Cs580
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  • 1. Line Drawing and Generalization Sung Yong Shin Dept. of Computer Science KAIST CS580 Computer Graphics
  • 2. Outline
    • overview
    • line drawing
    • circle drawing
    • curve drawing
  • 3. 1. Overview application data structures/models application programs graphics systems display devices graphics primitives OpenGL pixel information
  • 4. Geometric Primitives
    • Building blocks for a 3D object
    Application programs must describe their service requests to a graphics system using geometric primitives !!! points, lines, polygons Why not providing data structures directly to the graphics system?
  • 5.
    • display lists
    • evaluators
    • per-vertex operations & primitive assembly
    • pixel operations
    • rasterization
    • texture assembly
    • per-fragment operations
    OpenGL Rendering Pipeline per-vertex operations & primitive assembly rasterization frame buffer texture assembly display lists evaluators pixel operations per-fragment operations geometric data pixel data
  • 6. Continuity 8-connectivity (king – continuity) 4-connectivity (rook – continuity)
  • 7. 2. Line Drawing f f : L 1 => L 2  quality degradation!! Line drawing is at the heart of many graphics programs.  Smooth, even, and continuous as much as possible !!! Simple and fast !!! ( x 1 , y 1 ) ( x 2 , y 2 )
  • 8. Bresenham’s Line Drawing Algorithm via Pragram Transformation
    • additions / subtractions only
    • integer arithmetic
    • not programmers’ point of view but system developers’ point of view
  • 9. var yt : real;  x,  y, xi, yi : integer; for xi := 0 to  x do begin yt := [  y/  x]*xi; yi := trunc(yt+[1/2]); display(xi,yi); end; var yt : real;  x,  y, xi, yi : integer; yt := 0; for xi := 0 to  x do begin yi := trunc(yt + [1/2]); display(xi,yi); yt := yt+[  y/  x] end; Eliminate multiplication !!!  x  y y = mx, m = [  y/  x] * **  x ≥  y ∴ m ≤ 1  x,  y: positive integers (0,0) ( ∆x, ∆y)
  • 10. Motivation(Cont’) var ys : real;  x,  y, xi, yi : integer; ys := 1/2; for xi := 0 to dx do begin yi := trunc(ys); display(xi,yi); ys := ys+[  y/  x] end; var ysf : real;  x,  y, xi, ysi : integer; ysi := 0; ysf := 1/2; for xi := 0 to  x do begin display(xi,ysi); if ysf+[  y/  x] < 1 then begin ysf := ysf + [  y/  x]; end else begin ysi := ysi + 1; ysf := ysf + [  y/  x-1]; end; end; integer part fractional part *** ****
  • 11. Motivation(Cont’) var  x,  y, xi, ysi, r : integer; ysi := 0; for xi := 0 to  x do begin display(xi,ysi); if then begin end else begin ysi := ysi + 1; end; end;
  • 12. Motivation(Cont’) var  x,  y, xi, ysi, r : integer; ysi := 0; r := 2*  y -  x; for xi := 0 to  x do begin display(xi,ysi); if r < 0 then begin r := r + [2*  y]; end else begin ysi := ysi + 1; r := r + [2*  y -2*  x ]; end; end; Bresenham’s Algorithm !!! No multiplication/ division. No floating point operations.
  • 13. Line-Drawing Algorithms Assumptions
  • 14. DDA(Digital Differential Analyzer) Algorithm basic idea Take unit steps with one coordinate and calculate values for the other coordinate i.e. or discontinuity !! Why?
  • 15. DDA(Cont’) procedure dda (x1, y1, x2, y2 : integer); var  x,  y, k : integer; x, y : real begin  x := x2 - x1;  y := y2 - y1; x := x1; y := y1; display(x,y); for k := 1 to  x do begin x := x + 1; y := y + [  y/  x]; display(round(x),round(y)); end { for k } end; { dda } expensive !! no *’s Assumption : 0  m <1, x1<x2
  • 16. Bresenham’s Line Algorithm
    • basic idea
      • Find the closest integer coordinates to the actual line path using only integer arithmetic
      • Candidates for the next pixel position
    Specified Line Path Specified Line Path
  • 17. Bresenham’s Line algorithm(Cont’)
  • 18. Bresenham’s Line algorithm(Cont’) =1
  • 19. Bresenham’s Line algorithm(Cont’) procedure bres_line (x1, y1, x2, y2 : integer); var  x,  y, x,y,p,incrE,incrNE : integer; begin  x := x2 – x1;  y := y2 – y1; p := 2*  y -  x; incrE := 2*  y; incrNE := 2*(  y -  x); x := x1; y := y1; display(x,y); while x < x2 do begin if p<0 then begin p := p + incrE; x := x + 1; end; { then begin } else begin p := p + incrNE; y := y + 1; x := x + 1; end; { else begin } display (x, y); end { while x < x2 } end; { bres_line} Homework #2 Extend this algorithm for general cases.
  • 20. Midpoint Line Algorithm Van Aken, “An Efficient Ellipse - Drawing Algorithm” IEEE CG & A, 4(9), 24-35, 1984. Current pixel Choices for next pixel
  • 21. Midpoint Line Algorithm (Cont’)
  • 22. Midpoint Line Alg. (Cont’)
  • 23. Midpoint Line Alg. (Cont’) Midpoint Line Algorithm(Cont’) procedure mid_point (x1, y1, x2, y2,value : integer); var  x,  y,incrE,incrNE,p,x,y : integer; begin  x := x2 – x1;  y := y2 – y1; p := 2*  y -  x; incrE := 2*  y; incrNE := 2*(  y-  x); x := x1; y := y1; display(x,y); while x  x2 do begin if p < 0 then begin p := p + incrE; x := x + 1; end; { then begin } else begin p := p + incrNE; y := y + 1; x := x + 1; end; { else begin } display(x,y); end; { while x  x2 } end; { mid_point }
  • 24. Geometric Interpretation any slope Bresenhams’s algorithm
  • 25. Geometric Interpretation(Cont’) x y
  • 26. Aliasing Effects staircases (or jaggies) intensity variation
    • line drawing
  • 27. animation texturing popping-up
  • 28. Anti-aliasing Lines
    • Removing the staircase appearance of a line
      • Why staircases?
