When we come to compress images in the real world,
we do this quite a lot.
So, when I took this the original image,
I actually had a perfect representation of the image,
at least at that particular scale:
I had a tiff.
The file format there means that every single pixel in the file
is registered as either being on or off, black or white.
On the left-hand side, you can see what happens
if I use my unfortunately non-patented compression algorithm,
where here, instead of compressing it using 10x10 grids,
I do a much less aggressive compression
where I only take little 3x3 grids.
If you compare how ugly the 10x10 grid compression is,
to the 3x3 grid compression,
you might think, "Hey, that's pretty good."
And, in fact, by doing that, by going from a grid that's 1x1,
by keeping track of every pixel,
and instead going to a grid that's 3x3,
I decreased the file size by a factor of 9.
What I also can do is take that tiff file,
open it up, and max preview,
and now ask the computer to do some coarse graining for me.
because we spend a lot of time trying to figure out how to compress images,
how to reduce their size,
how to throw away the information that people don't need,
in order to allow the image to be transmitted faster,
to be stored more conveniently.
So, on the right-hand side here,
we also have a coarse graining of the Alice image,
but now using a coarse-graining algorithm called the JPEG.
No one's ever referred to the JPEG as a coarse-graining algorithm,
at least they don't do it that much.
What you can see is both images look reasonably good.
In fact, they have different properties, so let's go zoom in here on Dinah.
What you can see on the left-hand side is the majority vote coarse graining.
One of the features of the majority vote coarse graining, by the way, you will see,
is that each pixel is still a bit black or white.
If I zoom in again,
and I guess Dinah's looking here much more like a rat now than she is like a cat,
but if you zoom in on Dinah on the right-hand side,
you can see now the JPEG image has made different choices
about what to keep and what to throw away.
Importantly, one of the ways the JPEG works
is not in what we call real space, in what the physicists call real space,
but instead what's called Fourian Space.
We'll talk a little bit more
about the distinction between those two ways to represent an image,
but for now the simplest way to think of it is this:
On the right-hand side,
what I did was I took the representation of the image in the spatial field.
So, I took the entire array,
and I turned it by taking chunks that were locally connected to each other.
I took little local chunks from the image,
and, for each of those chunks, I did a little compression scheme.
I said all these differences here don't matter,
you don't have to keep track
of all the either 9 or 100 pixels within that square,
I'm going to summarize it, I'm going to coarse grain it,
I'm going to simplify it, going to losslessly compress it –
all these words are equivalent,
different ways and different fields,
or rather different ways
that different fields have discovered to talk about this –
I'm going to simplify all of those pixels in a particular way.
On the right-hand side, what the JPEG does is the following:
it does a transformation on this image,
it represents this image in a very different way.
What it does is it represents the image
in terms of the fluctuations that occur in it.
So, it takes the long-wavelength parts of the image,
the fact that in the center it's darker than it is around the edges,
and it puts that in one pixel.
Then it also takes the high-frequency component,
the wiggles where the image is going from black to white
very quickly along the line.
So, here, for example, you can see
on the back of the armchair where Alice is sitting,
very fine grid lines.
What the JPEG does, is it says,
"There's a patch here where things are oscillating very quickly,
so I'm going to put that over
in this high-frequency part of the image data."
It's also true of Alice's hair.
If you look at Alice's hair there, she has these sort of kinky curls,
and those curls
have a high-frequency component the JPEG records.
So, what the JPEG does is,
it represents that image now,
instead of representing it spatially –
so stuff on the left is physically stuff that was on the left
in Tenniel's original recording –
it puts the low-frequency components in one part
and the high-frequency components in the other part.
And then it does two things.
First of all blurs them,
so it does a kind of chunking coarse graining on those components,
and then also it just entirely cuts off
all the high-frequency components in the image –
because there are parts of that image
that you yourself are not sensitive to,
and those parts are where the image starts fluctuating back and forth very quickly.
In fact, you can see it:
on the right-hand side, the JPEG looks smoother
than in the compression I've done.
In fact, what's so clever about the JPEG
is it actually respects the way
in which the human eye records information.
Amazingly enough, of course,
when the human eyes gets data from the real world,
and makes an image on the back of your retina,
it appears almost like a set of pixels.
But before it actually wants to transmit that image back into your brain
to make decisions,
it does itself a series of coarse grainings
using Gabor functions in the particular sensitivity of the neurons.
It does a particular set of coarse grainings
that, in fact, the JPEG knows about.
So the things that your retina is going to throw out anyway
as it transmits it backwards,
the JPEG has already thrown out on your behalf.