Perspective Projection

Transcription

1 Perspectie Projection (Com S 477/77 Notes) Yan-Bin Jia Aug 9, 7 Introduction We now moe on to isualization of three-dimensional objects, getting back to the use of homogeneous coordinates. Current display deices such as computer monitors, LCD screens, and printers are two-dimensional. Cameras also hae D image screens. It is necessary to obtain a planar iew of an object which gies three-dimensional realism. Visualization of an object is achieed by a sequence of operations called the iewing pipeline.. A projection is applied which maps the object to a new flat object in a specified plane known as the iewplane.. A coordinate system is defined in the iewplane and the description of the flat object in the coordinates is obtained.. The flat object is mapped to the computer screen by means of a two-dimensional deice coordinate transformation. We will first look at projections of the plane onto a line, and subsequently projections of threedimensional space onto a plane. Projections of the Plane Let us consider the problem of projecting the plane onto a line l contained in the plane. Let be a point not on the line. The perspectie projection from onto l is the transformation which maps any point p to the point p which is the intersection of the lines p and l, as illustrated in Figure. The point is called the iewpoint or center of perspectiity, and the line l is called the iewline. Theorem The perspectie projection from a iewpoint (in homogeneous coordinates) onto a iewline ector l is a twodimensional transformation gien by the matrix M l T (l )I, where I is the identity matrix. l p p Figure : Perspectie Projection

2 Proof The image point p of a point p is the intersection of the iewline l with the line l through p and the iewpoint. In homogeneous coordinates, the line l has the line ector p, and therefore intersects l at the point p l ( p) (l p) (l )p (l p) (l )I p l T p (l )I p ) (l T (l )I p. Thus p Mp where M l T (l )I. p l p The matrix M is called the projection matrix of the perspectie projection from onto l. Lines through the iewpoint are called projectors. When the iewpoint is a point at infinity, the projection is called a parallel projection. As shown in Figure, a parallel projection has iewpoint (,,), that is, the infinity point in the direction (, ). The projectors are parallel lines in the Cartesian plane with direction (, ). It is common practice to use the term perspectie projection to refer to a non-parallel projection. (,, ) infinity Figure : Parallel Projection Example. We determine the perspectie projection of the triangle with ertices ( ( ), 4 4, and ) onto the line x+y 4 from the iewpoint ( ). The homogeneous iewpoint is (,,) T, the line ector l (,, 4) T with l 48. We calculate the projection matrix as follows M l T (l )I The images of the ertices are obtained by multiplying M to their homogeneous coordinates: The Cartesian coordinates of the ertex images are ( / 4/, /7 8/7, and /9 4/9). They are shown in Figure (a).

3 p p (4, 4) p p (, ) (, ) p p (, ) (, ) (, ) l : x + y 4 (4, 4) (a) l : x + y 4 (b) Figure : Perspectie and parallel projections of a triangle. Example. Now let us determine the parallel projection of the same triangle in Example onto the line x+y 4 in the direction of the y-axis. The iewpoint is thus (,,) T, the point at infinity in the direction of the y-axis. We hae l (,, 4) T and l. From () the projection matrix is determined to be M 4 And the homogeneous coordinates of the images of the triangle ertices are contained in the matrix product The Cartesian coordinates are therefore (, 4 4, and /), as shown in Figure (b). Projections of Space Analogous to a iewline in the projection of the plane, a iewplane is inoled in the projection of three-dimensional space. Let n (a,b,c,d) be the plane ector of a iewplane described by the general equation ax + by + cz + d, and be a point not on the iewplane. The perspectie projection from onto the iewplane n is a transformation that maps any point p, onto the intersection point p of the line p and the plane. If is a point at infinity then the projection is called a parallel projection. The two projections are illustrated in Figure 4. Theorem The projection with homogeneous iewpoint and iewplane with plane ector n is the three-dimensional transformation gien by the matrix M n T (n )I 4, where I 4 is the 4 4 identity matrix. Proof Let p with p be a point to be projected. By Lemma from the handouts Homogeneous Coordinates of Space, eery point on the line through p and has the homogeneous.

4 p p p p (at infinity) Viewplane Viewplane (a) (b) Figure 4: (a) Perspectie projection and (b) parallel projection. coordinates of the form αp+β for some α and β such that α or β. The line intersects the iewplane if αp+β lies on the plane for some α and β, that is, when n (αp+β). Thus The following two cases arise: α(n p)+β(n ). (). When n p, the point p is on the iewplane and its projected image is itself. Meanwhile, we hae that ) Mp (n T (n )I 4 p (n p) (n )I 4 p (n )p. Thus, Mp is a multiple of p, or equialently, p itself in homogeneous coordinates.. When n p, from () we obtain that α β(n )/(n p). Substituting for α, the point of intersection has homogeneous coordinates p αp+β ( ) β(n )/(n p) p+β. Multiplying the coordinates by the scalar n p and diiding by β gies the alternatie homogeneous coordinates in matrix form: Hence we also hae M n T (n )I 4. p (n p) (n )p ) (n T (n )I 4 p. 4

