COMP 558 lecture 5 Sept. 22, 2010

Transcription

1 Up to now, we have taken the projection plane to be in ront o the center o projection. O course, the physical projection planes that are ound in cameras (and eyes) are behind the center o the projection. For this reason, it will be convenient or us to consider two coordinate systems: (,Y o, ) represents the coordinates o a point in the scene, and (X i,y i,z i ) represents a point behind the center o projection. (Subscript o is or object, and i is or image.) Since the images on the projection plane behind the center o projectoin are upside-down and backwards, we orient the axis o X i to be opposite to and similarly or Y i vs Y o and Z i versus. See igure. The two coordinate systems share the same origin. (Xo, Zo) Xo Zo Zi (Xi, Zi) Xi Non-pinhole cameras The model that we have been discussing up to now assumes that we are projecting towards a single point the center o projection. I we project to a plane behind the origin, we have a pinhole camera. The idea o a pinhole camera is that we are allowing light to pass through a tiny hole in the Z = 0 plane and orming an image on a plane inside a black box. What happens i we open the pinhole so that the opening has a width A (which we reer to as the aperture)? Suppose the light passes through the aperture and arrives at a plane at depth behind the aperture. For simplicity suppose the point we are considering in the scene lies on a surace o constant depth Z 0 i.e. a wall that is oriented parallel to the. The resulting image will be blurred. We can think o the resulting blurring in two ways. See igures below. Each point on the sensor plane (X i,z i ) (where Z i = in this case) will receive light rom an area o points on the scene plane, namely rom the point that are visible through the aperture. (This is sometimes known as reverse projection.) Xo Reverse Projection Forward Projection Xi Zo Zo Alternatively, consider the point (, ) that projects to (X i,) through the center o the aperture, which we take to be the origin. This (X i,) is a single imaged point when the aperture

2 goes to zero but or a inite aperture, there is a set o rays rom (, ) that pass through the aperture and reach an area on the. (This is sometimes called orward projection.) Thin lens model Real cameras (and real eyes) indeed have apertures. These serve to allow more light to reach the sensors than a pinhole camera would. Cameras and eyes also have lenses, which redirect the light and ocus it. We will consider a simple model o the optics o lenses called the thin lens model. The thin lens model assumes that, or any point on an object (,Y o, ), the light rays that diverge rom that point and that pass through the lens all converge at some image point (X i,y i,z i ) behind the lens. Such points (,Y o, ) and (X i,y i,z i ) are called conjugate pairs. Note that this is just a model. Real lenses behave this way only approximately, and only when certain conditions are met you ll need to check out an Optics text i you want to understand what these conditions are. The relationship between the coordinates o a conjugate pair can be derived as ollows. Consider irst the case o a point (,Y o, ) = (0,0, ) which is the point at ininity in the direction o the optical axis or (0,0,,0) in homogeneous coordinates. The rays leaving this point and arriving at the Z = 0 plane are all parallel and they pass through the lens and converge at a point (X i,y i,z i ) = (0,0,) which is also (by symmetry) on the optical axis. This constant is called the ocal length o the lens. This constant depends on the curvature o the two sides o the lens and on the material o the lens (e.g. glass vs. water vs...). Note: does not depend on the distance rom the lens to the, since obviously the lens does not know where the is. Thus, we are using dierently rom how we used it in the previous lectures! This will make more sense later. Xo Zo Zi Xi One can relate the variables o a conjugate pair by assuming that the ray that passes through 2

