Hello everyone and welcome back to computer vision lecture series.

This is lecture 7 part 1.

In this lecture we are going to talk about epipolar geometry.

We are going to introduce some concepts of stereo vision and then calculate some disparity

map and then see how we can evolve or how does the epipolar geometry evolves from that

perspective.

And then in the second part we are also going to look into, so at the end of this part you

will see that we are going to compute some essential matrix and subsequently introduce

the fundamental matrix as well.

And from that point on in the next part we will explore how to estimate the fundamental

matrix using an 8 point algorithm as well as how we can do some robust model fitting

with ransack.

So let's begin.

This is a simple stereo system.

Here our aim is to recover some depth by two different images.

So you capture two different images of the same scene and so when you move around and

take another image without changing the camera and knowing the distance that you moved, you

are capturing a bit different image which is usually called stereo image pair.

So you have these two image pairs and how do you recover depth from it is a simple triangulation

problem which you have to solve.

So O and O dash here represent camera centers of this image captures that you generated

from these two different locations.

So F is constant in both the cameras.

This is the X that was projected in the image plane for the first image and X dash is the

projection of the same point in the world in the second image.

And Z is the distance of your camera system from the camera center from the world point.

Basically we assume parallel optical axis.

So you did not have any rotation or movement in any other direction of these two cameras

when you captured these two images.

And we already know the camera parameters.

So this is the kind of geometrical construction that we can come up with.

We know the F, we know the T distance which is the separation between the camera centers.

We also know the Z, Z is the distance of the world coordinate or the world point P from

the axis joining the two camera centers.

So OL and OR are left and right cameras.

You know these simple terminologies.

Here we see a simple mathematical construction where you see P1, P and PR to be one triangle

and OL, P and OR to be another triangle and they are similar triangle.

And using the properties of similar triangle you can with similar triangles the ratios

of their angles and sides are the same.

So using that idea you just calculate or solve for Z and you will know the distance between

the point and the camera center.

So using the disparity or you can find the disparity either way.

So you can solve one with the other and that is how you compute the disparity using a stereo

system.

So our goal usually is to recover depth by finding image coordinate X that corresponds

to X in these two different image planes.

But we have assumed a lot of things in this.

We have assumed that we know the camera parameters, we know the relationship in this case also.

So with the stereo setup we know and we have enforced parallel axis, the same camera system