Hello everyone, welcome back to computer vision lecture series.

This is lecture 5 part 5.

We will continue talking about cameras and optics and specifically this part we are talking

about projective geometry and we are going to study the characteristics of the projections.

So let's go ahead.

In the last part we saw how we can model our projections which we generate from real world

to the image plane through pinhole camera model and that is our basic model to study

the process of generating the projections.

In this part we are going to talk about perspective projections which is the follow up of the

pinhole camera model and also we will discuss about the intrinsic and extrinsic properties

of the camera parameters and how adding one or removing one and what are the assumptions

being made and what are the free parameters to be estimated for the camera.

A little recap.

Projection what is projection?

Projection is basically having a world coordinates of x, y and z a point in the real world coordinate

being mapped into image coordinates in this image plane as p here.

So what happens here?

Let's follow this ray of light.

So from here from this point yellow point the light passes through the center of the

optical center or the camera center.

This can be a pinhole or a lens in general.

The ray of light passes through and it's projected on the image plane here.

What are the relationship of these distances that are formed or these objects that are

formed on the image plane and in the real world plane?

That is given by our intercept theorem.

Intercept theorem basically states that when you have this rays of light and you have this

camera axis going through here and you know that there is a real world plane here and

this is the image plane here and intercept theorem basically gives you the proportion

of the line segments formed between these two intersecting lines.

Here x is the real world point point in the real world and u is the its corresponding

point formed in the image plane.

Similarly y is a point is a point in the real world and v is the point formed in the image

plane.

So essentially the three dimensional world coordinates are converted into two dimensional

image coordinates.

That's what projection means.

In projections we are basically studying the geometry of our real world because we are

also mapping different objects into image plane and we want to study what kind of characteristics

the objects or the formed in the image plane still have or which and which properties have

they lost.

So in this example of the image plane here there is an image of the from the previous

part.

It's not easy to say the height of this person.

Same is here.

So both of these lines are same length but it's not easy to say what will be their heights.

So similar example another example is if you have these two objects here it's difficult

to say which is closer just because it is a perspective projection.

So essentially when we have mapped real world coordinates into an image plane these projections

do not have they do not preserve length and so if they don't preserve length they do not

preserve the area as well.