CSE 167: Introduction to Computer Graphics Lecture #4: Vertex Transformation


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1 CSE 167: Introduction to Computer Graphics Lecture #4: Vertex Transformation Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2013
2 Announcements Project 2 due Friday, October 11 Compare online homework scores to your notes and let us know if they do not match To get graded early, please see course assistants during office hours New: Friday grading with two signup boards in labs 260 and 270 Changes to office hours in lab: see course web page 2
3 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 3
4 View Volumes View volume = 3D volume seen by camera Orthographic view volume Camera coordinates Perspective view volume Camera coordinates World coordinates World coordinates 4
5 Projection Matrix Camera coordinates Projection matrix Canonical view volume Viewport transformation 5 Image space (pixel coordinates)
6 Orthographic View Volume Specified by 6 parameters: Right, left, top, bottom, near, far Or, if symmetrical: Width, height, near, far 6
7 Orthographic Projection Matrix In OpenGL: glortho(left, right, bottom, top, near, far) 7 2 right + left 0 0 right left right left 2 top + bottom 0 0 P ortho (right,left,top,bottom,near, far) = top bottom top bottom 2 far + near 0 0 far near far near width P ortho (width,height,near, far) = height No equivalent in OpenGL 2 far + near 0 0 far near far near
8 Perspective View Volume General view volume Camera coordinates Defined by 6 parameters, in camera coordinates Left, right, top, bottom boundaries Near, far clipping planes Clipping planes to avoid numerical problems Divide by zero Low precision for distant objects Usually symmetric, i.e., left=right, top=bottom 8
9 Perspective View Volume Symmetrical view volume y y=top FOV z z=near Only 4 parameters Vertical field of view (FOV) Image aspect ratio (width/height) Near, far clipping planes z=far aspect ratio= tan(fov / 2) = top near right left top bottom = right top 9
10 Perspective Projection Matrix General view frustum with 6 parameters Camera coordinates In OpenGL: glfrustum(left, right, bottom, top, near, far) 10
11 Perspective Projection Matrix Symmetrical view frustum with field of view, aspect ratio, near and far clip planes Camera coordinates y y=top FOV z=near z z=far aspect tan(fov / 2) P persp (FOV,aspect,near, far) = tan(fov / 2) near + far 0 0 near far In OpenGL: gluperspective(fov, aspect, near, far) 11 2 near far near far
12 Canonical View Volume Goal: create projection matrix so that User defined view volume is transformed into canonical view volume: cube [1,1]x[1,1]x[1,1] Multiplying corner vertices of view volume by projection matrix and performing homogeneous divide yields corners of canonical view volume Perspective and orthographic projection are treated the same way Canonical view volume is last stage in which coordinates are in 3D Next step is projection to 2D frame buffer 12
13 Viewport Transformation After applying projection matrix, scene points are in normalized viewing coordinates Per definition within range [1..1] x [1..1] x [1..1] Next is projection from 3D to 2D (not reversible) Normalized viewing coordinates can be mapped to image (=pixel=frame buffer) coordinates Range depends on window (view port) size: [x0 x1] x [y0 y1] Scale and translation required: 13 D( x 0, x 1, y 0, y 1 )= ( x 1 x 0 ) x 0 + x 1 0 ( y 1 y 0 ) 2 0 y 0 + y ( ) 2 ( ) 2
14 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 14
15 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Object space M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix 15
16 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Object space World space M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix 16
17 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space 17
18 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space Canonical view volume 18
19 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Image space M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix Object space World space Camera space Canonical view volume 19
20 Complete Vertex Transformation Mapping a 3D point in object coordinates to pixel coordinates: Pixel coordinates: M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix 20
21 The Complete Vertex Transformation Object Coordinates Model Matrix World Coordinates Camera Matrix Camera Coordinates Canonical View Volume Coordinates Window Coordinates Projection Matrix Viewport Matrix 21
22 Complete Vertex Transformation in OpenGL Mapping a 3D point in object coordinates to pixel coordinates: OpenGL GL_MODELVIEW matrix M: Objecttoworld matrix C: camera matrix P: projection matrix D: viewport matrix OpenGL GL_PROJECTION matrix 22
