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Today we're going to talk about single view geometry and

Â let's start with a picture.

Â A simple picture like this, a picture of a street scene.

Â We see the lane marker, we see the distance, we see the building.

Â What else we see?

Â We see the lights.

Â We see that the lane markings are getting shorter and shorter.

Â How about the camera man himself?

Â Do we see ourself in the picture?

Â While we don't see ourself physically, but we do see ourself oriented in the street,

Â in the sense that, we are facing straight into the street lane markers.

Â Do we also see, know we are looking down or looking up.

Â You probably can guess this, that we are looking up rather than looking down,

Â because the horizons are up from where we're looking from the street.

Â And the information tells us how we are oriented in this space,

Â just from a single picture itself.

Â So this course will start looking at single pictures and

Â moving to sequence of pictures and videos.

Â From those we will try to infer the geometry of the scene, as well as,

Â how the camera man is looking into the scene.

Â All this started with the simple fact that we live in the three dimensional world.

Â That when we measure the world we tend to measure in x, y,

Â z location, where three numbers are x, y, and z.

Â But when we take a picture of it,

Â that three dimensional location has become a two dimensional plane.

Â And that plane could be the camera, or it could be our eyeballs, but

Â in short, we have only a two dimensional description of the world.

Â We had lost a third dimension in the process.

Â So that's the first fact.

Â The second fact is that when we take a picture it matters how we

Â are oriented to the world.

Â Depends how we orient ourself to the world, the image of that point,

Â project to different points in image play.

Â So those two facts are what we'll start using for the rest of the classes,

Â is simple facts, but they will become more interesting as we go on.

Â 3:34

This is pretty clumsy as you can imagine this take two person to draw our pictures.

Â In the 1600s there was another device made up of two parallel rods that

Â simplify the process, that requires only a single person to take a picture.

Â Here's a video of it.

Â So the rods are no longer horizontal or vertical.

Â They're made of two parallel rods that attach to a vertical rod.

Â The person looking through the object with line of sight lining up with the top rod.

Â And then painted with the bottom rods on the piece of paper.

Â Yet another device that we have is called a Bi-Dimensional Perspectograph,

Â and that's a much more interesting complicated device.

Â This allow you to draw a picture of a three dimensional

Â shape on the same piece of paper, on the same plane.

Â Here's a video illustrating this process of

Â taking a picture of this rectangular object.

Â And what we're trying to do is draw a picture as if this rectangle is a scene in

Â a vertical play, but we'll have this mechanical device to help us draw it.

Â On a horizontal paper.

Â This device required two linkage straight rods pivoting around a point,

Â and it's actually tracing through a set of geometrical constructions, points

Â of this rectangle on the vertical plane as seen As is through a picture plane.

Â But again we going to imagine this paper plane is placed horizontal on the table.

Â My colleague Costos will actually give you a more precise construction of this

Â geometrically.

Â So now of course we don't have to go through this complicated process of making

Â a picture.

Â We can just take our cell phones, go out.

Â And just click, with a click we can take a picture.

Â And we will start understanding, given this single picture, how do we see

Â the world, and how does the world reflect the geometry of the camera person himself?

Â For example,

Â we would like to know what are the lengths of This vertical post in the building.

Â If I tell you one of the posts is made about two meters tall,

Â and we want to know how tall is the other post?

Â Well we could probably guess,

Â because they're made of horizontal lines of the same length.

Â We probably can guess they're are about two meters as well But

Â how about the person standing between the two posts?

Â How tall is he?

Â We can probably infer a little bit less than two meters, but

Â exactly how tall is he?

Â We won't actually figure this out from just this single picture.

Â Another thing we want to know is exactly the length of the window, the height of

Â the window Is the person taller than the window or shorter than the window.

Â If I will can only infer this from the single picture.

Â This line of work in fact was started by Antonio Criminisi.

Â And was documented in the newspaper in 1999 So fairly reason work.

Â In this work he showed that from the single picture,

Â he's able to reconstruct three-dimensional measurements in the picture.

Â He was able to measure how wide the floor is, how tall the piano is,

Â how big the room is.

Â As a result he can do his three-dimensional simulation of this

Â world seen from a single picture.

Â So how can we do this?

Â Without even thinking about too much about the math, how do we do this?

Â The first idea is imagine we are looking at this picture of a hallway.

Â And we would like to measure distance on the floor.

Â So how will we actually go about measuring this distance on the floor?

Â Any ideas?

Â But what ideas we have is we know this floor is made of tiles.

