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Scan Conversion

Scan Conversion. CS123. Scan Converting Lines. Line Drawing First problem statement: Draw a line on a raster screen between two points Why is this a difficult problem? What is “drawing” on a raster display? What is a “line” in raster world? Efficiency and appearance are both important

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Scan Conversion

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  1. Scan Conversion CS123 Scan Conversion - 10/14/2014

  2. Scan Converting Lines Line Drawing • First problem statement: Draw a line on a raster screen between two points • Why is this a difficult problem? • What is “drawing” on a raster display? • What is a “line” in raster world? • Efficiency and appearance are both important Problem Statement • Given two points P and Q in XY plane, both with integer coordinates, determine which pixels on a raster screen should be on in order to best approximate a unit-width line segment starting at P and ending at Q Scan Conversion - 10/14/2014

  3. What is Scan Conversion? • Final step of rasterization (process of taking geometric shapes and converting them into an array of pixels stored in the framebuffer to be displayed) – converting from “random scan”/vector specification to a raster scan where we specify which pixels are on based on a stored array of pixels • Takes place after clipping occurs • All graphics packages do this at end of the rendering pipeline • Takes triangles (or higher-order primitives) and maps them to pixels on screen • For 3D rendering also takes into account other processes, like lighting and shading, but we’ll focus first on algorithms for line scan conversion Scan Conversion - 10/14/2014

  4. Scan-converting a Line: Finding the next pixel Special cases: • Horizontal Line: • Draw pixel P and increment x coordinate value by 1 to get next pixel. • Vertical Line: • Draw pixel P and increment ycoordinate value by 1 to get next pixel. • Diagonal Line: • Draw pixel P and increment both x and ycoordinate by 1 to get next pixel. • What should we do in the general case? • For slope <= 1, increment x coordinate by 1 and choose pixel on or closest to line. Slopes in other octants by reflection (e.g., in second octant, increment Y) • But how do we measure “closest”? Scan Conversion - 10/14/2014

  5. Vertical Distance • Why can we use vertical distance as a measure of which point (pixel center) is closer? • … because vertical distance is proportional to actual distance • Similar triangles show that true distances to line (in blue) are directly proportional to vertical distances to line (in black) for each point • Therefore, point with smaller vertical distance to line is closest to line Scan Conversion - 10/14/2014

  6. Strategy 1 – Incremental Algorithm (1/3) Scan Conversion - 10/14/2014

  7. Strategy 1 – Incremental Algorithm (2/3) Scan Conversion - 10/14/2014

  8. Strategy 1 – Incremental Algorithm (3/3) - Issues void Line(intx0,int y0, intx1,int y1){ int x,y; floatdy= y1 – y0; float dx = x1 – x0; float m =dy/ dx; y = y0; for(x =x0;x < x1; ++x){ WritePixel( x,Round(y) ); y = y + m; } } Since slope is fractional, need special case for vertical lines (dx = 0) Rounding takes time Scan Conversion - 10/14/2014

  9. Strategy 2 – Midpoint Line Algorithm (1/3) • Assume that line’s slope is shallow and positive (0 < slope < 1); other slopes can be handled by suitable reflections about principal axes • Call lower left endpoint and upper right endpoint • Assume that we have just selected pixel at • Next, we must choose between pixel to right (E pixel), or one right and one up (NE pixel) • Let Q be intersection point of line being scan-converted and vertical line Scan Conversion - 10/14/2014

  10. Strategy 2 – Midpoint Line Algorithm (2/3) NE pixel Previous pixel Q Midpoint M E pixel Scan Conversion - 10/14/2014

  11. NE pixel E pixel Strategy 2- Midpoint Line Algorithm (3/3) For line shown, algorithm chooses NE as next pixel. Now, need to find a way to calculate on which side of line midpoint lies Scan Conversion - 10/14/2014

  12. General Line Equation in Implicit Form • Line equation as function: • Line equation as implicit function: • Avoids infinite slopes, provides symmetry between x and y • So from line eqn, • Properties (proof by case analysis): • when any point is on line • when any point is above line • when any point is below line • Our decision will be based on value of function at midpoint m= at Scan Conversion - 10/14/2014

  13. Decision Variable Scan Conversion - 10/14/2014

  14. Incrementing Decision Variable if E was chosen: Scan Conversion - 10/14/2014

  15. If NE was chosen: Scan Conversion - 10/14/2014

  16. Summary (1/2) Scan Conversion - 10/14/2014

  17. Summary (2/2) Scan Conversion - 10/14/2014

  18. Example Code void MidpointLine(int x0,int y0,int x1,int y1){ intdx=(x1 - x0),dy=(y1 - y0); int d=2* dy - dx; int incrE=2* dy; int incrNE=2*(dy - dx); int x= x0,y= y0; WritePixel(x, y); while(x < x1){ if(d <=0) d = d + incrE; // East Case else{ d = d + incrNE;++y; } // Northeast Case ++x; WritePixel(x, y); } } Scan Conversion - 10/14/2014

