1. C O M P U T E R G R A P H I C S Guoying Zhao 1 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 1 / 50 Computer Graphics Implementation II

2. C O M P U T E R G R A P H I C S Guoying Zhao 2 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 2 / 50 Polygon Filling –Scan-line Conversion Approaches –Area Filling Approaches Antialiasing Clipping

3. C O M P U T E R G R A P H I C S Guoying Zhao 3 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 3 / 50 Polygon Filling Scan-line Conversion Area Filling

4. C O M P U T E R G R A P H I C S Guoying Zhao 4 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 4 / 50 Polygon Filling Vertex-edge representation  Pixel set representation P0 P1 P2 P3 P4 P5 P6 P7

5. C O M P U T E R G R A P H I C S Guoying Zhao 5 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 5 / 50 Scan-line Conversion Check pixel by pixel P0 P1 P2 P3 P4 P5 P6 P7

6. C O M P U T E R G R A P H I C S Guoying Zhao 6 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 6 / 50 How to judge a point inside or outside a polygon? Shoot a radial from the point to intersect with the polygon edges; if there are odd number of intersection points, the point is inside the polygon; if even number, outside. Odd point: the intersection point is polygon’s vertex (special case).

7. C O M P U T E R G R A P H I C S Guoying Zhao 7 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 7 / 50 Scan-line Conversion Improved (limit the pixel sets into the bounding box of the polygon for speeding up) P0 P1 P2 P3 P4 P5 P6 P7

8. C O M P U T E R G R A P H I C S Guoying Zhao 8 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 8 / 50 Scan-line Conversion Scan-line polygon-fill algorithm –Taking full advantage of the coherence properties of pixels –Three coherence properties Area coherence Scan-line coherence Edge coherence

9. C O M P U T E R G R A P H I C S Guoying Zhao 9 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 9 / 50 Area Coherence The screen region between two scan-lines is partitioned into some trapezoids by the polygon. (1) Two types of trapezoid: the one inside the polygon and the one outside polygon. (2) The two types of trapezoids are arranged alternately.

10. C O M P U T E R G R A P H I C S Guoying Zhao 10 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 10 / 50 Scan-line Coherence Suppose the intersection points of scan-line y=e and polygon edge e i (P i-1 P i ) is x ei. Suppose the intersection points sequence arranged by the x- increase is x ei1, x ei2, x ei3 … x ein. According to area coherence, we can get: (1)n is even number (2)On the scan-line, only the segments (x eik, x eik+1 ), k=1,3,5,…n–1) are inside the polygon. Scan-line coherence is the reflection of area coherence on a scan-line.

11. C O M P U T E R G R A P H I C S Guoying Zhao 11 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 11 / 50 Edge Coherence Suppose the intersection point sequence on y=e is x ei1, x ei2, …x ein ; point sequence on y=e-1 is x di1, x di2, … x din. If edge e r (P r-1 P r ) intersects with both y=e and y=e-1; the corresponding points x er and x dr have following relationship: x er = x dr + 1/m r Thus, we can calculate the intersection points on y=e from the points on y=e-1 Edge coherence is the reflection of area coherence on edges.

12. C O M P U T E R G R A P H I C S Guoying Zhao 12 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 12 / 50 Polygon Filling Scan-line Conversion Area Filling

13. C O M P U T E R G R A P H I C S Guoying Zhao 13 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 13 / 50 Area Filling Area Filling: To start from a given interior position (seed) and paint outward from this point until we encounter the specified boundary conditions. The “area” should be identified with its interior color or boundary color. –Colorate all the interior pixels to a specified color –Colorate all the boundary pixels to the boundary color

14. C O M P U T E R G R A P H I C S Guoying Zhao 14 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 14 / 50 种子填充 4-connected Neighbourhood 8-connected Neighbourhood Two kinds of connectivity

15. C O M P U T E R G R A P H I C S Guoying Zhao 15 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 15 / 50 4-connected area and 8-connected area 4-connected area: Giving any two interior points A and B, we can travel from A to B by the 4-directions moving: right, left, up and down moving. To filling the 4-connected area, we only need to test its 4- direction neighbors 4-connected area 8-connected area

16. C O M P U T E R G R A P H I C S Guoying Zhao 16 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 16 / 50 4-connected area and 8-connected area The boundary of 8- connected area must be 4-connected The boundary of 4- connected area is 8- connected

