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Rendering and Rasterization

Rendering and Rasterization. Lars M. Bishop (lars@essentialmath.com). Rendering Overview . Owing to time pressures, we will only cover the basic concepts of rendering Designed to be a high-level overview to help you understand other references

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Rendering and Rasterization

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  1. Rendering and Rasterization Lars M. Bishop (lars@essentialmath.com)

  2. Rendering Overview • Owing to time pressures, we will only cover the basic concepts of rendering • Designed to be a high-level overview to help you understand other references • Details of each stage may be found in the references at the end of the section • No OpenGL/D3D specifics will be given Essential Math for Games

  3. Rendering: The Last Step • The final goal is drawing to the screen • Drawing involves creating a digital image of the projected scene • Involves assigning colors to every pixel on the screen • This may be done by: • Dedicated hardware (console, modern PC) • Hand-optimized software (cell phone, old PC) Essential Math for Games

  4. Destination: The Framebuffer • We store the rendered image in the framebuffer • 2D digital image (grid of pixels) • Generally RGB(A) • The graphics hardware reads these color values out to the screen and displays them to the user Essential Math for Games

  5. Top-level Steps We have a few steps remaining • Tessellation: How do we represent the surfaces of objects? • Shading: How do we assign colors to points on these surfaces? • Rasterization: How do we color pixels based on the shaded geometry? Essential Math for Games

  6. Representing Surfaces: Points • The geometry pipeline leaves us with: • Points in 2D screen (pixel) space (xs, ys) • Per-point depth value (zndc) • We could render these points directly by coloring the pixel containing each point • If we draw enough of these points, maybe we could represent objects with them • But that could take millions of points (or more) Essential Math for Games

  7. Representing Surfaces: Line Segments • Could connect pairs of points with line segments • Draw lines in the framebuffer, just like a 2D drafting program • This is called wireframe rendering • Useful in engineering and app debugging • Not very realistic • But the idea is good – use sets of vertices to define higher-level primitives Essential Math for Games

  8. Representing Surfaces: Triangles • We saw earlier that three points define a triangle • Triangles are planar surfaces, bounded by three edges • In real-time 3D, we use triangles to join together discrete points into surfaces Essential Math for Games

  9. A Simple Cube Cube represented by 8 points (trust us) Cube represented by triangles (defined by 8 points) Essential Math for Games

  10. The Power of Triangles • Flexible • We can approximate a wide range of surfaces using sets of triangles • Tunable • A smooth surface can be approximated by more or fewer triangles • Allows us to trade off speed and accuracy • Simple but Effective • We already know how to transform and project the three vertices that define a triangle Essential Math for Games

  11. Vertices – “Heavy” Points • As we will see, we often need to store additional data at each distinct point that define our triangles • We call these data structures vertices • Vertices provide a way of storing the added values we need to compute the color of the surface while rendering Essential Math for Games

  12. Common Per-vertex Values • Position • The “points” we’ve been transforming • Color • The color of the surface at/near the point • Normal vector • A vector perpendicular to the surface at/near the point • Texture coordinates • A value used to apply a digital image (akin to a decal on a plastic model) to the surface Essential Math for Games

  13. Efficient Rendering ofTriangle Sets • Indexed geometry allows vertices to be shared by several triangles • Uses an array of indices into the array of vertices • Each set of three indices determine a triangle • Uses much fewer verts than (3 x Tris) Essential Math for Games

  14. Indexed Geometry Example 2 1 3 6 0 4 5 Configuration 18 Individual vertices (exploded view) 7 shared vertices Index list for shared vertices (0,1,2),(0,2,3),(0,3,4),(0,4,5),(0,5,6),(0,6,1) Essential Math for Games

  15. Even More Efficient • Triangle Strips (tristrips) • Every vertex after the first two defines a triangle: (i-2, i-1, i) • Tristrips use much shorter index lists • (2 + Tris) versus (3 x Tris) • You may not quite get this stated efficiency • You may have to include degenerate triangles to fit some topologies • For example, try to build our earlier 6-tri fan… Essential Math for Games

  16. Tristrip Example 0 2 4 6 1 3 5 7 Index list for shared vertices (0, 1, 2, 3, 4, 5, 6, 7) Essential Math for Games

  17. Triangle Set Implementation • Most 3D hardware is optimized for tristrips • However, transformed vertices are often “cached” in a limited-size cache • Indexed geometry is not “perfect” – using a vertex in two triangles may not gain much performance if the vertex is kicked out of the cache between them • Lots of papers available from the hardware vendors on how to optimize for their caches Essential Math for Games

  18. Geometry Representation Summary • Generally, we draw triangles • Triangles are defined by three screen-space vertices • Vertices include additional information required to store or compute colors • Indexed primitives allow us to represent triangles more efficiently Essential Math for Games

  19. Shading - Assigning Colors • The next step is known as shading, and involves assigning colors to any and all points on the surface of each triangle • There are several common shading techniques • We’ll present them from least complex (and expensive) to most complex Essential Math for Games

  20. Triangle Shading Methods • Per-triangle: • Called “Flat” shading • Looks faceted • Per-vertex: • Called “Smooth” or “Gouraud” shading • Looks smooth, but still low-detail • Per-pixel: • Image-based texturing is one example • Programmable shaders are more general Essential Math for Games

  21. Flat Shading • Uses colors defined per-triangle directly as the color for the entire triangle Essential Math for Games

  22. Gouraud (Smooth) Shading • Gouraud shading defines a smooth interpolation of the colors at the three vertices of a triangle • The colors at the vertices (C0,C1,C2) define an affine mapping from barycentric coordinates (s,t) on a triangle to a color at that point: Essential Math for Games

