny and oily at this point.
If you take the difference between these two images here, you can get an image lit from the entire sphere of light of just the shine off of Emily's skin.
I don't think any photograph like this had ever been taken before we had done this.
And this is very important light to capture, because this is the light that reflects off the first surface of the skin.
It doesn't get underneath the translucent layers of the skin and blur out.
And, as a result, it's a very good cue to the detailed shape of the skin-pore structure and all of the fine wrinkles that all of us have, the things that actually make us look like real humans.
So, if we use information that comes off of this specular reflection, we can go from a traditional face scan that might have the gross contours of the face and the basic shape, and augment it with information that puts in all of that skin pore structure and fine wrinkles.
And, even more importantly, since this is a photometric process that only takes three seconds to capture, we can shoot Emily in just part of an afternoon, in many different facial poses and facial expressions.
So, here you can see her moving her eyes around, moving her mouth around.
And these we're actually going to use to create a photo-real digital character.
If you take a look at these scans that we have of Emily, you can see that the human face does an enormous amount of amazing things as it goes into different facial expressions.
You can see things.
Not only the face shape changes, but all sorts of different skin buckling and skin wrinkling occurs.
You can see that the skin pore structure changes enormously from stretched skin pores to the regular skin texture.
You can see the furrows in the brow and how the microstructure changes there.
You can see muscles pulling down at flesh to bring her eyebrows down.
Her muscles bulging in her forehead when she winces like that.
In addition to this kind of high-resolution geometry, since it's all captured with cameras, we've got a great texture map to use for the face.
And by looking at how the different color channels of the illumination, the red and the green and the blue, diffuse the light differently, we can come up with a way of shading the skin on the computer.
Then, instead of looking like a plaster mannequin, it actually looks like it's made out of living human flesh.
And this is what we used to give to the company Image Metrics to create a rigged, digital version of Emily.
We're just seeing the coarse-scale geometry here.
But they basically created a digital puppet of her, where you can pull on these various strings, and it actually moves her face in ways that are completely consistent with the scans that we took.
And, in addition to the coarse-scale geometry, they also used all of that detail to create a set of what are called "displacement maps" that animate as well.
These are the displacement maps here.
And you can see those different wrinkles actually show up as she animates.
So the next process was then to animate her.
We actually used one of her own performances to provide the source data.
So, by analyzing this video with computer vision techniques, they were able to drive the facial rig with the computer-generated performance.
So what you're going to see now, after this, is a completely photo-real digital face.
We can turn the volume up a little bit if that's available.
Emily: Image Metrics is a markerless, performance-driven animation company.
We specialize in high-quality facial animation for video games and films.
Image Metrics is a markerless, performance-driven animation company.
We specialize in high quality facial animation for video games and films.
Paul Debevec: So, if we break that down into layers, here's that diffuse component we saw in the first slide.
Here is the specular component animating.
You can see all the wrinkles happening there.
And there is the underlying wireframe mesh.
And that is Emily herself.
Now, where are we going with this here?
We've gone a little bit beyond Light Stage 5.
This is Light Stage 6, and we're looking at taking this technology and applying it to whole human bodies.
This is Bruce Lawmen, one of our researchers in the group, who graciously agreed to get captured running in the Light Stage.
And let's take a look at a computer-generated version of Bruce, running in a new environment.
And thank you very much.
(Applause)