Polygon Filling - cg.tuwien.ac.at filePolygon Filling Werner Purgathofer. Werner Purgathofer / Einf....

Einführung in Visual Computing186.822

Polygon Filling

Werner Purgathofer

Werner Purgathofer / Einf. in Visual Computing 2

flexible data structure

Linked Lists

…x1 x2

data pointer(link to next element)

…x1 x2 …x1 x2 …x1 x2

list

list^.next list^.next^.next

list^ list^.next^


Linked Lists: Removal of First Element

1 2 3 4 5list

list = list^.next


Linked Lists: Inserting New First Element

1 2 3 4 5

0new

new^.next = list

list


Linked Lists: Inserting New First Element

1 2 3 4 5

0new

new^.next = listlist = new

list


Linked Lists: Exchanging Elements 2 & 4

1 2 3 4 5

last1 last2

list

help = last1^.next^.next

help



1 2 3 4 5

last1 last2

list

help = last1^.next^.nextlast1^.next^.next = last2^.next^.next

help



1 2 3 4 5

last1 last2

list

help = last1^.next^.nextlast1^.next^.next = last2^.next^.nextlast2^.next^.next = help

help



1 2 3 4 5

last1 last2

list

help = last1^.next^.nextlast1^.next^.next = last2^.next^.nextlast2^.next^.next = helphelp = last1^.next

help



1 2 3 4 5

last1 last2

list

help = last1^.next^.nextlast1^.next^.next = last2^.next^.nextlast2^.next^.next = helphelp = last1^.nextlast1^.next = last2^.next

help



1 2 3 4 5

last1 last2

list

help = last1^.next^.nextlast1^.next^.next = last2^.next^.nextlast2^.next^.next = helphelp = last1^.nextlast1^.next = last2^.nextlast2^.next = help

help


odd-even rule nonzero-winding- number rule

Repetition: What is Inside a Polygon?

???


area-filling algorithms“interior”, “exterior” for self-intersecting polygons?odd-even rulenonzero-winding-number rulesame result for simple polygons

Inside-Outside Tests


inside/outside switches at every edgestraight line to the outside:

even # edge intersections = outside odd # edge intersections = inside

What is Inside?: Odd-Even Rule

1234

1234

123

12

1


point is inside if polygon surrounds itstraight line to the outside:

same # edges up and down = outsidedifferent # edges up and down = inside

What is Inside?: Nonzero Winding Number


for polygon area (solid-color, patterned)scan-line polygon fill algorithm

intersection points located and sortedconsecutive pairs define interior spanattention with vertex intersectionsexploit coherence (incremental calculations)

flood fill algorithm

Fill-Area Primitives


Scan-Line Fill: Sorted Edge Table

The sorted edge table contains all polygon edges sorted by lowest y-value

A

B

C

D

E

F

yF xA 1/mAF yB xA 1/mAB

yE xF 1/mFE

yB xC 1/mCB yD xC 1/mCD

yE xD 1/mDE

ymax xstart 1/m


sort all edges by smallest y-valueedge entry:

[max. y-value, x-intercept, inverse slope]

active-edge listfor each scan linecontains all edges crossed by that scan lineincremental update

consecutive intersection pairs (spans) filled

Scan-Line Fill: Sorted Edge Table



Sorted Edge Table

The sorted edge table contains all polygon edges sorted by lowest y-value

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE

ymax xstart 1/m


Sorted Edge Table / Active Edge List

Active Edge List When processing from bottom to top, keep a list of all active edges

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


incremental update!yF xA 1/mAF yB xA 1/mAB

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List



A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


yE xF 1/mFE yB xA 1/mAB

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List



A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE

yE xF 1/mFE yB xA 1/mAB yB xC 1/mCB yD xC 1/mCD


Active Edge List


yE xF 1/mFE yB xA 1/mAB yB xC 1/mCB yD xC 1/mCD

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


yE xF 1/mFE yD xC 1/mCD

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


yE xF 1/mFE yE xD 1/mDE

A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


Active Edge List


A

B

C

D

E

F


yE xF 1/mFE


yE xD 1/mDE


incremental update of intersection point

1+=kk+1

yymxx kk+11+=

slope of polygon boundary line: m

(for 2 successive scanlines)

Scan-Line Fill: Incremental Update

yk + 1yk(xk , yk)

(xk+1, yk+1)


intersection points along scan lines that intersect polygonvertices

→ either special handling (1 or 2 intersections?)

