S Summary of Under. CG related to f U d CG l t d t CS680 - PowerPoint PPT Presentation

CS680: Scalable Global Illumination S Summary of Under. CG related to f U d CG l t d t CS680 Sung-Eui Yoon ( 윤성의 ) ( 윤성의 ) C Course URL: URL http://jupiter.kaist.ac.kr/~sungeui/CG

Overview of Computer Graphics Overview of Computer Graphics ● We will discuss various parts of computer W ill di i t f t graphics Modelling Simulation & Rendering Image Computer vision inverts the process Computer vision inverts the process Image processing deals with images 2

Lecture 2: Screen Space & World Space Space 3

Mapping from World to Screen Mapping from World to Screen Screen W World ld Window 4

Screen Space Screen Space ● Graphical image is presented by setting colors (width-1,0) (0,0) for a set of discrete samples for a set of discrete samples called “pixels” ● Pixels displayed on screen in windows ● Pixels are addressed as 2D arrays ● I ndices are “screen- space” coordinates space coordinates (0,height-1) (width-1, height-1) 5

OpenGL Coordinate System OpenGL Coordinate System 6

Pixel Independence Pixel Independence ● Often easier to structure graphical objects Often easier to structure graphical objects independent of screen or window sizes ● Define graphical objects in “world space” ● Define graphical objects in world-space 1.25 meters 500 cubits 2 meters 800 cubits 7

Lecture: 2D Transformation Lecture: 2D Transformation 8

2D Geometric Transforms 2D Geometric Transforms ● Functions to map F ti t points from one place to another ● Geometric transforms can be applied to ● Drawing primitives (points, lines, conics, triangles) ● Pi Pixel coordinates of an l di t f image Demo Demo 9

Translation Translation ● Translations have the following form: T l ti h th f ll i f t  x '  x x' = x + t x     x           y y' = y + t y y t t y y y '         y         y ● inverse function: undoes the translation: x = x' - t x y = y' - t y ● identity : leaves every point unchanged x' = x + 0 x = x + 0 y' = y + 0 10

2D Rotations 2D Rotations ● Another group - rotation about the origin: Another group rotation about the origin: 11

Rotations in Series Rotations in Series ● We want to rotate the object 30 degree We want to rotate the object 30 degree and, then, 60 degree  x '  cos(60) - sin(60) cos(30) - sin(30) x                sin(60) cos(60) sin(30) cos(30) y y '             We can merge multiple rotations into one rotation matrix x ' cos(90) - sin(90) x              sin(90) sin(90) cos(90) cos(90) y y y y '                 12

Euclidean Transforms Euclidean Transforms ● Euclidean Group E lid G ● Translations + rotations ● Rigid body transforms ● Rigid body transforms ● Properties: ● Preserve distances P di t ● Preserve angles ● How do you represent these functions? ● How do you represent these functions? 13

Problems with this Form Problems with this Form ● Translation and rotation considered Translation and rotation considered separately ● Typically we perform a series of rotations and ● Typically we perform a series of rotations and translations to place objects in world space ● I t’s inconvenient and inefficient in the previous form ● I nverse transform involves multiple steps ● How can we address it? ● How can we represent the translation as a matrix multiplication? 14

Homogeneous Coordinates Homogeneous Coordinates ● Consider our 2D plane as a subspace within Consider our 2D plane as a subspace within 3D (x y) (x, y) ( (x, y, z) ) 15

Matrix Multiplications and Homogeneous Coordinates Homogeneous Coordinates ● Can use any planar subspace that does not contain C l b th t d t t i the origin ● Assume our 2D space lies on the 3D plane z = 1 Assume our 2D space lies on the 3D plane z 1 ● Now we can express all Euclidean transforms in matrix form: form: 16

Scaling Scaling ● S is a scaling factor g  x '  s x     0 0             y ' ' s y    0 0          1          1 0 0 1     17