  23. Gouraud Shading Essential Math for Games

  24. Generating Source Colors • Both Flat and Gouraud shading require source colors to interpolate • There are several common sources: • Artist-supplied colors (modeling package) • Dynamic lighting Essential Math for Games

  25. Dynamic Lighting • Dynamic lighting assigns colors on a per-frame basis by computing an approximation of the light incident upon the point to be colored • Uses the vertex position, normal, and some per-object material color information • Dynamic lighting is detailed in most basic rendering texts, including ours Essential Math for Games

  26. Imaged-based Texturing • Extremely powerful shading method • Unlike Flat and Gouraud shading, texturing allows for sub-triangle, sub-vertex detail • Per-vertex values specify how to map an image onto the surface • Visual result is as if a digital image were “pasted” onto the surface Essential Math for Games

  27. Imaged-based Texturing • Per-vertex texture coordinates (or UVs) define an affine mapping from barycentric coords to a point in R2. • Resulting point (u,v) is a coordinate in a texture image: (0,1) (1,1) V Axis (0,0) (1,0) U Axis Essential Math for Games

  28. Texture Applied to Surface • The vertices of this cylinder include UVs that wrap the texture around it: V=1.0 V=0.0 Essential Math for Games

  29. Rasterization:Coloring the Pixels The final step in rendering is called rasterization, and involves • Determining which parts of a triangle are visible (Visible Surface Determination) • Determining which screen pixels are covered by each triangle • Computing the color at each pixel • Writing the pixel color to the framebuffer Essential Math for Games

  30. Computing Per-pixel Colors • Even though rasterization is almost universally done in hardware today, it involves some interesting and instructive mathematical concepts • As a result, we’ll cover this topic in greater detail • Specifically, we’ll look at perspective projections and texturing Essential Math for Games

  31. Conceptual Rasterization Order • Conceptually, we draw triangles to the framebuffer by rendering adjacent pixels in screen space one after the other • Rasterization draws pixel (x,y), then (x+1,y), then (x+2,y) etc. across each horizontal span covered by a triangle • Then, it draws the next horizontal span Essential Math for Games

  32. Stepping in Screen Space • If we have a fast way to compute the color of a triangle at pixel (x+1,y) from the color of pixel (x,y), we can draw a triangle quite quickly • The same is true for computing texture coordinates • Affine mappings are perfect for this Essential Math for Games

  33. Affine Mappings on Triangles • Gouraud shading defines an affine mapping from world-space points in a triangle to colors • Texturing defines an affine mapping from world-space points on a triangle to texture coordinates (UVs) • Note, however, these are mappings from world space to colors and UVs Essential Math for Games

  34. Affine in Screen Space • For the moment, assume that we have an affine mapping from screen space (xs, ys) to color: • Note that the depth value is ignored Essential Math for Games

  35. Forward Differences • Note the difference in colors between a pixel and the pixel directly to the right: Wow – that’s nice! Essential Math for Games

  36. Forward Differencing • This simple difference leads to a useful trick: • Given the color for a base pixel in a triangle, the color of the other pixels are: • To step from pixel to adjacent pixel (i=1 and/or j=1) is just an addition! • This is called forward differencing Essential Math for Games

  37. Perspective • Perspective projection is by far the most popular projection method in games • It looks the most like reality to us • Perspective rendering does involve some interesting math and surprising visual results Essential Math for Games

  38. Affine Mapping of Texture UVs • Over the next few slides, we’ll apply a texture to a pair of triangles • We’ll use screen-space affine mappings to compute the texture coordinates for each pixel • First, let’s look at a special case: a pair of triangles parallel to the view plane Essential Math for Games

  39. Polygon Parallel to View:Affine Interpolation Wire-frame view Textured view CORRECT Essential Math for Games

  40. Looking good so far… • That worked correctly – maybe we can get away with using affine mappings (and thus forward differencing) for all of our texturing… • Not so fast – next, we’ll tilt the top of the square away from the camera… Essential Math for Games

  41. Polygon Tilted in Depth: Affine Interpolation Wire-frame view Textured view WRONG! Essential Math for Games

  42. What Happened? • We were interpolating using an affine mapping in (2D) screen space. • Perspective doesn’t preserve affine maps • Let’s look at the inverse mapping: namely, we’ll derive the mapping from world space to screen space • If this mapping isn’t affine, then the inverse (screen to triangle) can’t be affine, either Essential Math for Games

  43. Example – Projecting a Line • We’ll project a 2D line segment (y,z) into 1D using perspective: Projection of any point (not necessarily on line) to 1D: Line in world space: Essential Math for Games

  44. Projecting a Line Segment Y-axis View Plane D Z-axis P Essential Math for Games

  45. Special Case – Parallel to View Y-axis View Plane D=(DY,0) Z-axis P Affine when projected -That’s why this case worked Essential Math for Games

  46. General Case Y-axis D Z-axis P Not affine when projected – That’s why this case was wrong Essential Math for Games

  47. Correct Interpolation • The perspective projection breaks our nice, simple affine mapping • Affine in world space becomes projective in screen space • Per-vertex depth matters • The correct mapping from screen space to color or UVs is projective: Essential Math for Games

  48. Perspective-correct Stepping • The per-pixel method for stepping our projective mapping in screen space is: • Affine forward diff to step numerator • Affine forward diff to step denominator • Division to compute final value • This requires an expensive per-pixel division, or even several divisions Essential Math for Games

  49. Polygon Tilted in Depth: Correct Perspective Wire-frame view Textured view CORRECT! Essential Math for Games

  50. Colors versus Texture UVs • Textures need to be perspective-correct because the detailed features in most textures make errors obvious • This is why the example images that we have shown involve texturing • Gouraud-shaded colors are more gradual and can often get by without perspective correction Essential Math for Games

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