Hello everyone and welcome back to Computer Vision lecture series.

This is lecture 3 part 4.

We are going to continue talking about how we think in terms of frequency domain.

In the last part of this lecture we saw some examples of sampling and one of the problems

of sampling called aliasing.

And now we are going to continue talking about more in terms of frequency domain.

So let's go ahead.

Another way of thinking about frequency is when you look at an image how many different

values of different pixels is there and how frequent those values occur.

And this gives you an idea of where we are going in this direction.

And in Fourier domain or in frequency domain we are basically handling images using frequencies.

And we are going to see what advantages does this analysis bring for images specifically

and computer vision in general.

We are going to talk about Fourier series, transforms, filtering in this domain as well

as deconvolution and one specific example of JPEG compression using this frequency

analysis.

So what is Fourier series?

Fourier series is basically a way of representing any univariate function in terms of weighted

sum of its sinusoidal waves of different frequencies.

Here in this example the basic building block here you can see is a sin omega t plus b cos

omega t.

This is a combination of two different sinusoidal waves with amplitudes a and b and the same

frequency omega.

When we have such frequencies when we add a lot of them we can represent our univariate

function.

An example is given on the right hand side.

Let's say our target function is this.

We start by adding f0 which is a static value constant value of 1 and then we have one sinusoidal

wave with frequency f1.

We add it to the f0 and so it shifts its amplitude from minus 1 to 1 to 0 to 2.

Similarly we add another frequency sinusoidal frequency a sinusoidal wave frequency f2 and

f3 and so on and so forth such that we are able to approximately generate the target

function mentioned here.

So basically a and b here are the amplitudes or the intensities of these sinusoidal waves

and omega is the frequency value of each of these sinusoidal waves.

A very concrete example or more simpler example of representing a function is this.

Here we have a signal g of t we can represent it in a combination of two sinusoidal waves

one with a frequency of f which is this one and another with the frequency of 3f.

It's clear that this wavelength when repeated three times is having a higher frequency here

and when you combine both of them we are able to generate our original we are able to represent

this function using a combination of these two sinusoidal waves.

When we look at the coefficients they are basically the values or the intensity of each

and every sinusoidal component.

Here the first component has a value of 1 which is represented by this bar here whereas

the second component with frequency 3f has amplitude of 1 by 3 which is represented here

and so this is a coefficient profile of the various intensities each sinusoidal component

has for a given for representing a given function.

How do we represent a square wave spectrum?

It is actually very difficult almost impossible if I may say to represent a square wave spectra.

It's not possible to have such a spectra in practice.