Last time? Rasterization MIT EECS 6.837 Frédo Durand and Barb Cutler MIT EECS 6.837, Cutler and Durand Point and segment Clipping Planes as homogenous vectors (duality) In homogeneous coordinates before division Outcodes for efficient rejection Notion of conveity Polygon clipping via walking Line rasterization, incremental computation MIT EECS 6.837, Cutler and Durand 2 High-level concepts for 6.837 Linearity Homogeneous coordinates Conveity Discrete vs. continuous Incremental computation Trying things on simple eamples Scan Conversion (Rasterization) Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Rasterizes objects into piels Interpolate values as we go (color, depth, etc.) Visibility / Display MIT EECS 6.837, Cutler and Durand 3 MIT EECS 6.837, Cutler and Durand 4 Visibility / Display Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display Each piel remembers the closest object (depth buffer) Almost every step in the graphics pipeline involves a change of coordinate system. Transformations are central to understanding 3D computer graphics. MIT EECS 6.837, Cutler and Durand 5 Today Line scan-conversion Polygon scan conversion smart back to brute force Visibility MIT EECS 6.837, Cutler and Durand 6
Scan Converting 2D Line Segments Given: Segment endpoints (integers, y; 2, y2) Identify: Set of piels (, y) to display for segment (2, y2) Line Rasterization Requirements Transform continuous primitive into discrete samples Uniform thickness & brightness Continuous appearance No gaps (2, y2) Accuracy Speed (, y) (, y) MIT EECS 6.837, Cutler and Durand 7 MIT EECS 6.837, Cutler and Durand 8 Algorithm Design Choices Assume: m = dy/d, < m < Eactly one piel per column fewer disconnected, more too thick (2, y2) Naive Line Rasterization Algorithm Simply compute y as a function of Conceptually: move vertical scan line from to 2 What is the epression of y as function of? Set piel (, round (y())) (2, y2) y = y + ( y2 y) 2 = y + m( ) (, y) y (, y) dy m = d MIT EECS 6.837, Cutler and Durand 9 MIT EECS 6.837, Cutler and Durand Efficiency Computing y value is epensive y = y + m( ) Observe: y += m at each step (m = dy/d) y(+) y() (2, y2) y(+) m y() + (, y) + MIT EECS 6.837, Cutler and Durand Bresenham's Algorithm (DDA) Select piel vertically closest to line segment intuitive, efficient, piel center always within.5 vertically Same answer as naive approach MIT EECS 6.837, Cutler and Durand 2 2
Bresenham's Algorithm (DDA) Observation: If we're at piel P ( p, y p ), the net piel must be either E ( p +, y p ) or NE ( p +, y p +) Why? Bresenham Step Which piel to choose: E or NE? Choose E if segment passes below or through middle point M Choose NE if segment passes above M NE E NE M E NE M E MIT EECS 6.837, Cutler and Durand 3 MIT EECS 6.837, Cutler and Durand 4 Bresenham Step Use decision function D to identify points underlying line L: F> p D(, y) = y-m-b F= M positive above L F< zero on L negative below L D(p,p y ) = vertical distance from point to line MIT EECS 6.837, Cutler and Durand 5 Bresenham's Algorithm (DDA) Decision Function: D(, y) = y-m-b Initialize: error term e = D(,y) On each iteration: update : ' = + update e: e' = e + m if (e.5): y' = y (choose piel E) if (e >.5): y' = y + (choose piel NE) e' = e - MIT EECS 6.837, Cutler and Durand 6 e e Summary of Bresenham initialize, y, e for ( = ; 2; ++) plot (,y) update, y, e NE E Line Rasterization We will use it for ray-casting acceleration March a ray through a grid Generalize to handle all eight octants using symmetry Can be modified to use only integer arithmetic MIT EECS 6.837, Cutler and Durand 7 MIT EECS 6.837, Cutler and Durand 8 3