        • raster effect !!!
        • need some compensation in line-drawing algorithms for this raster effect?
      • How to anti-alias?
      • well, …
      • increasing resolution .
      • However, ...
  • 29. Increasing resolution memory cost memory bandwidth scan conversion time display device cost
  • 30. Anti-aliasing Line (Cont’)
    • super sampling
    • (postfiltering)
    • area sampling
    • (prefiltering)
    • stochastic sampling
    Cook, ACM Trans. CG, 5(1), 307-316, 1986. Anti-aliasing Techniques
  • 31. Area Sampling (Prefiltering)
  • 32. Filters box filter cone filter unweighted area sampling weighted area sampling
  • 33. Table look-up for f ( D , t ) The table is invariant of the slope of line! Why? The search key for the table is D! Why? D
  • 34. Intensity Functions f ( D , t ) (assumption : t=1)
  • 35. How to compute D (assumption : t=1)
  • 36. How to compute D , incrementally
  • 37. IF (Cont’) Similarly, M D 1 - v 1 + v v
  • 38. IF (Cont’)
    •  x := x2 - x1;
    •  y := y2 - y1;
    • p := 2 *  y -  x; {Initial value p 1 as before}
    • incrE := 2 *  y; {Increment used for move to E}
    • incrNE := 2 * (  y -  x); {Increment used for move to NE}
    • two_v_  x := 0; {Numerator; v = 0 for start pixel}
    • invDenom := 1 / (2 * Sqrt(  x *  x +  y *  y)); {Precomputed inverse denominator}
    • two_  x_invDenom := 2 *  x * invDenom; {Precomputed constant}
    • x := x1;
    • y := y1;
    • IntensifyPixel (x, y, 0); {Start pixel}
    • IntensifyPixel(x, y + 1, two_  x_invDenom); {Neighbor}
    • IntensifyPixel(x, y - 1, two_  x_invDenom); {Neighbor}
    • while x < x2 do
    • begin
    • if p < 0 then
    • begin {Choose E}
    • two_v_  x := p +  x;
    • p := p + incrE;
    • x := x + 1
    • end
    • else
    • begin {Choose NE}
    • two_v_  x := p -  x;
    • p := p + incrNE;
    • x := x + 1;
    • y := y + 1;
    • end ;
    • {Now set chosen pixel and its neighbors}
    • IntensifyPixel (x, y, two_v_  x * invDenom);
    • IntensifyPixel (x, y + 1, two_  x_invDenom - two_v_  x * invDenom);
    • IntensifyPixel (x, y - 1, two_  x_invDenom + two_v_  x * invDenom)
    • end {while}
  • 39. 3. Circle Drawing
  • 40. symmetry using symmetry (0,r) r
  • 41. Midpoint Circle Algorithm Current pixel Choices for next pixel
  • 42. MCA (Cont’) (0, R)
  • 43. MCA (Cont’)
  • 44. MCA (Cont’) procedure MidpointCircle (radius,value : integer); var x,y : integer; P: real; begin x := 0; { initialization } y := radius; P := 5/4 - radius; CirclePoints(x,y,value); while y > x do begin if P < 0 then { select E } P : = P + 2*x + 3; x := x + 1; end else begin { select SE } P := P + 2*(x - y) + 5; x := x + 1; y := y - 1; end CirclePoints(x,y,value) end { while } end; { MidpointCircle } * ** d = P - 1/4 P = d + 1/4 * d = 1 - radius ** d < -1/4  d < 0 why? *** d := d + 2*x + 3 **** d := d + 2(x-y) + 5 *** **** (0, R) y=x
  • 45. MCA (Cont’) procedure MidpointCircle (radius,value : integer); { Assumes center of circle is at origin. Integer arithmetic only } var x,y,d : integer; begin x := 0; { initialization } y := radius; d := 1 - radius; CirclePoints(x,y,value); while y > x do begin if d < 0 then { select E } d := d + 2*x + 3; x := x + 1; end else begin { select SE } d := d+2*(x - y) + 5; x := x + 1; y := y - 1; end CirclePoints(x,y,value) end { while } end; { MidpointCircle } Can you go further?
  • 46. MCA (Cont’)
  • 47. MCA (Cont’) procedure MidpointCircle (radius,value : integer); { This procedure uses second-order partial differences to compute increments in the decision variable. Assumes center of circle is origin. } var x,y,d,deltaE,deltaSE : integer; begin x := 0; { initialization } y := radius; d := 1 - radius; deltaE := 3; deltaSE := -2*radius + 5; CirclePoints(x,y,value); while y > x do begin if d < 0 then { select E } d := d + deltaE; deltaE := deltaE + 2; deltaSE := deltaSE + 2; x := x + 1; end else begin { select SE } d := d + deltaSE; deltaE := deltaE + 2; deltaSE := deltaSE + 4; x := x + 1; y := y - 1 end CirclePoints(x,y,value) end { while } end; { MidpointCircle }
  • 48. Drawing Ellipse Homework #3 Modify the midpoint circle drawing algorithm to draw ellipses.
  • 49. 4. Curve Drawing
    • parametric equation
    • discrete data set
      • curve fitting
      • piecewise linear