Srno4_ conti

8/4/2019 Srno4_ conti....

1/30

Scan conversion

The scan conversion is digitation of picturedefinition into set of pixel intensity value forstorage in frame buffer .

E.g :- line drawing is accomplished bycalculating intermediate positions along theline path between two specific endpoint

positions.

8/4/2019 Srno4_ conti....

2/30

8/4/2019 Srno4_ conti....

3/30

The Problem (cont)

But what happens when we try to draw this on apixel based display?

How do we choose which pixels to turn on?

8/4/2019 Srno4_ conti....

4/30

Considerations

Considerations to keep in mind: The line has to look good

Avoid jaggies It has to be lightening fast!

How many lines need to be drawn in a typical scene? This is going to come back to bite us again and again

8/4/2019 Srno4_ conti....

5/30

Line Equations

Lets quickly review the equations involved indrawing lines

x

y

y0

yend

x end x 0

Slope-intercept line

equation:b xm y

where:

0

0

x x y ym

end

end

00 xm yb

8/4/2019 Srno4_ conti....

6/30

Lines & Slopes

The slope of a line ( m) is defined by its start andend coordinatesThe diagram below shows some examples of

lines and their slopes

m = 0

m = - 1/ 3

m = - 1/ 2

m = -1

m = -2 m = -4

m =

m = 1/ 3

m = 1/ 2

m = 1

m = 2 m = 4

m = 0

8/4/2019 Srno4_ conti....

7/30

A Very Simple Solution

We could simply work out the corresponding y coordinate for each unit x coordinate

Lets consider the following example:

x

y

(2, 2)

(7, 5)

2 7

2

5

8/4/2019 Srno4_ conti....

8/30

A Very Simple Solution (cont)

1

2

3

4

5

0

1 2 3 4 5 60 7

8/4/2019 Srno4_ conti....

9/30

8/4/2019 Srno4_ conti....

10/30


x

y

(2, 2)

(7, 5)

2 3 4 5 6 7

2

5

5

3

27

25m

5

42

5

32b

First work out m and b:

Now for each x value work out the y value:

5

32

5

43

5

3)3( y

5

13

5

44

5

3)4( y

5

435

455

3)5( y 5

245

465

3)6( y

8/4/2019 Srno4_ conti....

11/30

X y (y=mx+b)

2 2

3 =>(3/5)*3+(4/5)=>2.6

4 =>(3/5)*4+(4/5)=>3.2

5 =>(3/5)*5+(4/5)=>3.8

6 =>(3/5)*6+(4/5)

=>4.47 =>(3/5)*7+(4/5)

=>5

8/4/2019 Srno4_ conti....

12/30


Now just round off the results and turn on thesepixels to draw our line

35

32)3( y

35

13)4( y

45

43)5( y

45

24)6( y

0 1 2 3 4 5 6 7 8

0

12

3

4

5

6

7

8/4/2019 Srno4_ conti....

13/30

Which point will be scan converted bydirected method for a line whose end pointsare p1(2,3) and p2(8,5).

8/4/2019 Srno4_ conti....

14/30

8/4/2019 Srno4_ conti....

15/30

Which point will be scan converted bydirected method for a line whose end pointsare p1(2,3) and p2(5,9).

Sol: m=9-3/5-2=2 As slope is +ve y is incremented by 1 and

compute x by x=(y-b)/m

8/4/2019 Srno4_ conti....

16/30


However, this approach is just way too slow

In particular look out for:

The equation y = mx + b requires themultiplication of m by x Rounding off the resulting y coordinates

We need a faster solution

8/4/2019 Srno4_ conti....

17/30

A Quick Note About Slopes

In the previous example we chose to solve theparametric line equation to give us the y coordinate for each unit x coordinateWhat if we had done it the other way around?So this gives us:

where: andm

b y x

0

0

x x y y

mend

end 00 xm yb

8/4/2019 Srno4_ conti....

18/30

A Quick Note About Slopes (cont)

Leaving out the details this gives us:

We can see easily thatthis line doesnt look very good!We choose which wayto work out the linepixels based on theslope of the line

0 1 2 3 4 5 6 7 8

0

1

2

3

4

5

6

7

43

23)3( x 5

3

15)4( x

8/4/2019 Srno4_ conti....

19/30

A Quick Note About Slopes (cont)

If the slope of a line is between -1 and 1 then wework out the y coordinates for a line based on itsunit x coordinates

Otherwise we do the opposite x coordinates arecomputed based on unit y coordinates

m = 0

m = - 1/ 3

m = - 1/ 2

m = -1

m = -2 m = -4

m =

m = 1/ 3

m = 1/ 2

m = 1

m = 2 m = 4

m = 0

8/4/2019 Srno4_ conti....

20/30

8/4/2019 Srno4_ conti....

21/30

The DDA Algorithm

The digital differential analyzer (DDA) algorithmtakes an incremental

approach in order to speedup scan conversion

Simply calculate yk+1 based

on yk

The original differential analyzerw a s a p h y s i c a l m a c h i n edeveloped by Vannevar Bush atMIT in the 1930 s in order tosolve ordinary differentiale q u a t i o n s .

M o r e i n f o r m a t i o n h e r e .
http://scoter2.union.edu/~hemmendd/Encyc/Articles/Difanal/difanal.htmlhttp://scoter2.union.edu/~hemmendd/Encyc/Articles/Difanal/difanal.html

8/4/2019 Srno4_ conti....

22/30

The DDA Algorithm (cont)

Consider the list of points that we determinedfor the line in our previous example:

(2, 2), (3, 2 3/ 5), (4, 3 1/ 5), (5, 3 4/ 5), (6, 4 2/ 5), (7, 5)

Notice that as the x coordinates go up by one,the y coordinates simply go up by the slope of the line

This is the key insight in the DDA algorithm

8/4/2019 Srno4_ conti....

23/30


When the slope of the line is between -1 and 1begin at the first point in the line and, byincrementing the x coordinate by 1, calculate the

corresponding y coordinates as follows:

When the slope is outside these limits, increment

the y coordinate by 1 and calculate thecorresponding x coordinates as follows:

m y y k k 1

m x x k k 1

1

8/4/2019 Srno4_ conti....

24/30


Again the values calculated by the equationsused by the DDA algorithm must be rounded tomatch pixel values

( x k , y k) ( x k+1, y k+m)

( x k , round( y k))

( xk+1, round( y

k+m))

( x k , y k) ( x k+1 / m , y k+1)

(round( x k), yk)

(round( x k+ 1 / m ), y k+1)

8/4/2019 Srno4_ conti....

25/30

DDA Algorithm Example

Lets try out the following examples:

x

y

(2, 2)

(7, 5)

2 7

2

5

x

y (2, 7)

(3, 2)

2 3

2

7

8/4/2019 Srno4_ conti....

26/30

DDA Algorithm Example (cont)

7

2

3

4

5

6

1 2 3 4 5 60 7

8/4/2019 Srno4_ conti....

27/30

The DDA Algorithm Summary

The DDA algorithm is much faster than ourprevious attempt

In particular, there are no longer anymultiplications involved

However, there are still two big issues: Accumulation of round-off errors can make the

pixelated line drift away from what was intended The rounding operations and floating point

arithmetic involved are time consuming

8/4/2019 Srno4_ conti....

28/30

8/4/2019 Srno4_ conti....

29/30

8/4/2019 Srno4_ conti....

30/30

Documents

Srno4_ conti