  19. (0, 17) (0, 17) (17, 0) (17, 0) Scan Converting Circles Scan Conversion - 10/14/2014

  20. (x0 + a, y0 + b) R (x0, y0) (x-x0)2 + (y-y0)2 = R2 Version 3 — Use Symmetry Symmetry: • If is on circle centered at • Then and are also on the circle • Hence there is 8-way symmetry • Reduce the problem to finding the pixels for 1/8 of the circle. Scan Conversion - 10/14/2014

  21. (x0, y0) Using the Symmetry Scan Conversion - 10/14/2014

  22. The Incremental Algorithm – a Pseudocode Sketch Scan Conversion - 10/14/2014

  23. What we need for the Incremental Algorithm Scan Conversion - 10/14/2014

  24. The Decision Variable • If negative, midpoint inside circle, choose E • vertical distance to the circle is less at than at • If positive, opposite is true, choose SE Scan Conversion - 10/14/2014

  25. The right decision variable? Scan Conversion - 10/14/2014

  26. Alternate Phrasing (1/3) Scan Conversion - 10/14/2014

  27. Alternate Phrasing (2/3) Scan Conversion - 10/14/2014

  28. Alternate Phrasing (3/3) • The radial distance decision is whether is positive or negative. • The vertical distance decision is whether is positive or negative; and are ¼ apart. • The integer is positive only if ¼ is positive (except special case where : remember you’re using integers). Scan Conversion - 10/14/2014

  29. Incremental Computation Revisited (1/2) Scan Conversion - 10/14/2014

  30. Incremental Computation (2/2) • If we move E, update d = f(M)by adding • If we move SE, update dby adding • Forward differences of a 1st degree polynomial are constants and those of a 2nd degree polynomial are 1st degree polynomials • this “first order forward difference,” like a partial derivative, is one degree lower Scan Conversion - 10/14/2014

  31. Second Differences (1/2) • The function is linear, hence amenable to incremental computation: • Similarly Scan Conversion - 10/14/2014

  32. Second Differences (2/2) Scan Conversion - 10/14/2014

  33. Midpoint Eighth Circle Algorithm MidpointEighthCircle(R){ /* 1/8th of a circle w/ radius R */ intx =0, y =R; intdeltaE=2 * x +3; intdeltaSE=2 *(x - y)+5; floatdecision=(x + 1) * (x + 1)+(y -0.5) * (y -0.5)– R*R; WritePixel(x, y); while (y > x ){ if(decision >0){ // Move East x++;WritePixel(x, y); decision +=deltaE; deltaE+=2;deltaSE+=2;// Update deltas } else{ // Move SouthEast y--; x++;WritePixel(x, y); decision +=deltaSE; deltaE+=2;deltaSE+=4;// Update deltas } } } Scan Conversion - 10/14/2014

  34. Analysis • Uses floats! • 1 test, 3 or 4 additions per pixel • Initialization can be improved • Multiply everything by 4: No Floats! • Makes the components even, but sign of decision variable remains same Questions • Are we getting all pixels whose distance from the circle is less than ½? • Why is y > xthe right criterion? • What if it were an ellipse? Scan Conversion - 10/14/2014

  35. Other Scan Conversion Problems • Aligned Ellipses • Non-integer primitives • General conics • Patterned primitives Scan Conversion - 10/14/2014

  36. Aligned Ellipses Scan Conversion - 10/14/2014

  37. Direction Changing Criterion (1/2) Scan Conversion - 10/14/2014

  38. Direction Changing Criterion (2/2) • In our case, and so we check for • This, too, can be computed incrementally Scan Conversion - 10/14/2014

  39. Problems with aligned ellipses • Now in ENE octant, not ESE octant. Used the wrong direction because the ellipse rapidly changes slope within a pixel • This problem is an artifact of aliasing, remember filter? Scan Conversion - 10/14/2014

  40. Patterned Lines P Q P Q Patterned line from P to Q is not same as patterned line from Q to P. • Patterns can be cosmeticor geometric • Cosmetic: Texture applied after transformations • Geometric: Pattern subject to transformations Cosmetic patterned line Geometric patterned line Scan Conversion - 10/14/2014

  41. Geometric vs. Cosmetic + Cosmetic (Real-World Contact Paper) Geometric (Perspectivized/Filtered) Scan Conversion - 10/14/2014

  42. Non-Integer Primitives and General Conics Scan Conversion - 10/14/2014

  43. Generic Polygons • What's the difference between these two solutions? Under which circumstances is the right one "better"? Balsa Video - http://www.youtube.com/watch?v=GXi32vnA-2A Scan Conversion - 10/14/2014

  44. Scan Converting Arbitrary Solids • Rapid Prototyping is becoming cheaper and more prevalent • 3D printers lay down layer after layer to produce solid objects • We have an RP lab across the street in Barus and Holley • Printing food - http://www.youtube.com/watch?v=55NvbBJzDpU • Bio-printing - http://www.youtube.com/watch?v=-RgI_bcETkM Scan Conversion - 10/14/2014

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