Frame Buffer Frame Buffer ● Contains an image for the final Contains an image for the final visualization ● Color buffer depth buffer etc ● Color buffer, depth buffer, etc. ● Buffer initialization B ff i iti li ti ● glClear(GL_COLOR_BUFFER_BI T); ● glClearColor (..); glClearColor ( ); ● Buffer creation ● glutI nitDisplayMode (GLUT_DOUBLE | l tI itDi l M d (GLUT DOUBLE | GLUT_RGB); ● Buffer swap ● Buffer swap ● glutSwapBuffers(); 18

Lecture: Modeling Transformation Transformation 19

The Classic Rendering Pipeline The Classic Rendering Pipeline ● Object primitives defined by ● Object primitives defined by vertices fed in at the top ● Pixels come out in the display at ● Pixels come out in the display at the bottom ● Commonly have multiple ● Commonly have multiple primitives in various stages of rendering 20

Modeling Transforms Modeling Transforms ● Start with 3D models defined in ● Start with 3D models defined in modeling spaces with their own t t t m , m ,..., m    modeling frames: 1 2 n ● Modeling transformations orient models within a common coordinate frame t t w w called world space , called world space  ● All objects, light sources, and the camera live in world space ● Trivial rejection attempts to eliminate eliminate objects that cannot possibly be seen ● An optimization 21

Illumination Illumination ● I lluminate potentially visible objects ● I lluminate potentially visible objects ● Final rendered color is determined by object’s orientation, its material object s orientation, its material properties, and the light sources in the scene 22

Viewing Transformations Viewing Transformations ● Maps points from world space to ● Maps points from world space to eye space: V   t t e w  ● Viewing position is transformed to the origin ● Viewing direction is oriented along Vi i di i i i d l some axis 23

Clipping and Projection Clipping and Projection ● We specify a volume called a viewing ● We specify a volume called a viewing frustum ● Map the view frustum to the unit cube ● Map the view frustum to the unit cube ● Clip objects against the view volume, thereby eliminating geometry not visible in thereby eliminating geometry not visible in the image ● Project objects j j into two-dimensions ● Transform from eye space to normalized device coordinates coordinates 24

Rasterization and Display Rasterization and Display ● Transform normalized device ● Transform normalized device coordinates to screen space ● Rasterization converts objects pixels ● Rasterization converts objects pixels - Almost every step in the rendering pipeline involves a change of coordinate pipeline involves a change of coordinate systems! - Transformations are central to understanding 3D computer graphics 25

Lecture: Interaction Lecture: Interaction 26

Primitive 3D Primitive 3D ● How do we specify 3D objects? How do we specify 3D objects? ● Simple mathematical functions, z = f(x,y) ● Parametric functions (x(u v) y(u v) z(u v)) ● Parametric functions, (x(u,v), y(u,v), z(u,v)) ● I mplicit functions, f(x,y,z) = 0 ● Build up from simple primitives ● Point – nothing really to see ● Point nothing really to see ● Lines – nearly see through ● Planes – a surface 27

Simple Planes Simple Planes ● Surfaces modeled as connected planar Surfaces modeled as connected planar facets ● N (> 3) vertices each with 3 coordinates ● N (> 3) vertices, each with 3 coordinates ● Minimally a triangle 28

Specifying a Face Specifying a Face ● Face or facet Face or facet Face [v0.x, v0.y, v0.z] [v1.x, v1.y, v1.z] … [vN.x, vN.y, vN.z] ● Sharing vertices via indirection Vertex[0] = [v0.x, v0.y, v0.z] Vertex[1] = [v1.x, v1.y, v1.z] Vertex[2] = [v2.x, v2.y, v2.z] v 3 v 0 v 2 v 2 : : v 1 Vertex[N] = [vN.x, vN.y, vN.z] Face v0 v1 v2 Face v0, v1, v2, … vN vN 29

Vertex Specification Vertex Specification ● Where Wh ● Geometric coordinates [x, y, z] ● Attributes ● Attributes ● Color values [r, g, b] ● Texture Coordinates [u, v] ● Orientation ● I nside vs. Outside ● ● Encoded implicitly in ordering Encoded implicitly in ordering ● Geometry nearby ● Often we’d like to “fake” a more complex shape than our true f faceted (piecewise-planar) model d ( i i l ) d l ● Required for lighting and shading in OpenGL 30