3 the origin (the center o the lens) does not change direction and so, by similar triangles, we have X i Z i = Z 0. Another useul relationship comes rom similar triangles. There are two similar triangles in ront o the lens, giving X i = Z 0 and there are similar triangles behind the lens, giving = X i Z i. By rewriting each equation in terms o X i and then perorming a ew lines o algebra (do it!), one can isolate a relationship between,z i,, namely the thin lens ormula: + Z i = Notice that i, then Z i. In particular, i an object is very ar away then all the rays rom that object (which will be roughly parallel when they arrive at the lens) will converge at the depth Z i = behind the lens. We saw this earlier or the special case o a point on the optical axis, but now we see according to the model this property holds or all points at ininity. Another interesting observation comes when we rewrite the thin lens equation as: Z i = Z 0 Z 0. We now see that the transormation rom (,Y o, ) to (X i,y i,z i ) can be written: (X i,y i,z i ) = ( Z 0, Y o Z 0, Z 0 Z 0 ). We can represent this transormation rom a scene point to its image point using homogeneous coordinates: X i Y i Z i Y o Note that this transorm is its own inverse i.e. any point is the conjugate point o its conjugate point, in the sense that There is a bending o the light at the lens reraction but it turns out that, at the origin, the bending at the ront and back ace cancel out 3

4 One inal point: Most cameras people use these days do not have a single lens but rather they have a set or system o lenses. These lenses are all centered onthe same optical axis, and the camera is designed so that the user can move them relative to each other. The main eect o this design is that it allows the user to change the ocal length. This is what an optical zoom camera does. Note that the position o the center o the lens i.e. the origin o the camera coordinate system does not necessarily correpond to the physical center o any o the lens elements. Rather, we should think o a virtual lens center or an equivalent thin lens o ocal length. From now on (and in Assignment ), we will pretend there is a single lens and continue to talk about its center. Sensor plane and blur Suppose we put a at distance Z s rom the lens center. The conjugate points will lie at some depth Z s in the scene, according to namely = Z s + Z s Zs = Z s Z s deines the ocal plane in the scene. Points that do not lie at depth Zs will not be in ocus, in that the rays rom such points will not converge on a single point in the. Rather they will converge on a single point that is either in ront o or behind the. Thus, rays leaving a object point will strike an area on the. This is illustrated below. I the lens aperture is a disk, then the rays rom (,Y o, ) will arrive at a roughly disk shaped region on the sensor. This disk is oten called the circle o conusion. It is a roughly a circle because the lens aperture is (roughly) circularly shaped. This is easiest to imagine i you consider the scene point (0,0, ). since the scene and lens would all be rotationally symmetric about the optical axis, the blur region would also be rotationally symmetric. Let s next derive an expression or the blur width X i. We now consider a point at depth such that Zs. Let X i be the diameter o the blur disk. (In the igure, the blur disk is a dark line on the. For a 2D, it is a disk.) Let A be the diameter o the lens aperture. Then, by similar triangles, A Z i = X i Z s Z i and so X i = A Z s = A Z s ( Z i ) For a given photograph, all the terms on the right hand side are constant except or, and so we see that blur width is a constant plus another constant times. (Veriy that i = Zs, then the blur width is 0.) This is a very simple relationship. For example, recall rom lecture that i there is a plane in the scene then inverse depth varies linearly across the image coordinates (x,y). Z Hence the blur width would vary linearly across the image as well. We will return to this in Part 3 o the course. 4

5 ocal plane Zs * Zs Xi ocal plane -number (-stop) As we will see in an upcoming lecture, the amount o light that arrives at a point on the sensor plane depends on the number o rays that arrive at that point, and this in turn depends on the angle subtended by the lens. The angle subtended by the lens is approximately A radians, where A is the width o the aperture and is the distance rom the aperture to the sensor. (More precisely, the angle is approximately atan( A ) radians. The inverse o this ratio is called the -number (or 2 2 -stop): N A While is very common in photography to reer to the ocal length o the lens explicitly (in units o mm), it is much less common to reer to aperture explicitly. Instead, one reers to the aperture as a particular /# where # denotes a particular -number (i.e. N). For example, /4 reers to an aperture (in mm units) that corresponds to a particular ocal length and -number o 4. This is conusing or novice photographers, and indeed even experienced photographers sometimes write /4 when talking about the -number 4, when in act /4 reers to an aperture. 5