Hello everyone and welcome back to Computer Vision Lecture Series.

This is lecture 3 part 1.

In this lecture we are going to talk about image filters, specifically linear filters.

We are also going to talk about how these different kind of linear filters as well as

we are going to see some examples of those linear filters.

We will cover filters like mean filters, Gaussian filters and which filters are considered convolutional

filters.

Similarly, we also talk about the characteristics of linear filters and also one main characteristic

called separability.

So what is the formal definition of filtering mathematically?

Before we talk about mathematical definition, let's see what we really mean in real life

what filtering means.

Let's say you have you're going out in a summer afternoon to enjoy the for a bike ride.

Let's say for example and then you need shades to protect yourself from direct sunlight or

too bright reflection from the sun and you wear those shades.

The shades are acting as a filter for direct sunlight for you.

These are physical filters.

You might have also heard about Instagram filters or filters used in Photoshop and in

software like Photoshop and you are right that they are also considered filters and

they are specifically in the domain of image filters because they are operating on image

levels.

So here is the point where we discuss what is the formal definition of filtering in image

filtering.

Basically, you have a filter of size K cross L and you have an image of size M cross N

and when you do an operation or a neighborhood of the image and you calculate some values

or that neighborhood that computation this computation function is called filtering and

you do this for every each and every location across the image and you find and you generate

a new image basically.

Why is filtering important?

Because it helps to enhance images for example saturation improvement contrast enhancement

histogram equalization resizing denoising things like that.

You can also extract important information from images using filtering for example edges

textures some patterns you can also detect some templates predefined templates that you

are looking into.

So this is the basics of filtering.

What are linear filters now?

We want to know what specifically we mean when we say that a filter is linear.

So in order for a filter to be linear it has to satisfy two important properties or characteristics.

First is linearity here f1 and f2 are filters and i is an image.

This property says that if you have two different filters and if you combine them linearly and

then do you do a filtering operation that is equivalent to applying each and every individual

filters and converting and then doing a addition of these filtering operation individually.

And if this relationship holds then f1 is considered to have a linearity property.

Another important property is shift invariance.

So let's say if you shift a pixel location and then apply a filter or you apply a filter

and then you shift.

If this this property holds then the filters are considered to be shift invariant and if

you convert and if you have any filter which which follows or which satisfies both these

properties then they are they can be considered as linear filter.

So any linear shift invariant operator can be considered as convolution.