23 Complete Vertex Transformation in OpenGL GL_MODELVIEW, C 1 M Defined by the programmer. Think of the ModelView matrix as where you stand with the camera and the direction you point it. GL_PROJECTION, P Utility routines to set it by specifying view volume: glfrustum(), glperspective(), glortho() Think of the projection matrix as describing the attributes of your camera, such as field of view, focal length, etc. Viewport, D Specify implicitly via glviewport() No direct access with equivalent to GL_MODELVIEW or GL_PROJECTION 23
24 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 24
25 Rendering Pipeline Scene data Rendering pipeline Hardware and software which draws 3D scenes on the screen Consists of several stages Simplified version here Most operations performed by specialized hardware (GPU) Access to hardware through lowlevel 3D API (OpenGL, DirectX) All scene data flows through the pipeline at least once for each frame 25 Image
26 Rendering Pipeline Scene data Modeling and viewing transformation Shading Textures, lights, etc. Geometry Vertices and how they are connected Triangles, lines, points, triangle strips Attributes such as color Projection 26 Rasterization, visibility Image Specified in object coordinates Processed by the rendering pipeline onebyone
27 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Transform object to camera coordinates Specified by GL_MODELVIEW matrix in OpenGL User computes GL_MODELVIEW matrix as discussed Rasterization, visibility MODELVIEW matrix 27 Image
28 Rendering Pipeline Scene data Modeling and viewing transformation Shading Look up light sources Compute color for each vertex Projection Rasterization, visibility 28 Image
29 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Rasterization, visibility Project 3D vertices to 2D image positions GL_PROJECTION matrix 29 Image
30 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Draw primitives (triangles, lines, etc.) Determine what is visible Rasterization, visibility 30 Image
31 Rendering Pipeline Scene data Modeling and viewing transformation Shading Projection Rasterization, visibility 31 Image Pixel colors
32 Rendering Engine Scene data Rendering pipeline Rendering Engine: Additional software layer encapsulating lowlevel API Higher level functionality than OpenGL Platform independent Layered software architecture common in industry Game engines Graphics middleware 32 Image
33 Lecture Overview View Volumes Vertex Transformation Rendering Pipeline Culling 33
34 Culling Goal: Discard geometry that does not need to be drawn to speed up rendering Types of culling: View frustum culling Occlusion culling Small object culling Backface culling Degenerate culling 34
35 View Frustum Culling Triangles outside of view frustum are offscreen Done on canonical view volume Images: SGI OpenGL Optimizer Programmer's Guide 35
36 Videos Rendering Optimisations  Frustum Culling View Frustum Culling Demo 36
37 Bounding Box How to cull objects consisting of may polygons? Cull bounding box Rectangular box, parallel to object space coordinate planes Box is smallest box containing the entire object 37 Image: SGI OpenGL Optimizer Programmer's Guide
38 Occlusion Culling Geometry hidden behind occluder cannot be seen Many complex algorithms exist to identify occluded geometry Images: SGI OpenGL Optimizer Programmer's Guide 38
39 Video Umbra 3 Occlusion Culling explained 39
40 Small Object Culling Object projects to less than a specified size Cull objects whose screenspace bounding box is less than a threshold number of pixels 40
41 Backface Culling Consider triangles as onesided, i.e., only visible from the front Closed objects If the back of the triangle is facing the camera, it is not visible Gain efficiency by not drawing it (culling) Roughly 50% of triangles in a scene are back facing 41
42 Backface Culling Convention: Triangle is front facing if vertices are ordered counterclockwise p2 p1 p0 p0 Frontfacing OpenGL allows one or twosided triangles Onesided triangles: glenable(gl_cull_face); glcullface(gl_back) Twosided triangles (no backface culling): gldisable(gl_cull_face) p1 Backfacing p2 42
43 Backface Culling Compute triangle normal after projection (homogeneous division) Third component of n negative: frontfacing, otherwise backfacing Remember: projection matrix is such that homogeneous division flips sign of third component 43
44 Degenerate Culling Degenerate triangle has no area Vertices lie in a straight line Vertices at the exact same place Normal n=0 Source: Computer Methods in Applied Mechanics and Engineering, Volume 194, Issues
45 Rendering Pipeline Primitives Modeling and Viewing Transformation Shading Projection Scan conversion, visibility Culling, Clipping Discard geometry that will not be visible 45 Image
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