Computer Graphics Scan Conversion - Technion – …webcourse.cs.technion.ac.il/.../WCFiles/ScanConversion.pdfComputer Graphics Copyright 1996 C. Gotsman & G. Elber Computer Science

1

Computer Graphics

Copyright 1996 C. Gotsman & G. Elber

Computer Science Dept. Technion

Scan Conversion

Page 1

Scan Conversion:

Drawing Lines

on

Raster Display

bresenham

Scan Conversion - Center for Graphics and Geometric Computing,

Technion

2

Raster Display

Discrete grid of elements (frame buffers

pixels)

Shapes drawn by setting the “right”

elements

Frame buffer is scanned, one line at a

time, to refresh the image (as opposed

to vector display)

Properties

Difficult to draw smooth lines

Displays only a discrete approximation of

any shape

Refresh of entire frame buffer


Technion

3

Terminology

Pixel: Picture element

Smallest accessible element in picture

Usually rectangular or circular

Aspect Ratio: Ratio between physical

dimensions of pixel

(not necessarily 1 !!)

Dynamic Range: Ratio between minimal (not

zero!) and maximal light intensity emitted by

displayed pixel


Technion

4

Terminology (cont’d)

Resolution: number of distinguishable

rows and columns on device Measured in

Absolute values (1K x 1K)

Relative values (300 dots per inch)

Screen Space: discrete 2D Cartesian

coordinate system of screen pixels

Object Space: 3D Cartesian coordinate

system of the universe where the objects

(to be displayed) are embedded

1

2

6

6x8

2 81

y

z

x

drawing


Technion

5

Basic Line Drawing

Assume x1< x2 and line slope absolute value is 1

Questions:

Can this algorithm use integer arithmetic ?

Does it accumulate error ?

Is the error significant ?; end

; end

;

; ) )( , (

begin

do to from for

;

;

;

; , , , ,float

begin

) , , , (

21

1

12

12

2211

slopeyy

yx

xxx

yy

dxdy

slope

yydy

xxdx

slopeyxdydx

yxyx

RoundPlotPixel

Line


Technion

6

Recursive Line Drawing

Simple, recursive, integer, line drawing:

; end

; end

; ) , , , (

; ) , , , (

; ) , (

begin else

;return

) ) and (

or ) and ( ( if

;2/)(

; 2/)(

; ,int

begin

) , , , (

22

11

22

11

21

21

2211

yxyx

yxyx

yx

yyxx

yyxx

yyy

xxx

yx

yxyx

Line

Line

PlotPixel

Line

Questions:

How does the algorithm

work?

Does the algorithm

accumulate error ?

Is it significant ?

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2

Computer Graphics



Scan Conversion

Page 2


Technion

7

Recursive Line Drawing (cont’d)

More Potential Problems:

Line is not drawn sequentially

Function call for each pixel drawn

We want a faster and accurate algorithm !


Technion

8

Midpoint (Bresenham) Algorithm

Idea:

Define error function

Proceed along the line incrementally

Select direction that minimizes accumulated error

)+-(2 = ),(

0|,

cydxxdyyxd

cydxxdycbyaxyx

1and,12

121212

xx

yy

dx

dyyyxx

Assumptions:

Definitions

(x1,y1) (x,y)

d(x,y)

(x2,y2)


Technion

9

Midpoint (Bresenham) Algorithm (cont’d)

Given point P=(x,y), d(x,y) is the signed distance of P to (up to normalization factor)

d is zero for P d may serve as error function to be minimized

Starting point satisfies d(x1,y1) = 0

Each step moves right (east) or upper right

(northeast)

)+-(2 = ),( cydxxdyyxd


Technion

10

Midpoint Line Drawing (cont’d)

Sign of d(x1+1,y1+½) indicates whether to move east or

northeast

At (x1,y1)

dstart= d(x1+1,y1+½)=2dy-dx

Increment in d (after each step)

Move east

e= d(x+2,y+½) d(x+1,y+½) =

= 2((x+2)dy(y+½)dx+c))2((x+1)dy(y+½)dx+c)

= 2dy

Move northeast

ne= d(x+2,y+3/2) d(x+1,y+½) =

= 2((x+2)dy(y+3/2)dx+c))2((x+1)dy(y+½)dx+c)

= 2(dydx)

)+-(2 = ),( cydxxdyyxd


Technion

11

Midpoint Line Algorithm

; end

; end

; ) , (

; end

; 1

; 1

;

begin else

; end

; 1

;

begin

then) 0 ( if

do ) ( while

; ) , (

; )(2 ; 2

; 2

; ;

; ; xx

; , , , , , ,int

begin

) , , , (

2

1212

11

2211

yx

yy

xx

dd

xx

dd

d

xx

yx

dxdydy

dxdyd

yydyxxdx

yy

ddydxyx

yxyx

ne

e

nee

nee

PlotPixel

PlotPixel

Line

bresenham

fineline


Technion

12

Midpoint Examples

Question: Is there a problem with the MidPoint algorithm

(hint: horizontal vs. diagonal lines)?

Comment: Algorithm extends to higher order curves –

e.g. circles

Midpoint (squares) vs.

Recursive (squares) line drawingMidpoint line drawing

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3

Computer Graphics



Scan Conversion

Page 3


Technion

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Error function d can be viewed as explicit surface:

d(x,y)=2(xdy-ydx+c)

Implicit equation of origin-centered circle C is

x2+y2R2 = 0

Error functional

d(x,y) = x2+y2R2

vanishes on C

Can also be seen as an explicit surface

Error Function Intuition


Technion

14

Circle Midpoint For first quadrant, start from (x1,y1) = (0,R)

Move east or south-east

d(x,y) will be a threshold criteria at the midpoint

Because of symmetry - suffices to deal with

octant of first quadrant for which y>x


Technion

15

Circle Midpoint (cont’d)

At the beginning: dstart=d(x1+1,y1½)=d(1,R½)=5/4R

Moving east:

e = d(x+2,y½)d(x+1,y½)

= (x+2)2+(y½)2R2(x+1)2(y½)2+R2

= 2x+3

Moving south-east:

se = d(x+2,y3/2)d(x+1,y½)

= (x+2)2+(y3/2)2R2(x+1)2(y½)2+R2

= 2(xy)+5

e and se not constant anymore

Since d is incremented by integer values, one can use dstart=1R, yielding an integer algorithm

This last change has no effect on threshold criteria

d(x,y) = x2+y2R2


Technion

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Midpoint Circle AlgorithmCircleOctant2

PlotPixel

PlotPixel

(R)

begin

int

( );

while( ) do

if ( ) then

begin

end;

else begin

d d + 2(x - y)+ 5;

x x +1;

y y -1;

end;

( ) ( );

end;

x y d

x

y R

d R

x y

y x

d

d d x

x x

x y

, , ;

;

;

;

,

;

;

,

0

1

0

2 3

1

Generate other 7 octants by

mirroring (*)

Question: What will happen if we

change “while (y > x)” to

“while (y > 0)”?


Technion

17

Circle Midpoint Examples

Documents

Computer Graphics Scan Conversion - Technion – …webcourse.cs.technion.ac.il/.../WCFiles/ScanConversion.pdfComputer Graphics Copyright 1996 C. Gotsman & G. Elber Computer Science