1 / 44

Viewing

Viewing. Doug James’ CG Slides, Rich Riesenfeld’s CG Slides, Shirley, Fundamentals of Computer Graphics, Chap 7. Wen-Chieh (Steve) Lin Institute of Multimedia Engineering. Getting Geometry on the Screen. Given geometry in the world coordinate system, how do we get it to the display?

Download Presentation

Viewing

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Viewing Doug James’ CG Slides, Rich Riesenfeld’s CG Slides, Shirley, Fundamentals of Computer Graphics, Chap 7 Wen-Chieh (Steve) Lin Institute of Multimedia Engineering

  2. Getting Geometry on the Screen Given geometry in the world coordinate system, how do we get it to the display? • Transform to camera coordinate system • Transform (warp) into canonical view volume • Clip • Project to display coordinates • Rasterize ILE5014 Computer Graphics 10F

  3. Vertex Transformation Pipeline ILE5014 Computer Graphics 10F

  4. Vertex Transformation Pipeline glMatrixMode(GL_MODELVIEW) glMatrixMode(GL_PROJECTION) glViewport(…) ILE5014 Computer Graphics 10F

  5. OpenGL Transformation Overview glMatrixMode(GL_MODELVIEW) gluLookAt(…) glMatrixMode(GL_PROJECTION) glFrustrum(…) gluPerspective(…) glOrtho(…) glViewport(x,y,width,height) ILE5014 Computer Graphics 10F

  6. Viewing and Projection • Our eyes collapse 3-D world to 2-D retinal image (brain then has to reconstruct 3D) • In CG, this process occurs by projection • Projection has two parts: • Viewing transformations: camera position and direction • Perspective/orthographic transformation: reduces 3-D to 2-D • Use homogeneous transformations ILE5014 Computer Graphics 10F

  7. Pinhole Optics F P • Stand at point P, and look through the hole—anything within the cone is visible, and nothing else is • Reduce the hole to a point - the cone becomes a ray • Pin hole is the focal point, eye point or center of projection ILE5014 Computer Graphics 10F

  8. Perspective Projection of a Point Image W F I World • View planeorimage plane- a plane behind the pinhole on which the image is formed • sees anything on the line (ray) through the pinhole F • a point W projects along the ray through F to appear at I (intersection of WF with image plane) ILE5014 Computer Graphics 10F

  9. Image Formation Image W Image F I F World World • Projecting a shape • project each point onto the image plane • lines are projected by projecting end points only Camera lens Note: Since we don't want the image to be inverted, from now on we'll put F behind the image plane. ILE5014 Computer Graphics 10F

  10. Orthographic Projection World Image F • When the focal point is at infinity the rays are parallel and orthogonal to the image plane • When xy-plane is the image plane (x,y,z) -> (x,y,0) front orthographic view ILE5014 Computer Graphics 10F

  11. Multiview Orthographic Projection • Good model for CAD and architecture. No perspective effects. front side top ILE5014 Computer Graphics 10F

  12. Assume view transformation is done • Let’s start with a simple case where the camera is put at the origin, and looks along the negative z-axis! • Simple “canonical” views: orthographic projection perspective projection ILE5014 Computer Graphics 10F

  13. Orthographic Viewing Cube • (l,b,n) = (left, bottom, near) • (r,t,f) = (right, top, far) ILE5014 Computer Graphics 10F

  14. Orthographic Projection • Map orthographic viewing cube to the canonical view volume • 3D window transformation • [l, r] x [b, t] x [f, n]  [-1, 1]x[-1, 1]x[-1,1] ILE5014 Computer Graphics 10F

  15. Orthographic Projection (cont.) • 3D window transform (last class) • [l, r] x [b, t] x [f, n]  [-1, 1]x[-1, 1]x[-1,1] • zcanonical is ignored ILE5014 Computer Graphics 10F

  16. Map Image Plane to Screen y (1,1) x x (nx, ny) (-1,-1) y Screen Image Plane ILE5014 Computer Graphics 10F

  17. Map Image Plane to Screen • If y-axis of screen coord. points downward • [-1,1] x [-1, 1]  [-0.5, nx-0.5] x [ny-0.5, -0.5] translation scale reflection ILE5014 Computer Graphics 10F

  18. Orthographic Projection Matrix • Put everything together ILE5014 Computer Graphics 10F

  19. Arbitrary Viewing Position • What if we want the camera somewhere other than the canonical location? • Alternative #1: derive a general projection matrix. (hard) • Alternative #2: transform the world so that the camera is in canonical position and orientation (much simpler) ILE5014 Computer Graphics 10F

  20. Camera Control Values • All we need is a single translation and angle-axis rotation (orientation), but... • Good animation requires good camera control--we need better control knobs • Translation knob - move to the lookfrom point • Orientation can be specified in several ways: • specify camera rotations • specify a lookat point (solve for camera rotations) ILE5014 Computer Graphics 10F

  21. A Popular View Specification Approach • Focal length, image size/shape and clipping planes are in the perspective transformation • In addition: • lookfrom: where the focal point (camera) is • lookat: the world point to be centered in the image • Also specify camera orientation about the lookfrom-lookfat axis ILE5014 Computer Graphics 10F