17. C O M P U T E R G R A P H I C S Guoying Zhao 17 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 17 / 50 Recursive Method for Filling a 4-connected Area void AreaFill4(int x, int y, int fillCol, int boundaryCol) { int currentCol = getPixel(x,y); if( (currentCol != boundaryCol)&& (currentCol != fillCol)) {setPixel(x, y, fillCol); AreaFill4(x, y+1, fillCol, boundaryCol); AreaFill4(x, y-1, fillCol, boundaryCol); AreaFill4(x-1, y, fillCol, boundaryCol); AreaFill4(x+1, y, fillCol, boundaryCol); } The area is identified by the boundary color

18. C O M P U T E R G R A P H I C S Guoying Zhao 18 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 18 / 50 Recursive Method for Filling a 4-connected Area void FloodFill4(int x, int y, int fillCol, int interiorCol) { int currentCol = getPixel(x,y); if( currentCol == interiorCol ) {setPixel(x, y, fillCol); FloodFill4(x, y+1, fillCol, interiorCol); FloodFill4(x, y-1, fillCol, interiorCol); FloodFill4(x-1, y, fillCol, interiorCol); FloodFill4(x+1, y, fillCol, interiorCol); } The area is identified by the interior color. Called flood- fill algorithm

19. C O M P U T E R G R A P H I C S Guoying Zhao 19 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 19 / 50 Recursive Method for Filling a 4-connected Area A B C D E F G H I J L M N O P K

20. C O M P U T E R G R A P H I C S Guoying Zhao 20 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 20 / 50 Recursive Method for Filling a 4-connected Area Left pixel to 3 is the last one to be filled.

21. C O M P U T E R G R A P H I C S Guoying Zhao 21 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 21 / 50 How to expand the algorithm for 4 –connected area to the algorithm for 8-connect area?

22. C O M P U T E R G R A P H I C S Guoying Zhao 22 / 50 void AreaFill8(int x, int y, int fillCol, int boundaryCol) { int currentCol = getPixel(x,y); if((currentCol != boundaryCol) && (currentCol != fillCol)) {setPixel(x, y, fillCol); AreaFill8(x, y+1, fillCol, boundaryCol); AreaFill8(x, y-1, fillCol, boundaryCol); AreaFill8(x-1, y, fillCol, boundaryCol); AreaFill8(x+1, y, fillCol, boundaryCol); AreaFill8(x+1, y+1, fillCol, boundaryCol); AreaFill8(x+1, y-1, fillCol, boundaryCol); AreaFill8(x-1, y+1, fillCol, boundaryCol); AreaFill8(x-1, y-1, fillCol, boundaryCol); }

23. C O M P U T E R G R A P H I C S Guoying Zhao 23 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 23 / 50 Comparison Between Scan-line Conversion (A) and Area Filling (B) Basic idea: A changes the edge list representation into lattice representation. It uses the coherences of polygons. B does not change the representation of the area, but the color. It uses the connectivity of area. The requirements: For area filling, a seed point inside the area is needed. Boundary: For A, the number of the intersection points of each scan line with edges should be even. For B, the boundary of 4-connected area is closed 8-connected area and the boundary of 8-connected area is closed 4-connected area.

24. C O M P U T E R G R A P H I C S Guoying Zhao 24 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 24 / 50 Antialiasing Aliasing Problems of Raster Graphics Antialiasing Methods

25. C O M P U T E R G R A P H I C S Guoying Zhao 25 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 25 / 50 Aliasing Problems of Raster Graphics What’s aliasing? The distortion of information due to low- frequency sampling ( undersampling ) is called aliasing

26. C O M P U T E R G R A P H I C S Guoying Zhao 26 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 26 / 50 Aliasing Problems of Raster Graphics --- Jagged Boundaries 8 8 7 6 5 4 3 2 0 1 9 765432019 x y

27. C O M P U T E R G R A P H I C S Guoying Zhao 27 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 27 / 50 Aliasing Problems of Raster Graphics --- Shape Distortion Slim Primitives are lost

28. C O M P U T E R G R A P H I C S Guoying Zhao 28 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 28 / 50 Aliasing Problems of Raster Graphics --- Sparking The slim primitive sparks when it is moved

29. C O M P U T E R G R A P H I C S Guoying Zhao 29 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 29 / 50 Antialiasing Aliasing Problems of Raster Graphics Antialiasing Methods

30. C O M P U T E R G R A P H I C S Guoying Zhao 30 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 30 / 50 Antialiasing Methods Adopting area-sampling instead of point-sampling Supersampling

31. C O M P U T E R G R A P H I C S Guoying Zhao 31 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 31 / 50 Area-sampling Using transitional color scales on the edges

32. C O M P U T E R G R A P H I C S Guoying Zhao 32 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 32 / 50 Using Transitional Color Scales on the Edges

