Polygon Scan Conversion – 11b. 2 Rasterization Rasterization takes shapes like triangles and...

Polygon Scan Conversion – 11b

2

Rasterization

• Rasterization takes shapes like triangles and determines which pixels to fill.

3

Filling Polygons

First approach:

1. Polygon Scan-Conversion• Rasterize a polygon scan line by scan line,

determining which pixels to fill on each line.

4

Filling Polygons

Second Approach:

2. Polygon Fill• Select a pixel inside the polygon. Grow outward

until the whole polygon is filled.

5

Polygon TypesConvex

Not Convex

For every pair of points within a convex polygon, the line segment connecting them is also completely enclosed within the polygon.

6

Why Polygons?

• You can approximate practically any surface if you have enough polygons.

• Graphics hardware is optimized for polygons (and especially triangles)

7

Which Pixels to Fill?

• Pixels entirely inside polygon Yes

• Pixels partially inside ?

• Convention: only fill pixels whose centers are within the polygon

• This is an imperfect solution.

InsideOutside

Polygon edge

YesNo?

8

Spatial Coherence

• Spatial Coherence:• If one pixel is filled, its neighbors likely are also.

• Use spatial coherence to construct an efficient algorithm.

• Process each scan line in order from left to right.

9

Coherence

Scan-line

Edge

Span

10

Polygon Scan Conversion

Intersection Points

Other points in the span

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Polygon Scan Conversion

• Process each scan line1. Find the intersections of the scan line with all

polygon edges.

2. Sort the intersections by x coordinate.

3. Fill in pixels between pairs of intersections using an odd-parity rule.- Set parity even initially.

- Each intersection flips the

parity.

- Draw when parity is odd.

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Special Cases: Vertices• How do we count the intersecting vertex in the parity computation?

• Count it zero or two times.

Scan line

vertex

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Special Cases: Vertices• What about:

• Here it counts once.

Scan line

vertex

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Special Cases: Vertices

• Each edge has two vertices.• The ymin vertex is the vertex with the lower y

coordinate.

• Vertices only count when they are the lower ymin vertex for an edge.

15

Special Cases: Vertices

• Both cases now work correctly

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Horizontal Edges• What if neither vertex is lower? Which is ymin?

Don’t count either vertex in the parity calculation!

ScanLine

17

Top Spans of Polygons

• Effect of only counting ymin :• Top spans of polygons are not drawn

• If two polygons share this edge, it is not a problem.

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Shared Polygon Edges

• What if two polygons share an edge?

• Solution:• Span is closed on left and

open on right (xmin x < xmax)

• Scan lines closed on bottom and

open on top (ymin y < ymax)

Draw Last polygon wins

Orange last Blue last

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Polygon Scan-Conversion

• Process for scan converting polygons• Process one polygon at a time.

• Store information about every polygon edge.

• Compute spans for each scan line.

• Draw the pixels between the spans.

• This can be optimized using an edge table.

• One entry in the edge table per polygon edge.

• Perform scan conversion one scan line at a time.

• For each new scan line, consult the edge table to determine if it intersects any new edges.

This is the beginning of your code for project 1

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Computing Intersections

• For each scan line, we need to know if it intersects the polygon edges.

• It is expensive to compute a complete line-line intersection computation for each scan line.

• After computing the intersection between a scan line and an edge, we can use that information in the next scan line.

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Scan Line Intersection

Currentscan line

yi+1

Polygon Edges

Intersection points from

previous scan line

Intersection pointsneeded

Previousscan line

yi

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Scan Line IntersectionUse edge coherence to incrementally update the

x intersections.

We know

Each new scan line is 1 greater in y

We need to compute x for a given scan line,

min@max@

minmax ,yy xx

yymbmxy

11 ii yy

m

byx

(x@ymax,ymax)

(x@ymin,ymin)

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Scan Line Intersection

So,

and

then

This is a more efficient way to compute xi+1.

m

byx

m

byx i

ii

i

11 ,

11 ii yy

mx

mm

by

m

byx i

iii

1111

24

Edge Tables

• We will use two different edge tables:• Active Edge Table (AET)

• Stores all edges that intersect the current scan line.

• Global Edge Table (GET)• Stores all polygon edges.

• Used to update the AET.

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Active Edge Table

• Table contains one entry per edge intersected by the current scan line.

• At each new scan line:• Compute new intersections for all edges using the

formula.

• Add any new edges intersected.

• Remove any edges no longer intersected.

• To efficiently update the AET, we will maintain a global edge table (GET)

mxx ii

11

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Global Edge Table

• Table contains information about all polygon edges.

• It has one bucket for each scan line.• Each bucket stores a list of edges whose ymin value

is the scan line y value.

• Each edge is found in only one bucket.

• Each entry in the GET contains• The edge’s ymax value.

• The x@ymin value (the x value at the ymin point)

• The x increment value (1/m)

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Global Edge Table Example

GET Entries (ymax, x@ymin, 1/m)

(5, 2, ¼)

(6, 3, 3)

(8, 6, -3/2)

(8, 0, 3/4)

AB

BC

CD

DE

EA (4, 2, -2/3)

Why don’t we store ymin?

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

D

E

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

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Global Edge Table Example

GET

0

1

2

3

4

5

6

7

8

Indexed by

scan line

AB EA

(ymax, x@ymin, 1/m)

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

D

E

Place entries into the GET

based on the ymin values.