Line Rasterization vs. Grid Marching Questions? Line Rasterization: Best discrete approimation of the line Ray Acceleration: Must eamine every cell the line touches MIT EECS 6.837, Cutler and Durand 9 MIT EECS 6.837, Cutler and Durand 2 Circle Rasterization Generate piels for 2nd octant only Slope progresses from Analog of Bresenham Segment Algorithm MIT EECS 6.837, Cutler and Durand 2 Circle Rasterization Decision Function: D(, y) = 2 + y 2 R 2 Initialize: error term e = D(,y) R On each iteration: update : ' = + update e: e' = e + 2 + if (e.5): y' = y (choose piel E) if (e <.5): y' = y - (choose piel SE), e' = e + MIT EECS 6.837, Cutler and Durand 22 Philosophically Discrete differential analyzer (DDA): Perform incremental computation Work on derivative rather than function Gain one order for polynomial Line becomes constant derivative Circle becomes linear derivative Questions? MIT EECS 6.837, Cutler and Durand 23 MIT EECS 6.837, Cutler and Durand 24 4
Antialiased Line Rasterization Use gray scales to avoid jaggies Will be studied later in the course Today Line scan-conversion Polygon scan conversion smart back to brute force aliased antialiased MIT EECS 6.837, Cutler and Durand 25 Visibility MIT EECS 6.837, Cutler and Durand 26 2D Scan Conversion Geometric primitive 2D: point, line, polygon, circle... 3D: point, line, polyhedron, sphere... Primitives are continuous; screen is discrete 2D Scan Conversion Solution: compute discrete approimation Scan Conversion: algorithms for efficient generation of the samples comprising this approimation MIT EECS 6.837, Cutler and Durand 27 MIT EECS 6.837, Cutler and Durand 28 Brute force solution for triangles For each piel Compute line equations at piel center clip against the triangle Problem? Brute force solution for triangles For each piel Compute line equations at piel center clip against the triangle Problem? If the triangle is small, a lot of useless computation MIT EECS 6.837, Cutler and Durand 29 MIT EECS 6.837, Cutler and Durand 3 5
Brute force solution for triangles Improvement: Compute only for the screen bounding bo of the triangle How do we get such a bounding bo? Xmin, Xma, Ymin, Yma of the triangle vertices Can we do better? Kind of! We compute the line equation for many useless piels What could we do? MIT EECS 6.837, Cutler and Durand 3 MIT EECS 6.837, Cutler and Durand 32 Use line rasterization Compute the boundary piels Scan-line Rasterization Compute the boundary piels Fill the spans Shirley page 55 Shirley page 55 MIT EECS 6.837, Cutler and Durand 33 MIT EECS 6.837, Cutler and Durand 34 Scan-line Rasterization Requires some initial setup to prepare Today Line scan-conversion Polygon scan conversion smart Shirley page 55 back to brute force Visibility MIT EECS 6.837, Cutler and Durand 35 MIT EECS 6.837, Cutler and Durand 36 6
For modern graphics cards Triangles are usually very small Setup cost are becoming more troublesome Clipping is annoying Brute force is tractable Modern rasterization ComputeProjection Compute bbo, clip bbo to screen limits For all piels in bbo Compute line equations If all line equations> //piel [,y] in triangle Framebuffer[,y]=triangleColor MIT EECS 6.837, Cutler and Durand 37 MIT EECS 6.837, Cutler and Durand 38 Modern rasterization ComputeProjection Compute bbo, clip bbo to screen limits For all piels in bbo Compute line equations If all line equations> //piel [,y] in triangle Framebuffer[,y]=triangleColor Note that Bbo clipping is trivial Can we do better? ComputeProjection Compute bbo, clip bbo to screen limits For all piels in bbo Compute line equations If all line equations> //piel [,y] in triangle Framebuffer[,y]=triangleColor MIT EECS 6.837, Cutler and Durand 39 MIT EECS 6.837, Cutler and Durand 4 Can we do better? ComputeProjection Compute bbo, clip bbo to screen limits For all piels in bbo Compute line equations a+by+c If all line equations> //piel [,y] in triangle Framebuffer[,y]=triangleColor We don t need to recompute line equation from scratch MIT EECS 6.837, Cutler and Durand 4 Can we do better? ComputeProjection Compute bbo, clip bbo to screen limits Setup line eq compute a i d, b i dy for the 3 lines Initialize line eq, values for bbo corner L i =a i +b i y+c i For all scanline y in bbo For 3 lines, update Li For all in bbo Increment line equations: Li+=ad If all Li> //piel [,y] in triangle Framebuffer[,y]=triangleColor We save one multiplication per piel MIT EECS 6.837, Cutler and Durand 42 7
