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Rasterization

Rasterization. CS 418: Interactive Computer Graphics. UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN. Based on slide from Professor John Hart. Eric Shaffer. Fragment Pipeline. Rasterization is the process of converting vector graphics-style geometric primitives into a set of pixels.

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Rasterization

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  1. Rasterization CS 418: Interactive Computer Graphics UNIVERSITY OF ILLINOIS AT URBANA-CHAMPAIGN Based on slide from Professor John Hart Eric Shaffer

  2. Fragment Pipeline Rasterization is the process of converting vector graphics-style geometric primitives into a set of pixels

  3. How to Draw a Line • Jack Bresenham’s Algorithm • Given a line from point (x0,y0) to (x1,y1) • How do we know which pixels to illuminate to draw the line? • First simplify the problem to the first octant • Translate start vertex to origin • Reflect end vertex into first octant (x0-x1, y0-y1) (x1-x0, y1-y0) (x0,y0) (x1,y1)

  4. x’ = y y’ = -x x’ = y y’ = x x’ = -x y’ = y x’ = x y’ = y x’ = x y’ = -y x’ = -x y’ = -y x’ = -y y’ = x x’ = -y y’ = -x

  5. Line Rasterization 4 • How to rasterize a line from (0,0) to (4,3) • Pixel (0,0) and (4,3) easy • One pixel for each integer x-coordinate • Pixel’s y-coordinate closest to line • If line equal distance between two pixels, pick one arbitrarily but consistently 3 2 ? 1 ? 0 0 1 2 3 4

  6. Midpoint Algorithm • Which pixel should be plotted next? • East? • Northeast? NE E

  7. Midpoint Algorithm • Which pixel should be plotted next? • East? • Northeast? • Line equation y = mx + b m = (y1 – y0)/(x1 – x0) b = y0 – mx0 f(x,y) = mx + b – y f >0 f < 0 NE M E

  8. Midpoint Algorithm • Which pixel should be plotted next? • East? • Northeast? • Line equation y = mx + b m = (y1 – y0)/(x1 – x0) b = y0 – mx0 f(x,y) = mx + b – y • f(M) < 0  E • f(M)  0  NE NE M E

  9. Computing f (M) Fast If you know f (P), then you can compute f (M) easily: f(x,y) = mx + b – y M= P + (1,½) f(M) = f(x+1,y+½) = m(x+1) + b – (y+½) = mx + m + b – y – ½ = mx + b – y + m – ½ = f(P) + m – ½ NE M E P

  10. Preparing next f (P ) f(x,y) = mx + b – y The next iteration’s f (P ) isf (E ) or f (NE ) of this iteration F (E ) = f (x+1,y) = f (P ) + m F (NE ) = f (x +1,y +1) = f (P ) + m – 1 Also need f (P ) at start point: f (0,0) = b NE M E P

  11. Midpoint Increments f(M) = f(P) + m – ½ If choice is E, then next midpoint is ME f(ME) = f(x+2,y+½) = m(x+2) + b – (y+½) = f(P) + 2m – ½ = f(M) + m Otherwise next midpoint is MNE f(MNE) = f(x+2,y+1½) = m(x+2) + b – (y+1½) = f(P) + 2m – 1½ = f(M) + m – 1 Initialize: f(1, ½) = m + b – ½ NE 2 ENE NE MNE M ME E EE P

  12. Integer Math NE 2 f(ME) = f(M) + m f(MNE) = f(M) + m – 1 f(1, ½) = m + b – ½ b = 0 M = (y1 – y0)/(x1 – x0) = Dy/Dx Dx f(ME) = Dx f(M) + Dy Dx f(MNE) = Dx f(M) + Dy – Dx Dx f(1, ½) = Dy – ½ Dx ENE NE MNE M ME E EE P

  13. Integer Math NE 2 f(ME) = f(M) + m f(MNE) = f(M) + m – 1 f(1, ½) = m + b – ½ b = 0 m= (y1 – y0)/(x1 – x0)= Dy/Dx 2Dx f(ME) = 2Dx f(M) + 2Dy 2Dx f(MNE)=2Dx f(M)+2Dy–2Dx 2Dx f(1, ½) = 2Dy – Dx ENE NE MNE M ME E EE P

  14. Integer Math f(ME) = f(M) + m f(MNE) = f(M) + m – 1 f(1, ½) = m + b – ½ b = 0 m = (y1 – y0)/(x1 – x0)= Dy/Dx F(ME) = F(M) + 2Dy F(MNE) = F(M) + 2Dy – 2Dx F(1,½) = 2Dy – Dx NE 2 ENE NE MNE M ME E EE P

  15. Integer Math Bresenham Line Algorithm line(int x0,int y0,int x1,int y1) { intdx = x1 – x0; intdy = y1 – y0; int F = 2*dy – dx; intdFE = 2*dy; intdFNE = 2*dy – 2*dx; int y = y0; for (int x = x0, x < x1; x++) { plot(x,y); if (F < 0) { F += dFE; } else { F += dFNE; y++; } } } f(ME) = f(M) + m f(MNE) = f(M) + m – 1 f(1, ½) = m + b – ½ b = 0 m= (y1 – y0)/(x1 – x0)= Dy/Dx F(ME) = F(M) + 2Dy F(MNE) = F(M) + 2Dy – 2Dx F(1,½) = 2Dy – Dx

  16. PolygonRasterization • Ignore horizontal lines 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9

  17. PolygonRasterization 9 • Ignore horizontal lines • Sort edges by smallery coordinate 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  18. PolygonRasterization • For each scanline… • Add edges wherey = ymin • Sorted by x • Then by dx/dy 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  19. PolygonRasterization Plotting rules for when segments lie on pixels • Plot lefts • Don’t plot rights • Plot bottoms • Don’t plot tops 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  20. PolygonRasterization • y = 1 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  21. PolygonRasterization • y = 2 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  22. PolygonRasterization • y = 3 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  23. PolygonRasterization • y = 4 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  24. PolygonRasterization • y = 5 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  25. PolygonRasterization • y = 6 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  26. PolygonRasterization • y = 7 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  27. PolygonRasterization • y = 8 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  28. PolygonRasterization • y = 9 • Delete y = ymax edges • Update x • Add y = ymin edges • For each pair x0,x1, plot from ceil(x0)to ceil(x1) – 1 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

  29. Gouraud Shading Revisited • Flat shading • Per face normals • Color jumps across edge • Human visual perception accentuates edges • Smooth shading • Per vertex normals • Colors similar across edge • Edges become harder to discern Perception

  30. Gouraud Shading Revisited • Keep track of R, G, B at edge endpoints • Compute dR/dy, dG/dy and dB/dy per edge • Compute dR/dx, dG/dx and dB/dx at each scanline • Color each pixel R += dR/dx G += dG/dx B += dB/dx 9 8 D E 7 F 6 G 5 4 C 3 B 2 A 1 0 0 1 2 3 4 5 6 7 8 9

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