Â And the tiles are made of rectangular shapes meaning that the two lines on

Â a horizontal direction, they are parallel.

Â And two lines in the vertical direction are Parallel as well and

Â they form in 90 degree angles.

Â So one idea is we match and we can virtually change the single view picture

Â such that the lines which are supposed to be parallel, stay parallel.

Â Or we could do it, just imagine we can deform the picture into so

Â that Lies in the vertical directions.

Â Move slowly, not converging to a point but

Â it starts staying parallel to each other.

Â Once we create this image we can sort of go in to the picture in this

Â virtual view and start measuring distance and angles.

Â That's one idea that requires us imagining how the picture looks like

Â by moving lines, which are supposed to be parallel into parallel lines,

Â and the angle is supposed to be completely rectangular into 90 degrees.

Â Another idea we're going to use Is the idea of vanishing points.

Â Vanishing points are the concept of a point which is

Â at infinity actually can be seen.

Â And this is a little bit strange concept,

Â but in the image we actually see them all the time.

Â For example we have this picture of the building we saw earlier.

Â That we see the roof and the ground planes form a horizontal line.

Â And those set of lines are parallel in physical space because we built

Â the building this way.

Â In fact all the lines on the roof they are stayed parallel to each other.

Â And those parallel lines, if you look at it, they are tilted.

Â On the fact to converge to a single point.

Â And this point is called a vanishing point.

Â Similarly we can see others [INAUDIBLE].

Â There are planks on the building, and

Â those lines are staying parallel in the physical world.

Â If we're looking under the plank, they're still parallel.

Â But somehow when we take a picture of this The lines start converging,

Â if I could converge the single point, and is also called a vanishing point.

Â In fact there's many, many vanishing point in the world.

Â And, they lies in a particular lie, which we'll call the horizon,

Â called the vanishing line And this concept allow us to

Â reason about the position of the camera man to the scene.

Â As well as reason about the elements in a scene and

Â the lengths of those elements relative to each other.

Â Let's look at this situation more precisely.

Â Here is a image of the plane which is your camera centred in front a camera

Â center which is marked in red and we look it straight on to the world.

Â Imagine there's a ground plane in front of us ans we're tracing a line on

Â the ground plane.

Â As those points move in this physical space They are projected to

Â the vertical image plane, as is shown by straight line projections.

Â As you can see, as we trace our line out in the physical

Â space they project to a line in the image space.

Â As we move further and further out with equidistant steps

Â Those projection points getting closer and closer to each other.

Â And in fact they will converge slowly to a point, which is at infinity as if we

Â were looking straight out in the direction with this line through the camera.

Â And that's it's point of a vanishing point.

Â The interesting thing about vanishing point is first of all there are many

Â physical lines in a physical space, they are not touching each other so

Â long as they're in the same direction they will go out and

Â eventually meet each other at a point of infinity.

Â That's the vanishing point with that set of lines.

Â And this is a very strange concept because we physically have never seen

Â a vanishing point.

Â No matter how big your camera is, how good your lens is,

Â you can never see a point in infinity.

Â But what we see is intersection of those projection lines forming a physical

Â point corresponding to a point in infinity which is many, many light years away.

Â In fact, it's never going to be reaching that point but

Â is a physically situated in the picture.

Â Here's an overhead view In a two dimensional plane.

Â So imagine we have looking camera plane as a vertical plane, and

Â we're looking at a set of lines tracing out.

Â All the lines are physically away from each other,

Â they are not touching each other, but

Â in fact all other points on those lines parallel to each other will converge To

Â a point in infinity where a projected image is marked at the vanishing points.

Â And that's the main concept we will use in the next few lectures.

Â Now it's interesting to turn things around to ask the question,

Â what if we have Lines that appear parallel in the image plane,

Â how do they look like in the ground plane?

Â So, we can imagine the situation.

Â We have a set lines on the ground plane and when they take a picture of it,

Â this picture Actually in fact shows those lines not staying convergent or

Â divergent but those lines stay parallel to each other.

Â So who are those lines?

Â At this point I actually want you to take your cell phone out, and

Â do the following exercise.

Â And the exercise is fairly simple,

Â that you were to draw a set of radiating lines On a piece of paper,

Â single point conversion, reading straight out.

Â I would like you to take you cell phone and

Â simply move your cell phone such that long directions radiations and

Â keep the camera vertical to the paper until you start seeing.

Â Projection your lines start looking vertical as you see in the right.

Â And mark this point of where the camera is rather the conversion point is,

Â we'll come back to that exercise later.

Â