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Einfhrung in Visual Computing 186.822 Polygon Filling Werner Purgathofer Linked Lists flexible data structure x 1 x 2 pointer data (link to next element) list^ list^.next^ list x 1 x 2 x 1 x 2 x 1 x 2 list^.next

  1. Einführung in Visual Computing 186.822 Polygon Filling Werner Purgathofer

  2. Linked Lists flexible data structure … x 1 x 2 pointer data (link to next element) list^ list^.next^ list … … … x 1 x 2 x 1 x 2 x 1 x 2 list^.next list^.next^.next Werner Purgathofer / Einf. in Visual Computing 2

  3. Linked Lists: Removal of First Element list 1 2 3 4 5 list = list^.next Werner Purgathofer / Einf. in Visual Computing 3

  4. Linked Lists: Inserting New First Element list 1 2 3 4 5 new 0 new^.next = list Werner Purgathofer / Einf. in Visual Computing 4

  5. Linked Lists: Inserting New First Element list 1 2 3 4 5 new 0 new^.next = list list = new Werner Purgathofer / Einf. in Visual Computing 5

  6. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 1 2 3 4 5 help = last1^.next^.next Werner Purgathofer / Einf. in Visual Computing 6

  7. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 1 2 3 4 5 help = last1^.next^.next last1^.next^.next = last2^.next^.next Werner Purgathofer / Einf. in Visual Computing 7

  8. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 1 2 3 4 5 help = last1^.next^.next last1^.next^.next = last2^.next^.next last2^.next^.next = help Werner Purgathofer / Einf. in Visual Computing 8

  9. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 1 2 3 4 5 help = last1^.next^.next last1^.next^.next = last2^.next^.next last2^.next^.next = help help = last1^.next Werner Purgathofer / Einf. in Visual Computing 9

  10. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 2 3 4 5 1 help = last1^.next^.next last1^.next^.next = last2^.next^.next last2^.next^.next = help help = last1^.next last1^.next = last2^.next Werner Purgathofer / Einf. in Visual Computing 10

  11. Linked Lists: Exchanging Elements 2 & 4 last1 help last2 list 1 2 4 5 3 help = last1^.next^.next last1^.next^.next = last2^.next^.next last2^.next^.next = help help = last1^.next last1^.next = last2^.next last2^.next = help Werner Purgathofer / Einf. in Visual Computing 11

  12. Repetition: What is Inside a Polygon? odd-even rule nonzero-winding- number rule ??? Werner Purgathofer / Einf. in Visual Computing 12

  13. Inside-Outside Tests area-filling algorithms “interior”, “exterior” for self-intersecting polygons? odd-even rule nonzero-winding-number rule same result for simple polygons Werner Purgathofer / Einf. in Visual Computing 13

  14. What is Inside?: Odd-Even Rule inside/outside switches at every edge straight line to the outside: even # edge intersections = outside odd # edge intersections = inside 4 3 2 1 2 1 2 1 4 3 1 1 2 3 Werner Purgathofer / Einf. in Visual Computing 14

  15. What is Inside?: Nonzero Winding Number point is inside if polygon surrounds it straight line to the outside: same # edges up and down = outside different # edges up and down = inside Werner Purgathofer / Einf. in Visual Computing 15

  16. Fill-Area Primitives for polygon area (solid-color, patterned) scan-line polygon fill algorithm intersection points located and sorted consecutive pairs define interior span attention with vertex intersections exploit coherence (incremental calculations) flood fill algorithm Werner Purgathofer / Einf. in Visual Computing 16

  17. Scan-Line Fill: Sorted Edge Table E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A The sorted edge table contains all polygon edges sorted by lowest y-value y max x start 1/m Werner Purgathofer / Einf. in Visual Computing 17

  18. Scan-Line Fill: Sorted Edge Table sort all edges by smallest y-value edge entry: [max. y-value, x-intercept, inverse slope] y B x C 1/m CB y D x C 1/m CD active-edge list for each scan line contains all edges crossed by that scan line incremental update consecutive intersection pairs (spans) filled Werner Purgathofer / Einf. in Visual Computing 18

  19. Sorted Edge Table E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A The sorted edge table contains all polygon edges sorted by lowest y-value y max x start 1/m Werner Purgathofer / Einf. in Visual Computing 19

  20. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List When processing from bottom to top, keep a list of all active edges Werner Purgathofer / Einf. in Visual Computing 20

  21. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List incremental update! y F x A 1/m AF y B x A 1/m AB Werner Purgathofer / Einf. in Visual Computing 21

  22. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y F x A 1/m AF y B x A 1/m AB Werner Purgathofer / Einf. in Visual Computing 22

  23. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y B x A 1/m AB Werner Purgathofer / Einf. in Visual Computing 23

  24. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y B x A 1/m AB Werner Purgathofer / Einf. in Visual Computing 24

  25. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y B x A 1/m AB Werner Purgathofer / Einf. in Visual Computing 25

  26. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y B x A 1/m AB y B x C 1/m CB y D x C 1/m CD Werner Purgathofer / Einf. in Visual Computing 26

  27. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y B x A 1/m AB y B x C 1/m CB y D x C 1/m CD Werner Purgathofer / Einf. in Visual Computing 27

  28. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y D x C 1/m CD Werner Purgathofer / Einf. in Visual Computing 28

  29. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y D x C 1/m CD Werner Purgathofer / Einf. in Visual Computing 29

  30. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y E x D 1/m DE Werner Purgathofer / Einf. in Visual Computing 30

  31. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List y E x F 1/m FE y E x D 1/m DE Werner Purgathofer / Einf. in Visual Computing 31

  32. Sorted Edge Table / Active Edge List E y E x D 1/m DE D B y B x C 1/m CB y D x C 1/m CD C F y E x F 1/m FE y F x A 1/m AF y B x A 1/m AB A Active Edge List Werner Purgathofer / Einf. in Visual Computing 32

  33. Scan-Line Fill: Incremental Update incremental update of intersection point slope of polygon boundary line: m 1 = + = + x x y y 1 m k+1 k k+1 k (for 2 successive scanlines) (x k+1 , y k+1 ) y k + 1 (x k , y k ) y k Werner Purgathofer / Einf. in Visual Computing 33

  34. Scan-Line Fill: Intersecting Vertices 2 intersection 1 points along scan lines that 1 2 intersect polygon 2 vertices → either special handling (1 or 2 intersections?) → or move vertices up or down by ε Werner Purgathofer / Einf. in Visual Computing 34

  35. Flood-Fill Algorithm pixel filling of area areas with no single color boundary start from interior point “flood” internal region 4-connected, 8-connected areas reduce stack size by eliminating several recursive calls Werner Purgathofer / Einf. in Visual Computing 35

  36. Flood-Fill: Boundary and Seed Point area must be distinguishable from boundaries seed point example: area defined within multiple color boundaries Werner Purgathofer / Einf. in Visual Computing 36

  37. Flood-Fill: Connectedness Definition: Definition: 4-connected means, that a 8-connected means, that a connection is only valid in connection is valid in these these 4 directions 8 directions Werner Purgathofer / Einf. in Visual Computing 37

  38. Examples for 4-connected and 8-connected a 4-connected area an 8-connected area has an 8-connected border has a 4-connected border Werner Purgathofer / Einf. in Visual Computing 38

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