33. C O M P U T E R G R A P H I C S Guoying Zhao 33 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 33 / 50 Area-sampling Exact area-sampling is time-consuming Some approximate algorithms are always used –Wu’s algorithm for drawing antialiasing lines –Pitteway and Watkinson’s algorithm for drawing antialiasing polygons

34. C O M P U T E R G R A P H I C S Guoying Zhao 34 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 34 / 50 Supersampling for Antialiasing Hardware method: adopting high resolution raster display Software method: sampling objects at a high resolution and displaying the results at a lower resolution –High resolution sampling: partition each pixel as several sub-pixels, such as the 3*3 partition. Then compute color for all the sub-pixels –Low resolution display: compute the pixel’s color by adding up all its sub-pixels’ color with a weighting mask

35. C O M P U T E R G R A P H I C S Guoying Zhao 35 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 35 / 50 Supersampling for Antialiasing 4 2 2 2 2 1 1 1 1 Weight mask for the 3*3 partition E B H F D I G C A 3*3 partition Pixel color = (Col A + 2*Col B + Col C + 2*Col D + 4*Col E + 2*Col F + Col G + 2*Col H + Col I ) / 16

36. C O M P U T E R G R A P H I C S Guoying Zhao 36 / 50 C O M P U T E R G R A P H I C S Guoying Zhao 36 / 50 Supersampling for Antialiasing 3 2 6 4 6 2 3 46 2 64 8 2 6 9 3 1 6 8 9 3 12 4 6 3 9 4 8 2 4 9 6 8 16 2 1 3 4 23 1 32 4 1 Weight mask for the 7*7 partition

37. C O M P U T E R G R A P H I C S Guoying Zhao 37 / 50 Clipping

38. C O M P U T E R G R A P H I C S Guoying Zhao 38 / 50 Clipping

39. C O M P U T E R G R A P H I C S Guoying Zhao 39 / 50 Line Clipping Clipping endpoints x min < x < x max and y min < y < y max point inside Endpoint analysis for lines: –if both endpoints in, do “trivial acceptance” –if one endpoint inside, one outside, must clip –if both endpoints out, don’t know Brute force clip: solve simultaneous equations using y = mx + b for line and four clip edges –slope-intercept formula handles infinite lines only –doesn’t handle vertical lines (X min, Y min ) (X max, Y max )

40. C O M P U T E R G R A P H I C S Guoying Zhao 40 / 50 Parametric Line Formulation For Clipping Parametric form for line segment X = x 0 + t(x 1 – x 0 ) 0 < t < 1 Y = y 0 + t(y 1 – y 0 ) P(t) = P 0 + t(P 1 – P 0 ) “true,” i.e., interior intersection, if s edge and t line in [0,1]

41. C O M P U T E R G R A P H I C S Guoying Zhao 41 / 50 Divide plane into 9 regions Compute the sign bit of 4 comparisons between a vertex and an edge –y max – y; y – y min ; x max – x; x – x min (>0 “0”; otherwise “1”) –point lies inside only if all four sign bits are 0, otherwise exceeds edge 4 bit outcode records results of four bounds tests: “1” First bit: outside halfplane of top edge, above top edge Second bit: outside halfplane of bottom edge, below bottom edge Third bit: outside halfplane of right edge, to right of right edge Fourth bit: outside halfplane of left edge, to left of left edge o1=o2 =0, accept; AND(o1, o2) <>0 (in same side), discard. Outcodes for Cohen-Sutherland Line Clipping in 2D Clip Rectangle

42. C O M P U T E R G R A P H I C S Guoying Zhao 42 / 50 Left plane 000001 (to left of) 000000 (to right of) Outcodes for Cohen-Sutherland Line Clipping in 3D Very similar to 2D Divide volume into 27 regions (Picture a Rubik’s cube) 6-bit outcode records results of 6 bounds tests First bit: outside back plane, behind back plane Second bit: outside front plane, in front of front plane Third bit: outside top plane, above top plane Fourth bit: outside bottom plane, below bottom plane Fifth bit:outside right plane, to right of right plane Sixth bit: outside left plane, to left of left plane Top plane 001000 (above) 000000 (below) Bottom plane 000000 (above) 000100 (below) Front plane 010000 (in front) 000000 (behind) Right plane 000000 (to left of) 000010 (to right of) Back plane 000000 (in front) 100000 (behind)

43. C O M P U T E R G R A P H I C S Guoying Zhao 43 / 50 If we can neither trivially reject/accept, divide and conquer subdivide line into two segments; then T/A or T/R one or both segments: –use a clip edge to cut line –use outcodes to choose edge that is crossed Edges where the two outcodes differ at that particular bit are crossed –pick an order for checking edges top – bottom – right – left –compute the intersection point the clip edge fixes either x or y can substitute into the line equation –iterate for the newly shortened line –“extra” clips may happen (e.g., E-I at H) Clip rectangle D C B A E F G H I Cohen-Sutherland Algorithm 1010 0100

