So far, we've been talking about one dimensional functions and

their Fourier transforms.

We've learned that to take a Fourier transform means to decompose

a complicated function into a series of simple sine waves.

The idea is that if you were to add those sine waves back up together,

adding them together you would recover the complicated function that you began with.

But in microscopy, we dont deal with one-dimensional functions,

rather we record two-dimensional images.

So the next subject is a two-dimensional Fourier transform.

So to introduce one-dimensional Fourier transforms, I showed you,

how you could take a series of one-dimensional sine waves, and

add them up to get a complex function.

To introduce two-dimensional Fourier transforms,

I'd like to start with the same thing.

I'd like to add some two-dimensional sine waves together

to start to form a more complicated picture.

In a one-dimensional sine wave, we have a variable.

And the value of the function rises and

falls in an oscillatory pattern as a function of that variable.

In a two-dimensional sine wave, we have two variables that define a plane.

And the value of the function either rises or

falls depending on the position within that plane.

So for instance,

a two-dimensional sine wave, I can represent with this piece of paper.

If each position in space here on this plane is

a position defined by an x and a y variable, the value of

the function I could then represent as the height of the piece of paper.

And so one two-dimensional sine wave looks like this in that it, it rises.

If this dimension is x and this dimension is y, this two-dimensional sine

wave rises at low x, and then as x increases, it goes through 0.

And then it has a minimum here.

And then it comes back to 0.

So that's a two-dimensional sine wave that varies with x.

A two-dimensional sine wave that varies with y would look, instead, like this,

because it would rise at low y, go back to 0, have a minimum,

and then come at, at, at higher values of y back to 0.

So this is a two-dimensional sine wave.

Now to look at more complicated two-dimensional sine waves,

I've written just a simple program that I can use to create a space and

then add sine waves to it.

So to create a space, here my dialog box says, New Image Dimension.

I'm going to pick a dimension of, let's say, 800 pixels.

And it gives me a space a plane of 800

pixels in x, and 800 pixels high in y.

And what I'll do is add a sine wave to that plane.

Now, two-dimensional sine waves are characterized by two parameters, h and k.

These are called Miller indices and we'll talk more about them later.

But they represent how many oscillations along x this wave has, that's the value h.

And how many oscillations there are in y, that's the index, k.

So for instance a, h equals 1 and k equals 0 wave with

an amplitude of 1 and a phase of 0, looks like this.

So what's happening here is the value of the function starts at 0 here

along this left side of the, of the box and then the value of the function arises,

which is represented by brighter pixels here, and

as it's a quarter the way across the box you have a crest arise here.

And then as we move to higher x values, it falls back to 0, and then you have

the minimum at three-quarters of the way across the box in x, you have a minimum,

represented by the darkest pixels, and then it rises again back up to zero.

This is the one-zero wave.

In the piece of paper, this was the first wave that I showed like this.

Okay, now let's look at a different wave.

I'll create another box, again of dimension 800, and let's add a sine wave.

Instead of the one, zero sine wave, this time let's add the zero, one.

So now it should oscillate no times along x and 1 oscillation along y,

and again, let's give it an amplitude of 1 and phase of 0.

Now this is the wave that we get.

Now in the way I've created the program, down here is the 0, 0 coordinate.

And y increases as you go up the box in this way.

And so, because of that, here along the x-axis,

are the value, the function starts at a value of 0.

And then as we move into positive values of y

at a quarter of the way across the box we reach a maximum of 1.

So that's the first crest.

And then the function falls to 0, and

then it approaches its first minimum, here three-quarters of the way across the box,

and then it recovers back to 0 as it goes across.

That's the zero, one wave.

Now let's look at another wave.