Last time we considered the Fourier series and we have seen that we can approximate any

periodic function by a linear combination of sine and cosine functions.

And if the sum of cosine and sine functions is finite, then we know that we can represent

the signal without loss of information in a digital computer system.

That's a very crucial result that we have seen last time.

Then we have seen how we can compute the coefficients of the Fourier series by using the property

that sine and cosine functions are orthonormal or orthogonal to each other.

You remember that?

And we will also have one or two exercise problems where you are required to compute

this type of integral.

It's very basic, but it takes a while to get used to this technical part.

And then we started out last week discussing representations of signals in computers.

What type of imagination should we develop to understand how signals are in our computer

and what type of processing methods we can apply.

And we have pointed out that image pre-processing or signal pre-processing or volume pre-processing

is important.

You remember the images with the cologne I have shown to you.

With a poor image quality you get from the detector and the high image quality you get

once you have applied proper filtering operations.

Then we started to discuss various types of digital filters.

So that was basically the story of our Thursday session.

And what we have to keep in mind that our signals are either vectors of numbers, of

digits or rectangles, matrices.

We call this structure a matrix, 2D structure or a 3D volume where each point in space gets

a digital value telling us for instance the physical density of the inner part of the

human body.

So I can for instance sample the head, the skull of a person, pixel or voxel by voxel

and measure the physical density in space and visualize this on the computer or on the

monitor.

How to compute these things we will learn later on.

For us it's important the signals we are considering are vectors, two dimensional structures or

three dimensional structures and these are grids and each grid point is associated with

an integer value and this integer value is associated with an intensity value for instance.

That's the picture.

This is the 1D, this is the 2D case and this is the 3D case.

You know what Einstein was saying you need to be equipped with the three I's.

You know the three I's that you have to develop over time if you want to be a good engineer

or scientist.

The three I's.

One thing is you can't change that anymore.

That's the intellect.

You have to be some kind of smart.

That's the intellect.

The second point is you have to develop a good intuition how to approach problems.

You know if I look at this problem I think I know in which direction it goes.

Without thinking about formulas and mechanisms and theory you need a good intuition to get

on the right track.

You need a good intuition how to approach problems.

And the third I you have to fulfill is imagination.

You have to keep good pictures in mind.