5 z z z x y x y x y (abc) Figure : (a) A prism under (b) a parallel projection and (c) a perspectie projection. Example. A prism shown in Figure (a) has ertices (,,) T, (,,) T, (,,) T, (,,) T, (,,) T, and (,,) T. Consider a parallel projection of the prism onto the plane in a direction parallel to the z-axis. The iewpoint is (,,,) T, and the plane ector is n (,,,) T. We determine the projection matrix: M ( ) Applying the projection to the ertices of the prism yields. Thus the images of the ertices under the projection are (,,) T, (,,) T, (,,) T, (,,) T, (,,) T, and (,,) T, as shown in Figure (b). Next, we consider a perspectie projection onto the plane z from the iewpoint (,,) T. Here (,,,) T and n (,,,) T. And the projection matrix is M Note that β must hold, otherwise n p. ( )

6 ˆ û Figure 6: Image of a robotic arm batting an in-flight dumbbell-like object to hit three pins (courtsey of Matthew Gardner). Video on YouTube at which, multiplied to the homogeneous coordinates of the prism ertices, yields Thus the image ertices are (,,) T, (,,) T, (,,) T, (,,) T, (,,)T, and (,,) T, as shown in Figure (c).. 4 Viewplane Coordinate Mapping So far the iew of an object in the iewplane is expressed in homogeneous coordinates that correspond to the -dimensional world coordinates. The next stage is to define a coordinate system on the iewplane and represent the object in terms of these new coordinates. Figure 6 shows an image taken by a high speed camera of a robotic arm about to bat a flying dumbbell-shaped object. The camera s image plane has a local coordinate system located at the upper left corner. What is transformation from a point, say, the center of the dumbbell, in the D world to image coordinates that is automatically carried out by the camera? Generally, the iewplane (u, )-coordinate system is specified in world coordinates by an origin q (q,q,q ) T, and two unit ectors ˆr (r,r,r ) T and ŝ (s,s,s ) T which indicate the directions of the u- and -axes, respectiely. See Figure 7. Consider a point on the iewplane with homogeneous world coordinates p (x,y,z,t) T and homogeneous iewplane coordinates p (u,,w) T. To obtain the Cartesian coordinates of p, we can simply project the ector qp 6

7 iewplane coordinates p ( ) u/w /w q +ŝ q p u q + ˆr x/t y/t z/t p x y z z Perspectie projection Viewplane coordinate mapping p p Mp x ŝ ˆr y world coordinates p Vp Figure 7: Viewplane and world coordinate systems. onto the unit ectors ˆr and ŝ using inner products; namely, u w w x/t y/t z/t x/t y/t z/t q q q q q q But let us try to sole the problem by a homogeneous transformation instead. More specifically, we would like to obtain p from p by a mapping V such that p Vp, where V is a 4 matrix. Rather than compute V directly, we determine a 4 matrix K such that p Kp and then express V in terms of K. The matrix K can be determined, up to scaling, from four non-collinear points on the iewplane, using their homogeneous world coordinates and corresponding iewplane coordinates. We choose the following four points: (a) the origin (q,q,q,) T of the iew plane; (b) the point at infinity (r,r,r,) T in the direction of the u-axis of the iewplane coordinate system; (c) the point at infinity (s,s,s,) T in the direction of the -axis of the iewplane coordinate system; and (4) the point (t,t,t,) (q +r +s,q +r +s,q +r +s,) T which is one-unit each in the u and directions from the origin. The homogeneous iewplane coordinates of these points are (,,) T, (,,) T, (,,) T, and (,,) T, respectiely. Then the corresponding points are mapped to each other as follows: q r s t q r s t q r s t r s q r s q r s q 7 r r r s s s,.

8 u Figure 8: Projected prism in Figure in iewplane coordinates. Hence the transformation matrix is K r s q r s q r s q. Clearly, the column ectors of K are linearly independent. Therefore rank(k). The iewplane coordinate mapping is an inerse of the transformation determined by the matrix K. SinceK is not a squarematrix there is nomatrix inerse K. We usealeft inerse L, for which LK I. From (K T K) K T K I we hae a left inerse L (K T K) K T. Here (K T K) exists because rank(k). Since p Kp, we see that Lp (K T K) K T (Kp ) (K T K) (K T K)p p. Hence the iewplane coordinate mapping is gien by the matrix V L (K T K) K T. () Obsere that the iewplane coordinate matrix V is determined only by the choice of origin (q,q,q ) T, and the directions (r,r,r ) T and (s,s,s ) T of the u- and -axes. Example 4. Consider the perspectie projection in Example of a prism onto the plane z from the iewpoint (,,) T. Let a (u,)-coordinate system on the iewplane be gien by the origin (,,) T, u-axis direction (,4,) T, and -axis direction ( 4,,) T. The unit ectors in the axis directions are ˆr (, 4,)T, and ŝ ( 4,,)T. The iewplane coordinate matrix V (K T K) K T that maps a point in the space to a point in the iewplane is obtained in the following steps: K T K

9 (K T K) V 46 6, , 4 4 In Example, we already determined the homogeneous world coordinates of the prism ertices. Their homogeneous iewplane coordinates are computed through a further multiplication by V: Hence the Cartesian iewplane coordinates of the ertices are ( / /,, 7/ /, / 7/, 6/ ( / 9/). References [] D. Marsh. Applied Geometry for Computer Graphics and CAD. Springer-Verlag, /), 9