  22. Implementing lookat/lookfrom/vup viewing scheme • Translate by lookfrom, bring focal point to origin • Rotate lookfrom - lookat to the z-axis with matrix R: • w = (lookfrom-lookat) (normalized) and z = [0,0,1] • rotation axis: a = (w × z)/|w × z| • rotation angle: cosθ = w•z and sinθ = |w × z| • Rotate about z-axis to get vup parallel to the y-axis ILE5014 Computer Graphics 10F

  23. It's not so complicated… y x vup y z x y lookfrom x z START HERE z w Translate LOOKFROM to the origin Rotate the view vector (lookfrom - lookat) onto the z-axis. y x Multiply by the projection matrix and everything will be in the canonical camera position z Rotate about z to bring vup to y-axis ILE5014 Computer Graphics 10F

  24. Translate Camera Frame • Let e=(ex, ey, ez) be the “lookfrom” position y x vup y z x lookfrom z START HERE w Translate LOOKFROM to the origin ILE5014 Computer Graphics 10F

  25. Rotate Camera Frame • You can derive these two rotation matrices based on what you learned in the last class y x y x z z Translate LOOKFROM to the origin Rotate the view vector (lookat -lookfrom) onto the z-axis. y x z Rotate about z to bring vup to y-axis ILE5014 Computer Graphics 10F

  26. Rotate Camera Frame (cont.) • Alternatively, we can view these two rotations as a single rotation that aligns u-v-w-axes to x-y-z-axes! v y x z w w: lookat – lookfrom v: view up direction u = v x w ILE5014 Computer Graphics 10F

  27. Recall Global vs. Local Coordinate… • x, y, u, v, o, e are all vectors in global system In global coord. (xp,yp) In local coord. (up,vp) ILE5014 Computer Graphics 10F

  28. Think about 3D case …. • Given two coordinate frames, how do you represent a vector specified in one frame in the other frame? ? w v u Z P e Y ? X ILE5014 Computer Graphics 10F

  29. Viewing Transformation • Put translation and rotation together w v Z u P Y e local coordinate world coordinate X ILE5014 Computer Graphics 10F

  30. Orthographic Projection Summary Given 3D geometry (a set of points a) • Compute view transformation Mv • Compute orthographic projection Mo • Compute M = MoMv • For each point ai, compute p = Mai ILE5014 Computer Graphics 10F

  31. A Simple Perspective Camera • Canonical case: • camera looks along the z-axis (toward negative z-axis) • focal point is the origin • image plane is parallel to the xy-plane at distance d • We call d the focal length, mainly for historical reasons Image plane Center of projection ILE5014 Computer Graphics 10F

  32. Geometry Eq. for Perspective Projection view plane y ys g e d z • Diagram shows y-coordinate, x-coordinate is similar • Point (x,y,z) projects to e: eye position g: gaze direction ILE5014 Computer Graphics 10F

  33. Perspective Projection Matrix • Projection using homogeneous coordinates: • transform (x,y,z) to • 2-D image point: • discard third coordinate • apply viewport transformation to obtain physical pixel coordinates Divide by 4th coordinate (the “w” coordinate) ILE5014 Computer Graphics 10F

  34. View Volume • Pyramid in space defined by focal point and window in the image plane (assume window mapped to viewport) • Defines visible region of space • Pyramid edges are clipping planes ILE5014 Computer Graphics 10F

  35. View Frustum • Truncated pyramid with near and far clipping planes • Why far plane? Allows z to be scaled to a limited fixed-point value (z-buffering) • Why near plane? Prevent points behind the camera being seen ILE5014 Computer Graphics 10F

  36. Why Canonical View Volume? • Rather than derive a different projection matrix for each type of projection, we can convert all projections to orthogonal projections with the default view volume • This strategy allows us to use standard transformations in the pipeline and makes for efficient clipping ILE5014 Computer Graphics 10F

  37. Taking Clipping into Account • After the view transformation, a simple projection and viewport transformation can generate screen coordinate. • However, projecting all vertices are usually unnecessary. • Clipping with 3D volume. • Associating projection with clipping and view volume normalization. ILE5014 Computer Graphics 10F

  38. Normalizing the Viewing Frustum • Solution: transform frustum to a cube before clipping • Converts perspective frustum to orthographic frustum • This is yet another homogeneous transform! ILE5014 Computer Graphics 10F

  39. Map Perspective View Volume to Orthographic View Volume • We already know how to map orthographic view volume to canonical view volume • Set view plane at the near plane ILE5014 Computer Graphics 10F

  40. Map Perspective View Volume to Orthographic View Volume • Map [x,y,n] to [x, y, n] • Map [xf/n,yf/n,f] to [x, y, f] ILE5014 Computer Graphics 10F

  41. Projection Matrix is not unique • Mp is not unique ILE5014 Computer Graphics 10F

  42. Properties of Perspective Transform • Lines and planes are preserved • Parallel lines (not parallel to the projection plan) won’t be parallel after transform • vanishing point: parallel lines intersect at the vanishing point vanishing point ILE5014 Computer Graphics 10F

  43. Orthographic Projection Summary Given 3D geometry (a set of points a) • Compute view transformation Mv • Compute orthographic projection Mo • Compute M = MoMv • For each point ai, compute p = Mai ILE5014 Computer Graphics 10F

  44. Perspective Projection Summary Given 3D geometry (a set of points a) • Compute view transformation Mv • Map perspective to orthographic Mp • Compute orthographic projection Mo • Compute M = MoMpMv • For each point ai, compute p = Mai ILE5014 Computer Graphics 10F

More Related