→ or move vertices up or down by ε

Scan-Line Fill: Intersecting Vertices

1

1

2

2

2


pixel filling of areaareas with no single color boundarystart from interior point“flood” internal region4-connected, 8-connected areasreduce stack size by eliminating several recursive calls

Flood-Fill Algorithm


area must be distinguishable from boundaries

example: area defined within multiple color boundaries

Flood-Fill: Boundary and Seed Point

seed point


Definition: 4-connected means, that a connection is only valid in these 4 directions

Definition: 8-connected means, that a connection is valid in these 8 directions

Flood-Fill: Connectedness


has an 8-connected borderan 8-connected areahas a 4-connected border

a 4-connected area

Examples for 4-connected and 8-connected


Simple Flood-Fill Algorithm

void floodFill4 (int x, int y, int new, int old){ int color;

/* set current color to new */getPixel (x, y, color);if (color = old) {

setPixel (x, y);floodFill4 (x–1, y, new, old); /* left */floodFill4 (x, y+1, new, old); /* up */floodFill4 (x+1, y, new, old); /* right */floodFill4 (x, y–1, new, old) /* down */

}}

12

43

recursionsequence


Bad Behavior of Simple Flood-Fill

12

43

recursionsequence


floodFill4 produces too high stacks (recursion!)solution:

incremental horizontal fill (left to right)recursive vertical fill (first up then down)

Span Flood-Fill Algorithm


Good Behavior of Span Flood-Fill

1

2recursionsequence


Span Flood-Fill Example

Stack:x



B

A1 2 3

Stack:xAB



A1 2 3

Stack:ABCDEF

F E

CD4 5 6 7 8 9 10 11



A1 2 3

Stack:ACDEFG

GE

CD4 5 6 7 8 9 10 11

12 13 14



A1 2 3

Stack:ACDEGH

E=H

CD4 5 6 7 8 9 10 11

12 13 1415 16 17 18



A1 2 3

Stack:ACDEHCD

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819 (E)



A1 2 3

Stack:ACDIJC

I

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

J20 21 22



A1 2 3

Stack:ACIJ

CI

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

20 21 2223



A1 2 3

Stack:ACIK

C

K

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

20 21 2223

24 25



A=N1 2 3

Stack:ACKL

MN

C

M

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

20 21 2223

24 2526 27 28 29 30

L



(A)1 2 3

Stack:ACLMNC

M

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

20 21 2223

24 2526 27 28 29 30

L

31 32 33



1 2 3

Stack:CLM

C4 5 6 7 8 9 10 11

12 13 1415 16 17 18

19

20 21 2223

24 2526 27 28 29 30

L

31 32 33

34 35



1 2 3

Stack:CL

C4 5 6 7 8 9 10 11

12 13 1415 16 17 18

19

20 21 2223

24 2526 27 28 29 30

31 32 33

34 35 36 37



1 2 3

Stack:C

4 5 6 7 8 9 10 1112 13 14

15 16 17 1819

20 21 2223

24 2526 27 28 29 30

31 32 33

34 35 36 37

38

finished!

3D Polygons

normally surface elements of 3D objectsattributes are surface attributes

color, material, transparency, texture, geometric microstructure, reflection properties, …

attributes and normal vectors often at the verticestriangles: linear interpolation with barycentric coordinates


Polygon Filling - cg.tuwien.ac.at filePolygon Filling Werner Purgathofer. Werner Purgathofer / Einf....

Documents

Transcript of Polygon Filling - cg.tuwien.ac.at filePolygon Filling Werner Purgathofer. Werner Purgathofer / Einf....