44. C O M P U T E R G R A P H I C S Guoying Zhao 44 / 50 ComputeOutCode(x0, y0, outcode0) ComputeOutCode(x1, y1, outcode1) repeat check for trivial reject or trivial accept pick the point that is outside the clip rectangle if TOP then x = x0 + (x1 – x0) * (ymax – y0)/(y1 – y0); y = ymax; else if BOTTOM then x = x0 + (x1 – x0) * (ymin – y0)/(y1 – y0); y = ymin; else if RIGHT then y = y0 + (y1 – y0) * (xmax – x0)/(x1 – x0); x = xmax; else if LEFT then y = y0 + (y1 – y0) * (xmin – x0)/(x1 – x0); x = xmin; if (x0, y0 is the outer point) then x0 = x; y0 = y; ComputeOutCode(x0, y0, outcode0) else x1 = x; y1 = y; ComputeOutCode(x1, y1, outcode1) until done y = y0 + slope*(x - x0) and x = x0 + (1/slope)*(y - y0) Pseudocode for the Cohen- Sutherland Algorithm

45. C O M P U T E R G R A P H I C S Guoying Zhao 45 / 50 Cyrus-Beck/Liang-Barsky Parametric Line Clipping-1 Use parametric line formulation P(t) = P 0 + (P 1 – P 0 )t Determine where the line intersects the infinite line formed by each edge by solving for t 4 times. Decide which of these intersections actually occur on the rectangle For any point P E i on edge E i

46. C O M P U T E R G R A P H I C S Guoying Zhao 46 / 50 Now solve for the value of t at the intersection of P 0 P 1 with the edge E i : N i [P(t) – P E i ] = 0 First, substitute for P(t): N i [P 0 + (P 1 – P 0 )t – P E i ] = 0 Next, group terms and distribute dot product: N i [P 0 – P E i ] + N i [P 1 – P 0 ]t = 0 Let D be the vector from P 0 to P 1 = (P 1 – P 0 ), and solve for t: Note that this gives a valid value of t only if the denominator of the expression is nonzero. For this to be true, it must be the case that: N i  0 (that is, the normal should not be 0; this could occur only as a mistake) D  0 (that is, P 1  P 0 ) N i D  0 (edge E i and line D are not parallel; if they are, no intersection). The algorithm checks these conditions. C-B/L-B Param. Line Clipping-2

47. C O M P U T E R G R A P H I C S Guoying Zhao 47 / 50 Eliminate t’s outside [0,1] on the line (does not to be processed) Which remaining t’s produce interior intersections? Can’t just take the innermost t values! (decide PE and PL) Move from P 0 to P 1 ; for a given edge, just before crossing: if N i D 0 Potentially Leaving (PL) Pick inner PE, PL pair: t E for P PE with max t, t L for P PL with min t, and t E > 0, t L < 1. If t L < t E, no intersection (Line 2) C-B/L-B Param. Line Clipping-3

48. C O M P U T E R G R A P H I C S Guoying Zhao 48 / 50 Pre-calculate N i and select P E i for each edge; for each line segment to be clipped if P 1 = P 0 then line is degenerate so clip as a point; else begin t E = 0; t L = 1; for each candidate intersection with a clip edge if Ni D  0 then {Ignore edges parallel to line} begin calculate t; {of line and clip edge intersection} use sign of N i D to categorize as PE or PL; if PE then t E = max(t E,t); if PL then t L = min(t L,t); end if t E > t L then return nil else return P(t E ) and P(t L ) as true clip intersections end Pseudocode for Cyrus-Beck/ Liang-Barsky Line Clipping Algorithm

49. C O M P U T E R G R A P H I C S Guoying Zhao 49 / 50 D = P 1 – P 0 = (x 1 – x 0, y 1 – y 0 ) Leave P E i as an arbitrary point on the clip edge; it’s a free variable and drops out Calculations for Parametric Line Clipping Algorithm (x 0 -x,y 0 - y max )(x, y max )(0,1)top: y = y max (x 0 -x,y 0 - y min )(x, y min )(0,-1)bottom: y = y min (x 0 - x max,y 0 -y)(x max,y)(1,0)right: x = x max (x 0 - x min,y 0 -y)(x min, y)(-1,0)left: x = x min P 0 -P E i PEiPEi Normal N i Clip Edge i

50. C O M P U T E R G R A P H I C S Guoying Zhao 50 / 50 Sutherland-Hodgman Polygon Clipping