8, 6, -3/2

6, 3, 3

8, 0, ¾

5, 2, ¼ 4, 2, -2/3

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

DE

CD

BC

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Active Edge Table Example

• The active edge table stores information about the edges that intersect the current scan line.

• Entries are• The ymax value of the edge

• The x value where the edge intersects the current scan line.

• The x increment value (1/m)

• Entries are sorted by x values.

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Active Edge Table Example

AETScan Line 3

1. The ymax value for that edge2. The x value where the scan line intersects

the edge.3. The x increment value (1/m)

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

D

E

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

AB

4, 2/3, -2/3 5, 5/2, 1/4

EA

(ymax, x, 1/m)

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Active Edge Table Example

AETScan Line 4

1. The ymax value for that edge2. The x value where the scan line intersects

the edge.3. The x increment value (1/m)

Scan Line 4

ymax = 4 for EA, so it gets removed from the AET.

New x value for AB is 5/2 + 1/4 = 11/4

In the GET, edge DE is stored in bucket 4,

so it gets added to the AET.

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

D

E

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

AB

4, 2/3, -2/3 5, 5/2, 1/4

EA

(ymax, x, 1/m)

8, 0, 3/4

DE

5, 11/4, 1/4

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Active Edge Table Example

AETScan Line 5

1. The ymax value for that edge2. The x value where the scan line intersects

the edge.3. The x increment value (1/m)

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

D

E

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

ABDE

(ymax, x, 1/m)

8, 0, 3/4 5, 11/4, 1/48, 3/4, 3/4

Increment x = 0 + 3/4

Remove AB,since ymax = 5.

Add BC since it is in the GET at ymin = 5.

6, 3, 3

BC

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Active Edge Table Example

Scan Line 6 – Your Turn

AB = (2, 1), (3, 5)BC = (3, 5), (6, 6)CD = (6, 6), (3, 8)DE = (3, 8), (0, 4)EA = (0, 4), (2, 1)

AET

DE

8, 3/4, 3/4 6, 3, 3

BC

D

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

E

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One More Example

• What is the global edge table for this polygon?• 1. Find Edges

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

AB = ?BC = ?CA = ?

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One More Example

• What is the global edge table for this polygon?• 2. Compute GET Entries

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

AB = (3, 3), (7, 8)BC = (7, 8), (1, 6)CA = (1, 6), (3, 3)

(ymax, x@ymin, 1/m)

(8, 3, 4/5) (8, 1, 3) (6, 3, -2/3)

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One More Example• What is the global edge table for this polygon?

• 3. Place entries into GET

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

AB = (3, 3), (7, 8)BC = (7, 8), (1, 6)CA = (1, 6), (3, 3)

(8, 3, 4/5) (8, 1, 3) (6, 3, -2/3)

GET

0

1

2

3

4

5

6

7

8

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One More Example• What is the global edge table for this polygon?

• 3. Place entries into GET

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

8

A

B

C

AB = (3, 3), (7, 8)BC = (7, 8), (1, 6)CA = (1, 6), (3, 3)

(8, 3, 4/5) (8, 1, 3) (6, 3, -2/3)

GET

0

1

2

3

4

5

6

7

8

CA

6, 3, -2/3 8, 3, 4/5

AB

8, 1, 3

BC

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Scan Line Algorithm

1. Add polygon edges to the Global Edge Table (GET)

2. Set y to the smallest y coordinate in the GET.

3. Initially, set the Active Edge Table (AET) to be empty.

4. Repeat until the AET and GET are empty:a. Add edges from the GET to the AET in which ymin = y.

b. Remove edges from the AET in which ymax = y.

c. Sort AET on x.

d. Fill pixels between pairs of intersections in the AET

e. For each edge in the AET, replace x with x + 1/m

f. Set y = y + 1 to move to the next scan line

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Filling Techniques

• A second approach to drawing polygons is to use a filling technique, rather than scan conversion.

• Pick a point inside the polygon, then fill neighboring pixels until the polygon boundary is reached.

• Boundary Fill Approach:• Draw polygon boundary.

• Determine an interior point.

• Starting at the interior point, do• If the point is not the boundary color or the fill color

• Set this pixel to the fill color.• Propagate to the pixel’s neighbors and continue.

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Propagating to Neighbors

• Most frequently used approaches:• 4-connected area

• 8-connected area

4-connected 8-connected

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Fill Problems

• Fill algorithms have potential problems

• E.g., 4-connected area fill:

Starting point Fill complete

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Fill Problems• Similarly, 8-connected can “leak” over to another

polygon

• Another problem: the algorithm is highly recursive• Can use a stack of spans to reduce amount of recursion

Starting point Fill Complete

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Pattern Filling

• Often we want to fill a region with a pattern, not just a color

• Define an n by m pixmap (or bitmap) that we wish to replicate across the region

5x4 pixmap

Object to be patterned

Final patterned object

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Pattern Filling

• How do we determine which color to color a point in the object?

• Use the MOD function to tile the pattern across the polygon

• For point (x, y)• Use the pattern color located at (x MOD m, y MOD

n)

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Triangles

• Always convex: No matter how you rotate a triangle, it only has one span per scan line.

• Any polygon can be decomposed into triangles. Convex Concave

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Triangles (cont.)

• Rasterization algorithms can take advantage of triangle properties.

• Graphics hardware is optimized for triangles.

• Because triangle drawing is so fast, many systems will subdivide polygons into triangles prior to scan conversion.