We've talked about in the past lectures about projective transformations and about vanishing points. Today, we are going to explain the geometric interpretation of the projecting transformations and how they relate to this vanishing point. Projective Transformation map planes to planes. They are also known as Homographies or Collineations. They represent the perspective projection from a ground plane to an image plane. They are an invertible 3x3 matrix. But it has only eight independent parameters. And if, for example, we know the coordinates of a point in the ground plane and the pixels in the image, then we can transform from one to another, back and forth. Let's take this example of a street, on Penn campus. We show a local coordinate system, with a yellow axis, but this can be any coordinate system, even the GPS coordinates. And we click at one point. The bottom point of this pole. In the image, it is 600,644 pixels. This is the coordinates in the ements plane. Now if we know the age, we can transform it back into the coordinates of the plane of the old. For example 10.2. 2.3, and visa versa, for any point we know to coordinates on the ground road. We can finds the pixels coordinates in the image. Let us look at the relation of the homography with the vanishing points. When we have vanishing points, these are intersections of parallel lines in the image. This means that the parallel lines originally do not intersect at all, but in the image they intersect and this is a real effect of a prospective projection. Let's imagine that we have parallel lines, parallel to the x axis. If we take an oblique image of this ground, they will intersect at the point A. Parallel lines to the y axis here on the ground, will dissect at the point B. Now if we take the vanishing point in the x direction, 1, 0, 0, and multiply it with the h matrix, we get the first column. So A can be represented by the first column of the collineation matrix. Or visa versa, if we know the collineation matrix, the first column has a very unique interpretation. Which is the vanishing point in the direction of the X axis on the ground plain. The same with the second column, it's in the rotation is nothing else than the vanishing point in the direction of the Y axis. Let us connect now these two vanishing points. This is well known as the horizon, the line of infinity. For any other set of parallel lines on the ground, they will intersect at a point in the horizon. For example, we see that the lines that are parallel to this corridor in the image. They will also intersect this orange horizon. How is the horizon related to the homography matrix because the point A has coordinate h1, the first column of the coordination. The point B is h2. We can learn that any lines with two points can be presented with a cross product. H1 cross H2. So the equation of this line is nothing else than H1 cross H2 and the inner product of the coordinates in the plane HY and 1. Now what is the meaning of this horizon geometrically? Let us imagine that we have this horizon line and we connect it with the projection center, which we see in the bottom. This makes a plane, and this plane is always parallel to the ground plane. Also, if we know the horizon and we know this plane, this plane will have a normal which will have coordinates H1 across H2. So this way we know also the normal for the ground plane. So the normal to the ground plane, is just the cross product of the first two columns. Or, if we just know the equation of the horizon, this gives us exactly the normal to the ground plane. What does normal to the ground plane mean? It means, really, how the ground plane is inclined. Let's take this example. When the horizon is exactly in the middle of the image, this means that our optical axis is exactly parallel to the ground plane. And another effect is an optical axis is really independent of whether we are turned to the left or to the right. In all cases, the horizon will remain the same. If we look downwards, the horizon is always on the plane parallel to the ground plane. This plane remains always the same. So if we look downwards, this plane will intersect with the line which is in the top of the image. For the same reason, if we look upwards, the plane will intersect our image plane at the line which is at the bottom of the image. If we start tilting out head the same way you see in flight simulator when you drive your plane and you always see the horizon, not only moving up and down, but also to the left or to the right. This means that we are rolling around the optical axis. So we see like this or like this. And this is the effect you had from the horizon. So to summarize, a homography contains a complete information about the horizon, as well as about two vanishing points on this horizon. We will see next time how using the notion of the cross ratio, we're going to be able to infer not only facts about the orientation, but also thoughts about real distances on this plane.
