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Scan Conversion

2001.1.10

Graphics Lab.

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ContentsScene DisplayWhat is Scan Conversion?

Line Drawing AlgorithmsCircle AlgorithmFilled-Area Primitives- Scan-Line Fill Algorithm

- Boundary Fill Algorithm- Flood-Fill Algorithm- Inside-Outside Test

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Scene Display

Scene Displayimage

pixel H/W Device(monitor)?

Scene Display - pixel arrays frame buffer load- basic geometric structures pixel patterns

scan convert

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Wh at is Scan Conversion?

Image display CRT H/W Device

scanning display image(approximately) pixel convert

* Graphics System,Package .

Constraint- Digital Device(raster graphics system)

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Wh at is Scan Conversion?

Assumptionpixel position : X : Scan line number

Y : Columnnumber

0 1 2 3 4 5012

34

ScanLine

Number

Pixel Column

Number

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L ine Drawing Algorit hm s Cartesian slope-intercept equation forstraight line

y = mx + b

x1 x2

y1

y2Slope m =x2 ±x1y2 ±y1

X I nterval xY I nterval y

y = m x -----(1)

x = -----(2)my

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L ine Drawing Algorit hm s DDA (D igital D ifferential A nalyzer)- left endpoint right endpoint

if slope m <= 1 thenunit x interval x ( x = 1)Yk+1 = Y k + m -------------------(3)

if slope m >= 1 thenunit y interval y ( y = 1)

Xk+1 = X k + -------------------(4)- right endpoint left endpoint

x = -1, y = -1

m1

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L ine Drawing Algorit hm s# include ³device.h´

# define ROUN D (a) ((int) (a+0.5))

void line DDA (int xa, int ya, int xb, int yb)

{ int dx = xb ± xa, dy = yb ± ya, steps, k;float x I ncrement, yincrement, x = xa, y = ya;

if (abs(dx) > abs(dy)) steps = abs(dx);else steps = abs(dy);xI ncrement = dx/(float) steps; y I ncrement = dy/(float) steps;

setPixel (ROUN D (x), ROUN D (y));for (k=0; k < steps ; k++) {

x += x I ncrement; y += y I ncrement;setPixel (ROUN D (x), ROUN D (y));}

}

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L ine Drawing Algorit hm s

DDA A lgorithm- : Cartesian slope equation .

-Roundoff error line pathdrift pixel position .

Cartesian slope equation ,

Rounding, floating point time-consuming

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L ine Drawing Algorit hm s Bresenham ¶s Line Algorithm

- only incremental integer calculation

- Specified Line Pathpixel Line Path

pixel position

10 11 12 13 14 1510

11121314

S pecifiedLine Path

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Y k+ 3

Y k+ 2

Y k+ 1

Y k

X k X k+ 1X k+ 2

y = mx + b

X k + 1

Y k

yY k + 1

d2

d1

y = m(X k +1) + bd1 = y - Y k

= m(X k + 1) + b ±Y k

d2 = (Y k + 1) ± y= (Y k + 1) ± m(X k + 1) ± b

I f d1 ± d2 > 0 then Y k + 1

else Y k

� Bresenham ¶s Line D rawing A lgorithm for |m| < 1

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L ine Drawing Algorit hm s D ecision Parameter P k = x(d1 ± d2)

= 2 y* X k - 2 x* Y k + c ----(5)P k+1 = 2 y* X k+1 - 2 x* Y k+1 + c

P k+1 - P k = 2 y(X k+1 ± X k ) - 2 x(Y k+1 - Y k )unit step x direction Xk+1 = X k + 1P k+1 = P k + 2 y - 2 x(Y k+1 - Y k )

Yk+1 ± Y k 0, 1 , (5)

first decision parameter P 0 = 2 y ± xstarting pixel position (X 0, Y 0), m = y / x

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L ine Drawing Algorit hm s� Bresenham ¶s Line D rawing A lgorithm for |m| < 1

1. I nput the two line endpoint and store the left endpointin (X 0, Y 0 ).

2. Load (X 0, Y 0 ) into the frame buffer, that is,plot the first point.3. Calculate constants x, y, and 2 y - 2 x, and

obtain the starting value for decision parameter asP 0 = 2 y ± x

4.A

t each Xk

along the line, starting at k= 0, perform thefollowing test: I f P k < 0, the next point to plot is(X k + 1, Y k ) and P k+1 = P k + 2 y

otherwise, the next point to plot is (X k + 1, Y k + 1) andP k+1 = P k + 2 y - 2 x

5. Repeat step 4 x times.

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L ine Drawing Algorit hm sExample

endpont (20,10), (30,18) , slope = 0.8

x = 10, y = 8initial decision parameter P 0 = 2 y ± x = 6

P k < 0 : P k+1 = P k + 2 yP k >= 0 : P k+1 = P k + 2 y - 2 xk

01234

P k

62

-21410

(X k+1 , X k+1 )