Adding Gouraud shading Interpolate colors of the 3 vertices Linear interpolation Adding Gouraud shading Interpolate colors of the 3 vertices Linear interpolation, e.g. for R channel: R=a R +b R y+c R Such that R[,y]=R; R[, y]=r; R[2,y2]=R2 Same as a plane equation in (,y,r) MIT EECS 6.837, Cutler and Durand 43 MIT EECS 6.837, Cutler and Durand 44 Adding Gouraud shading Interpolate colors ComputeProjection Compute bbo, clip bbo to screen limits Setup line eq Setup color equation For all piels in bbo Increment line equations Increment color equation If all Li> //piel [,y] in triangle Framebuffer[,y]=interpolatedColor Adding Gouraud shading Interpolate colors of the 3 vertices Other solution: use barycentric coordinates R=αR + β R + γ R 2 Such that P =α P + β P + γ P 2 P MIT EECS 6.837, Cutler and Durand 45 MIT EECS 6.837, Cutler and Durand 46 In the modern hardware Edge eq. in homogeneous coordinates [, y, w] Tiles to add a mid-level granularity Early rejection of tiles Memory access coherence Ref Henry Fuchs, Jack Goldfeather, Jeff Hultquist, Susan Spach, John Austin, Frederick Brooks, Jr., John Eyles and John Poulton, Fast Spheres, Shadows, Tetures, Transparencies, and Image Enhancements in Piel-Planes, Proceedings of SIGGRAPH 85 (San Francisco, CA, July 22 26, 985). In Computer Graphics, v9n3 (July 985), ACM SIGGRAPH, New York, NY, 985. Juan Pineda, A Parallel Algorithm for Polygon Rasterization, Proceedings of SIGGRAPH 88 (Atlanta, GA, August 5, 988). In Computer Graphics, v22n4 (August 988), ACM SIGGRAPH, New York, NY, 988. Figure 7: Image from the spinning teapot performance test. Triangle Scan Conversion using 2D Homogeneous Coordinates, Marc Olano Trey Greer http://www.cs.unc.edu/~olano/papers/2dh-tri/2dh-tri.pdf MIT EECS 6.837, Cutler and Durand 47 MIT EECS 6.837, Cutler and Durand 48 8
Take-home message The appropriate algorithm depends on Balance between various resources (CPU, memory, bandwidth) The input (size of triangles, etc.) Smart algorithms often have initial preprocess Assess whether it is worth it To save time, identify redundant computation Put outside the loop and interpolate if needed Questions? MIT EECS 6.837, Cutler and Durand 49 MIT EECS 6.837, Cutler and Durand 5 Today Line scan-conversion Polygon scan conversion Visibility How do we know which parts are visible/in front? smart back to brute force Visibility MIT EECS 6.837, Cutler and Durand 5 MIT EECS 6.837, Cutler and Durand 52 Ray Casting Maintain intersection with closest object Painter s algorithm Draw back-to-front How do we sort objects? Can we always sort objects? 3 4 5 7 2 6 MIT EECS 6.837, Cutler and Durand 53 MIT EECS 6.837, Cutler and Durand 54 9
Painter s algorithm Painter s algorithm Draw back-to-front How do we sort objects? Can we always sort objects? No, there can be cycles Requires to split polygons A A B C B Old solution for hidden-surface removal Good because ordering is useful for other operations (transparency, antialiasing) But Ordering is tough Cycles Must be done by CPU Hardly used now But some sort of partial ordering is sometimes useful Usuall front-to-back To make sure foreground is rendered first For transparency MIT EECS 6.837, Cutler and Durand 55 MIT EECS 6.837, Cutler and Durand 56 Visibility In ray casting, use intersection with closest t Now we have swapped the loops (piel, object) How do we do? Z buffer In addition to frame buffer (R, G, B) Store distance to camera (z-buffer) Piel is updated only if new z is closer than z-buffer value MIT EECS 6.837, Cutler and Durand 57 MIT EECS 6.837, Cutler and Durand 58 Z-buffer pseudo code Works for hard cases! Compute Projection, color at vertices Setup line equations Compute bbo, clip bbo to screen limits For all piels in bbo Increment line equations Compute curentz Increment currentcolor If all line equations> //piel [,y] in triangle If currentz<zbuffer[,y] //piel is visible Framebuffer[,y]=currentColor zbuffer[,y]=currentz A B C MIT EECS 6.837, Cutler and Durand 59 MIT EECS 6.837, Cutler and Durand 6
