The Computer Graphics Pipeline

Transcription

1 The Computer Graphics Pipeline vertices Vertex Processor Clipper and Assembler Rasterizer Fragment Processor pixels Programmable vs. Fixed Function Pipeline SIMD ECS 175 Chapter 2: The Computer Graphics Pipeline - Review 1

2 Chapter 3 1. Introduction 2. The Computer Graphics Pipeline 3. Object Representation 4. Object Transformation 5. 3D Projections, Camera, and Lighting 6. Scene Representation and Interaction 7. Advanced Texturing and Shading 2

3 Input - From Objects to Images How do we represent objects on the computer? Fulfill a number of requirements Low memory consumption Easy application of transformation operations (translate, clip, ) Closed representation (Object + Operation = Object) Easy to render Answer: Compose object of simpler base elements Vertices, Lines, Polygons (Triangles, Quads, ) Object represented as mesh/tessellation Additional optical properties (color) 3

4 Input - Basic Elements Vertex, Line, Triangle Drawing in OpenGL (fixed function - not using VBO): glbegin(gl_points); glbegin(gl_lines); glbegin(gl_triangles) glvertex2f(x,y); glvertex2f(x1,y1); glvertex2f(x1,y1); glend(); glvertex2f(x2,y2); glvertex2f(x2,y2); glend(); glvertex2f(x3,y3); glend(); 4

5 Input - Basic Elements Chrschn Wikimedia Commons 5

6 Remember the Graphics Pipeline vertices Vertex Processor Clipper and Assembler Rasterizer Fragment Processor pixels 3D Triangles Clipped Triangles 2D Triangles Fragments This chapter assumes a static 2D world After transformations and clipping What does our input look like? How do we show it on screen? 6

7 Rasterization Rasterization of 2D lines and triangles At this stage vertices are given in screen coordinates (2D) Attributes (color, depth, etc.) given per vertex 7

8 Rasterization Create fragments (potential pixels) Interpolate attributes (color, depth, etc.) from vertices Identify target pixels Integer grid centered at 0 or 0.5 (OpenGL) Here: pixels are squares 8

9 Rasterization DDA algorithm (digital differential analyzer) for lines y x = y 2 y 1 x 2 x 1 = m 0 m 1 y = m x float y = y 1 ; for(int x = x 1 ;x <= x 2 ;++x, y+=m) setpixel(x,round(y)); 9

10 Rasterization DDA algorithm Find best y for each x Slope condition is important (cover symmetric cases) 0 m 1 float y = y 1 ; for(int x = x 1 ;x <= x 2 ;++x, y+=m) setpixel(x,round(y)); 10

11 Rasterization Bresenham s algorithm avoids floating point operations 0 m 1 decide between y i and y i +1 y = y x x + b f(x, y) =x y + b x y x =0 decision based on sign of f(x +1,y+0.5) 11

12 Rasterization f(x, y) =x y + b x y x =0 D 0 = f(x 0 +1,y ) f(x 0,y 0 ) D 0 = y 1 2 x incremental decision variable if D >= 0 pick upper candidate D k+1 = D k + y if D k < 0 y x else multiply D by 2 to ensure integers 12

13 Rasterization Some examples 13

14 Rasterization Anti-aliasing (AA) Super sampling AA (SSAA) Multisample AA (MSAA) 14

15 Rasterization Rasterization of polygons not convex not simple Fill inside of polygon What is inside? Orientation: Clockwise, Counterclockwise 15

16 Rasterization What fragment positions are inside? odd-even test winding number 16

17 Rasterization Scanline algorithm Flood fill, Rasterizer interpolates attributes from vertices to fragments 17

18 What about colors? Colors (and other attributes) are specified at vertices How do we find values inside of polygons? 18

19 Interpolation Interpolation constructs new data within the range of known data points Construct a function with continuous domain from discrete data linear quadratic 19

20 Interpolation Mathematical formulation: Given (x i,f i ) ( positions and values ) Find function f defined on [x 0,x n ] such that f(x i )=f i 20

21 Interpolation We have already seen that there can be multiple solutions How do we pick the right function? Answer: Introduce constraints and/or make use of knowledge about data 21

22 Interpolation: Side Note Related concepts: Approximation Extrapolation 22

23 Interpolation: Linear Linear interpolation Interpolation conditions f(x 0 = 0) = f 0 f(x 1 = 1) = f 1 Linear interpolation Solution Alternative form f(x) =a + mx f(x) =f 0 +(f 1 f 0 )x f(u) =(1 u)f 0 + uf 1 23

24 Interpolation: Linear Linear interpolation Alternative form f(u) =(1 u)f 0 + uf 1 Values between x 0 and x 1 are mixtures of f 0 and f 1 u =0.2 80%f %f 1 24

25 Interpolation: Linear Basis functions/blending functions f(u) =(1 u)f 0 + uf 1 (1 u)+u =1 partition of unity f(u) =b 0 (u) f 0 + b 1 (u) f 1 f(u) = i b i (u) f i control point contributions are blended together 25

26 What about colors? Linear interpolation (component-wise) f RGB (t) =(f R (t),f G (t),f B (t)) 26