(21, 11)(22, 12)(23, 12)(24, 13)(25, 14)

k

56789

P k

62

-21410

(X k+1 , X k+1 )

(26, 15)(27, 16)(28, 16)(29, 17)(30, 18)

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20 25 3010

15

18

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L ine Drawing Algorit hm s� |m| > 1 , x, y

, |m| < 1 , unit step x directionXk+1 = X k + 1, yunit step y direction , x

� right endpoint left endpointx, y value decrease

� Horizontal Line( y = 0)Vertical Line( x = 0)diagonal Line(| x| = | y|)

F rame buffer load

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Circle Algorit hm s� Midpoint Circle A lgorithm

- Bresenham ¶s line A lgorithm circle

f circle (x, y) = X + Y - r2 2 2

Y k

Y k -1

X k X k+ 1Midpoint

f circle (x, y)< 0 inside circle= 0 on circle> 0 outside circle

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Circle Algorit hm s� Midpoint Circle A lgorithm

1. Input radius r and circle center ( X C,Y C), and obtain thefirst point on the circumference of a circle center on the

origin as (X

0,Y

0) = ( 0, r)2. Calculate the initial value of the Decision parameter asP0 = - r P 0 = 1 ± r (for r an integer)5

43. At each X k , starting k = 0 :

If P k < 0, next point ( X k+ 1, Y k ) and P k+ 1 = P k + 2X k+ 1 + 1else, next point ( X k+ 1, Y k -1) and P k+ 1 = P k + 2X k+ 1 + 1- 2 Y k+ 1

where 2X k+ 1 = 2X k + 2, 2Y k+ 1 = 2Y k - 2,4. Determine s ymmetr y points in the other seven octants.5. Move acutual circle center (x, y)6 . Repeat step 3 through 5 until x >= y.

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Circle Algorit hm sExample

r = 10, (X c,Y c) = ( 0, 0)

P0 = 1 ± r = 9(X 0,Y 0) = ( 0, 10) 2X 0 = 0, 2Y 0 = 20k 012345

6

P k

-9-6-16

-38

5

(X k+1 , X k+1 )(1, 10)(2, 10)(3, 10)(4, 9)(5, 9)(6, 8)

(7, 7)

2X k+1

2468

1012

14

2Y k+1

202020181816

14

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F illed-Area Pri m itives

Area filling on raster system- scan-line fill algorithm

: polygon, circle, ellipses, simple curves

- boundary fill algorithm: more complex boundaries,

interactive system

- flood fill algorithm

Inside ± Outside Test- odd-even rule- nonzero winding number rule

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Scan-Line Fill Algorithm

Check the intersection points of the scan line withthe polygon edgesVertices handling

Store the sorted edge table- maximum y value for that edge- x-intercept value at lower vertex for the edge- the inverse slop of the edge

Produce the active edge list

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Chec k the intersection points

y

x

Vertices handling

: scan line passing through a vertex intersects t wo edges.: trace around pol ygon boundar y

cloc k wise or countercloc k wise: observe relative change in vertex y coordinates as we

move one edge to the next.

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Scan line y´

Scan line y

Scan line y ´ intersect vertex two intersectingedge Scan line

intersection list

Scan line y ´ intersect vertex two intersectingedgehandling

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Scan-Line Fill Algorithmhandling : shortening lower edge to split vertices.- y value increasing : edge upper endpoint 1- y value decreasing : edge upper endpoint 1

Scan line y+ 1Scan line yScan line y-1

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Scan Line Y A

Scan Line Y D

Scan Line Y CC

C´D

A

B

01

Y A

Y D

Y C Y B X C 1/M CB

Y C X D1/M DC

Y E X D1/M DE

Y E X A1/M AE

Y B X A1/M AB

store Sorted edge tableproduce Active edge list

E

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B oundary Fill Algorithm

Point inside a region BoundaryI nput interior point(x, y)

Test neighboring positionsuntil all pixels up to boundary color- boundary color then stop- otherwise, fill color

4-connected boundary fill algorithm8-connected boundary fill algorithmHorizontal pixel spans across scan lines

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4-connected 8-connected

4 ± connected, 8-connected algorithm

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Start position 4-connected 8-connected

stac k ing of neighboring points .

horizontal pixel spans- horizontal pixel span beginning

position

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12

21

stac k

1

3

31

stac k

1

5

6

541

stac k

4

6

1

5

541

stac k

4

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F lood- F ill A lgorithmSingle color boundaryinterior fill method- start from a interior point- reassign all pixel values

: given interior color desired fill color- more than one interior color

all interior color same color reassign- 4 ± connected, 8 ± connected

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Inside-Outside Test

Odd ± even rule- polygon P polygon

line drawing.

- line polygon edge number counting.- odd number : P interior point- even number : P exterior point

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Inside-Outside Test

A

B

C

D

EF

GP 1P

1 1exterior

interior

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Inside-Outside Test

A

B

CD

EF

G

P left

Right(- 1)

interior

Pexterior

1 -1

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