What eactly do we store Floating point distance Can we interpolate z in screen space? i.e. does z vary linearly in screen space? z image Z interpolation X =/z Hyperbolic variation Z cannot be linearly interpolated z image z z MIT EECS 6.837, Cutler and Durand 6 MIT EECS 6.837, Cutler and Durand 62 Simple Perspective Projection Project all points along the z ais to the z = d plane, eyepoint at the origin Yet another Perspective Projection Change the z component Compute d/z Can be linearly interpolated homogenize homogenize * d / z y * d / z d = y z z / d = /d y z MIT EECS 6.837, Cutler and Durand 63 * d / z y * d / z d/z = y z / d = /d y z MIT EECS 6.837, Cutler and Durand 64 Advantages of /z Can be interpolated linearly in screen space Puts more precision for close objects Useful when using integers more precision where perceptible Integer z-buffer Use /z to have more precision in the foreground Set a near and far plane /z values linearly encoded between /near and /far Careful, test direction is reversed far near MIT EECS 6.837, Cutler and Durand 65 MIT EECS 6.837, Cutler and Durand 66
Integer Z-buffer pseudo code Compute Projection, color at vertices Setup line equations, depth equation Compute bbo, clip bbo to screen limits For all piels in bbo Increment line equations Increment curent_ovz Increment currentcolor If all line equations> //piel [,y] in triangle If current_ovz>ovzbuffer[,y]//piel is visible Framebuffer[,y]=currentColor ovzbuffer[,y]=currentovz MIT EECS 6.837, Cutler and Durand 67 Gouraud interpolation Gouraud: interpolate color linearly in screen space Is it correct? [R, z G, B] image [R, z G, B] MIT EECS 6.837, Cutler and Durand 68 Gouraud interpolation Questions? Gouraud: interpolate color linearly in screen space Not correct. We should use hyperbolic interpolation But quite costly (division) However, can now be done on modern hardware image [R, z G, B] [R, z G, B] MIT EECS 6.837, Cutler and Durand 69 MIT EECS 6.837, Cutler and Durand 7 The infamous half piel I refuse to teach it, but it s an annoying issue you should know about Do a line drawing of a rectangle from [top, right] to [bottom,left] Do we actually draw the columns/rows of piels? The infamous half piel Displace by half a piel so that top, right, bottom, left are in the middle of piels Just change the viewport transform MIT EECS 6.837, Cutler and Durand 7 MIT EECS 6.837, Cutler and Durand 72 2
The Graphics Pipeline Modern Graphics Hardware Modeling Transformations Illumination (Shading) Viewing Transformation (Perspective / Orthographic) Clipping Projection (to Screen Space) Scan Conversion (Rasterization) Visibility / Display MIT EECS 6.837, Cutler and Durand 73 MIT EECS 6.837, Cutler and Durand 74 Graphics Hardware High performance through Parallelism Specialization No data dependency Efficient pre-fetching G task parallelism R T F D data parallelism MIT EECS 6.837, Cutler and Durand 75 G R T F D G R T F D G R T F D Programmable Graphics Hardware G P R T F P D Geometry and piel (fragment) stage become programmable Elaborate appearance More and more general-purpose computation (GPU hacking) MIT EECS 6.837, Cutler and Durand 76 Modern Graphics Hardware About 4-6 geometry units About 6 fragment units Deep pipeline (~8 stages) Tiling (about 44) Early z-rejection if entire tile is occluded Piels rasterized by quads (22 piels) Allows for derivatives Very efficient teture pre-fetching And smart memory layout MIT EECS 6.837, Cutler and Durand 77 Current GPUs Programmable geometry and fragment stages 6 million vertices/second, 6 billion teels/second In the range of tera operations/second Floating point operations only Very little cache MIT EECS 6.837, Cutler and Durand 78 3
Computational Requirements Questions? [Akeley, Hanrahan] MIT EECS 6.837, Cutler and Durand 79 MIT EECS 6.837, Cutler and Durand 8 Net Week: Ray Casting Acceleration MIT EECS 6.837, Cutler and Durand 8 4