Now we're going to talk about vanishing points. Vanishing points is a concept we introduced last time, indicating the point at infinity. Here's a cartoon illustrating the danger of driving on a highway that has no vanishing points. So vanishing points are concepts that we can measure in image. Give an image like we see here, we can find the set of lines in the physical space that are parallel to each other. Here, we find many of the yellow marked lines which are the ground plane. And as well as in the building. And those set of planes, lines, are parallel too each other in physical space. And they physically intersect at a point in infinity, but virtually, in the image in front of us, they are going to intersect at a physical point that we can measure in the image. We can do that by trace those lines by hand and see where they intersect. There's also a concept that we will see later, called a horizon. This is marked in green. And yet later, we will see the concept of vertical vanishing points. In here, we see, instead of the lines which are perpendicular to the horizon that's marked in red. It's going up and down. We'll come back to this concept later. Here is another image. In the Renaissance it surprised how good you are as an artist by how well you can paint the picture, preserve the perspective the view of the scene. And here we see again a set of lines in a physical space. Which are traced back to a single point. That's the point called the vanishing point of those set of lines. Here's another scene that's on campus at the University of Pennsylvania and this is the building we work at. Like the two picture before, we really don't see the horizon physically but we can still measure the horizons, as well as the vanishing points. So we have two set of points here, one set of points are marked red, this is we are marked on the ground plane. Follow the bricks that we see, and those are two side of the bricks, they're laid out in parallel. And we mark those in the image, and we trace them physically in the picture until they intersect at a point. And this is the point at infinity associated with the set of lines on the red marked in red on the ground plane. We will do that again find another set of physical lines on the ground plane. This time we focus on the blue lines to mark this blue. Again those are tiles on the ground that form parallel lines. And we've traced those images of those tiles border lines and marked them as blue. And this time when we traced them they don't intersect in the picture itself. But they intersect, nonetheless, somewhere just outside the picture. And the intersection point is marked and this is the vanishing point associated with those blue lines on the ground. What we can do is that, connecting those two vanishing points and this is what we marked as a horizon. A horizon is what we called, the plane on the ground if we were receding to infinity, if there's no building behind us we can see the whole Earth right in front of us. This is where the ground plane will intersect at infinity. So, through this process we can go out, we can take many pictures, try to carefully find those lines which are parallel to each other in the physical space, mark them in the picture, and we start finding vanishing points and vanishing lines. Before we go on we want it pointed out that not all pictures should have the same vanishing points. In fact if you look at a cartoons, this is an interesting cartoon drawn by a Korean artist. You purposely draw the picture so that there are many different vanishing points. And this is very interesting, because he certainly is a good artist, he knows how to draw a picture with one vanishing point, but he created this illusion of a space by creating multiple vanishing points, one corresponds to the left eye, one corresponds to the right eye. This is a very interesting artistic creation. I just want to point out to you that not all pictures you'll have the same vantage point for artistic illustration purpose. Speaking of arts, here's a picture, speaking of art we have a picture in front of us illustrating how an artist drawn a picture of a three dimensional object. There's a screen in front of him illustrating what he sees. And there's a physical object behind the screen illustrating object in a 3D world. And then what we see here is he pointing finger out in two different directions. He's pointing at one vanishing point at left and one vanishing point on right. Those are points in the image itself, but the direction he's pointing from himself to this point form a ray. Which define the direction corresponding to the direction of all the lines in the physical space, that were converging to the same vanishing point. Again, if you point to the right where you form a vector for himself to the points and image where the vanishing point is that array extends out, if that array is going to be physically parallel to all of the arrays in the physical space which are parallel to each other. So it's important to note this lines do not necessarily sits on a plane. They are physical arrays in a space, that are distinct from each other, but yet they are going to be pointing the same direction. And that direction is exactly the direction he is pointing for himself to the vanishing point, to the space outwards. And that direction is what we call a vanishing point in image. So those are the physical concepts and intuitions in the next few slides is we will start understand how do we express this in the algebraic formulations. And we're going to do that through the concept of homogenous coordinates. So first I want to introduce concept of a projective plane and the projective plane is a concept that we're looking at a two dimensional image plane just like an artist have been looking at it. But then we have a point which is where the virtual observer is looking at this image plane. Why do we want to look in this concept of homogeneous coordinates of projective plane? This is going to be become a convenience as we go forward. You will see that this is a great way to understand how to write down a coordinate system of