Hello everyone and welcome back to computer vision lecture series.

This is lecture 7 part 2.

I decided to continue and do another part of this lecture that is the part 3.

After this you will have another part because mainly I wanted to focus on in this part about

and explain a little bit more about the fundamental matrix and the essential matrix because it

is not so clear sometimes and these mathematical concepts can be a bit difficult to grasp.

So I will also go through the geometric derivation of the fundamental matrix and then in this

part I will also explain 8 point algorithm how to compute the fundamental matrix and

some optimization of 8 point algorithm as well.

And in the next and the last part of this series and that is the third part we are going

to talk about RANSAC and HEW transforms some iterative optimization techniques to find

parameterized form of finding inliers and fitting algorithms basically.

So in this let us revisit the fundamental matrix.

Let us reconstruct our original setup.

In the left there is one image plane, on the right there is another image plane and we

assume a random plane pi in the real world where there lies a point capital X and its

corresponding small x in the left image is presented here on the right is x prime.

We say that for every x when it gets transformed to the other image there exists an x prime.

And there exists such points such pair of points for every x that lies in this image

plane.

And so we can say that there is a homography h pi which can map every x to x prime on the

other image using epipolar geometry.

Epipolar geometry and its epipolar constraints.

We will see how it is.

We can write and so because we know these pair of points exist we can say that these

points are projective points and they can be represented as x prime equals to applying

a homography of h pi to all the points in x.

In order to construct epipolar line l prime that passes through x prime we can consider

this case.

So in l prime can be written as e prime as a cross product of the epipole e prime and

x prime.

So we know x we know how to calculate x prime from that using the homography and we also

know that e prime is the projection of the other camera center.

So when we take the cross product of this we get the epipolar line l prime that passes

through x prime.

For simplicity and coming back to representing these cross products in matrices we represent

this cross product as e prime in the matrix form and the dot product with x prime.

We also know that x prime and x are homography can be recovered from homography.

We can write x prime as h pi of x and similarly for x we can write we can construct the epipolar

line l as the epipolar epipole matrix in this form with h pi and x prime.

In simplistic terms we can say that the f matrix or the fundamental matrix that transforms

one point to the other epipolar line can be written in this form.

So what is a fundamental matrix here?

We can see that it basically transforms every point in one plane or one image plane to its

corresponding epipolar line in the other image and this is very strong and very useful transformation.

We have already seen in the previous part of the lecture how calibrated and uncalibrated

cases can be solved using this epipolar constraint and this is very beautiful relationship.

It is very geometric, it is very grounded in analysis and so it is easier to implement

also.

We will see in the next slides how to implement that as well.