point in infinity. And also it's very convenient way for understanding how do we form lines from points. In the algebraic formulation rather than you tracing it through a lines. So the first thing we need to understand is when we look at a two dimensional image every point in an image has a location in terms of x y, two numbers. But we're going to think a little bit differently with this two points, x y. We're going to think of this x y, in fact, is viewed not as someone on the plane itself but as an artist looking at this image, some distance away from the image plane. In fact we call the artist location zero zero zero in it's three dimensional coordinates and we think of this image plane as located exactly one distance away from the origin. As such, a point in image is no longer a point. You can think of that point as always an a ray going from artist to that point into the world. It's the direction he's looking at. So every points is going to be associated with a ray going from the zero zero origin. To that point, x y in the image plane and extend further out into the space. So we can associate a point with that ray and the ray again is formed by the point and the origin which is the observer himself. To summarize what we have is every time we look at an image plane which is two dimensional object. Anytime we look at a point x,y, we're going to think of that's a homogeneous coordinates of three numbers, which is x, y and 1, 1 because we think that of image plane is exactly 1 distance away. Another way to think about that is every time we look at a point x,y, we think of a ray that's radiating out from the observer to the image and penetrating to the space. Think of image as a projector, projected array to the space. So every point is associated with a ray. As such, that any three numbers, x y z, is the same as another point xyz If there is a just scale factor of each other. Commonly we normalize the third elements to be one, for a con optimization. So, that's four points. How do we think about lines? If you have an image in front of you, there's a line in front of you. How do I represent this line in the same fashion? We're going to think of that as, again, is not just looking at the line image. You are going to think of the formation of that line with the observer at one distance away from that image plane. Now what does that form? We see that in this case this line we can think of is formed by the set of rays radiating out from an observer going along the line. And this set of rays all has a coordinate system recognized we saw before x, y, z. And this set of x, y, z satisfy the property that ax + by + cz = to 0. Or if you're right on the interim linear algebra, will have the vector of ABC that was XYZ equal to zero. So this is our concept of a line in the space in the homogeneous coordinate. And what does a, b, c mean exactly, some of you might ask? This picture shows the relationship a, b, c with this line. If you look at this, this set of rays forming radiating out from the origin, passing through the line, forms a plane as we think about the line going points is going out from the origin to the space. And this set of rays are going to scans through the lines, again like a projector. And while we scan through this lines this set of rays forms a plane. And this plane in fact has a surface normal. Orientating exactly what's shown here is a b c. And a b c the way defined it in previous slide. In fact nothing but the surface normal corresponds to the three dimensional vector radiating out from the origin. Perpendicular to set of the rays formed by the line and the origin itself. So no longer we see this a, b, c are mystery elements. They are geometrical entities representing exactly the surface normal of the set of planes. So some of you might say wow that's very interesting but how does it relate to the concept we know in terms of what we had learned in high school about line representations. On the following slides there is a simple relationship between the a b c representation and homogeneous coordinate representation of the line to the line equation that some of us are more familiar with. Which is in terms of angle of the lines and the distance of the line to the origin. So you can think of that line equation our typical represent in the full on fashion where you have row equals the x times cosine theta plus y times sin theta where theta itself is representing the direction or certain normal direction to the line. So if I'm horizontal line, the normal direction will be 90 degrees. And row is this distance of the line to the origin. And the equation here shows how a, b and c is really related to the concept rows which is distance to the origin and the theta which is normal to the line, oriented to the line, surface normal, line normal. So you can see the cosine of theta is related to a and sin of theta is related to b and row has something to do with the c elements. So a, b, s,c we saw before was a surface normal in a three dimensional plane, but again those a,b,c, are related to the geometrical concept that we see on the 2D plane. In the following slides we will go through some exercise, illustrating those points. Here we have a building in front of us, and we see a picture of the facade, and we measure one set of lines on this facade. And we measure this line to have an orientation of 0.3 times pi. In the normal direction to the line. So the line is this direction and the normal is perpendicular to it. And this line forms a horizontal axis with .3 times Pi that's the angle of this line. And further we measure the distance to the origin as illustrated. In the next we have another set of lines. Those lines are in fact parallel to each other, again in the image they do not appear parallel. Again we can figure out the normal direction of this line to the origin which is this direction and we'll look at angle from x-axis to this normal vector. That gave me the line orientation and we can measure the distance to this. We can convert those orientation and the distance to the formula of a and b and c, which is illustrated here in the slide.
