werner purgathofer · instruction in display list (random scan) entry in frame buffer (raster scan)...

Einführung in Visual Computing186.822

Rasterization

Werner Purgathofer

Rasterization in the Rendering Pipeline

Werner Purgathofer 2raster image in pixel coordinates

clipping + homogenization

object capture/creation

modelingviewing

projection

rasterizationviewport transformation

shading

vertex stage(„vertex shader“)

pixel stage(„fragment shader“)

scene in normalized device coordinates

transformed vertices in clip space

scene objects in object space

Important Graphics Output Primitives

in 2Dpoints, linespolygons, circles, ellipses & other curves (also filled)pixel array operationscharacters

in 3Dtriangles & other polygonsfree form surfaces

+ commands for properties: color, texture, …

Werner Purgathofer / Einf. in Visual Computing 3

Points and Lines

point plottinginstruction in display list (random scan)entry in frame buffer (raster scan)

line drawinginstruction in display list (random scan)intermediate discrete pixel positions calculated (raster scan)

“jaggies”, aliasing


Lines: Staircase Effect


stairstep effect (jaggies) produced when a line is generated as a series of pixel positions

Line-Drawing Algorithms


m =y1 − y0x1 − x0

b = y0 − m.x0

line path between two points:

line equation: y = m.x + b

}bx0

y0

y1

x1

x1 − x0

y1 − y0

DDA Line-Drawing Algorithm


δy = m. δx for |m|<1

line equation: y = m.x + b

x0

y0

y1

x1

δxm. δx

DDA Line-Drawing Algorithm

DDA (digital differential analyzer)


δy = m. δx for |m|<1

( δx = for |m|>1 )δym

for δx=1 , |m|<1 : yk+1 = yk + m

line equation: y = m.x + b sampling points with distance 1

x0

y0

y1

x11

m

DDA – Algorithm Principle


dx = x1 – x0; dy = y1 – y0;m = dy / dx;

x = x0; y = y0;drawPixel (round(x), round(y));

for (k = 0; k < dx; k++){ x += 1; y += m;

drawPixel (round(x), round(y)}

extension to other cases simplefor δx=1 , |m|<1

m =y1 − y0x1 − x0

yk+1 = yk + m

Bresenham’s Line Algorithmfaster than simple DDA

incremental integer calculationsadaptable to circles, other curves


y = m.(xk + 1) + b

decision for every column which of the two candidate pixels is selected

10 11 12 13

10

11

12

13

Bresenham’s Line Algorithm


section of the screen grid showing a pixel in column xk on scan line yk that is to be plotted along the path of a line segment with slope 0 < m < 1

xk xk+1 xk+2 xk+3

yk

yk+1

yk+2

y = mx + b

Bresenham’s Line Algorithm (1/4)

xk xk+1

yk

yk+1 dupper

dlower

y


13

dlower = y − yk == m.(xk + 1) + b − yk

dupper = (yk + 1) − y == yk + 1 − m.(xk + 1) − b

dlower − dupper = = 2m.(xk + 1) − 2yk + 2b − 1

y = m.xk+1 + b

xk+1

dupper

dlower

yk

yk+1

y

= m.(xk + 1) + b


14

m = ∆y/∆x

dlower − dupper = = 2m.(xk + 1) − 2yk + 2b − 1

pk = ∆x.(dlower − dupper) = 2∆y.xk − 2∆x.yk + c

decision parameter:

(∆x = x1 – x0, ∆y = y1 – y0)

→ same sign as (dlower − dupper)

xk+1

dupper

dlower

yk

yk+1

y



+ 0pk+1 = 2∆y.xk+1 − 2∆x.yk+1 + c + + pk – 2∆y.xk + 2∆x.yk – c =

= pk + 2∆y − 2∆x.(yk+1 − yk)

pk = ∆x.(dlower − dupper) = 2∆y.xk − 2∆x.yk + ccurrent decision value:

next decision value:

p0 = 2∆y − ∆x

starting decision value:



1. store left line endpoint in (x0,y0)2. draw pixel (x0,y0)3. calculate constants ∆x, ∆y, 2∆y, 2∆y − 2∆x,

and obtain p0 = 2∆y − ∆x4. at each xk along the line, perform test:

if pk<0 then draw pixel (xk+1,yk); pk+1 = pk+ 2∆yelse draw pixel (xk+1,yk+1); pk+1= pk+ 2∆y − 2∆x

5. perform “step 4” (∆x − 1) times.

Bresenham: Example


k pk (xk+1,yk+1)

(20,41)0 -4 (21,41)1 2 (22,42)2 -12 (23,42)3 -6 (24,42)4 0 (25,43)5 -14 (26,43)6 -8 (27,43)7 -2 (28,43)8 4 (29,44)9 -10 (30,44)

p0 = 2∆y − ∆xif pk<0 then draw pixel (xk+1,yk); pk+1 = pk+ 2∆yelse draw pixel (xk+1,yk+1); pk+1= pk+ 2∆y − 2∆x

21 22 23 24 25 26 27 28 29 302040414243444546 ∆x = 10, ∆y = 3


Attributes of Graphics Output Primitives

in 2Dpoints, linescharacters

in 2D and 3Dtrianglesother polygons((filled) ellipses and other curves)


color

type: solid, dashed, dotted,…

width, line caps, corners

pen and brush options

…

Points and Line Attributes


text attributesfont (e.g. Courier, Arial, Times, Roman, …)

proportionally sized vs. fixed space fontsstyles (regular, bold, italic, underline,…)size (32 point, 1 point = 1/72 inch)

string attributesorientationalignment (left, center, right, justify)

Character Attributes

vertical

horizontal

Displayed primitives generated by the raster algorithms discussed in Chapter 3 have a jagged, or stairstep, appearance.

Displayed primitives generated by the raster algorithms

discussed in Chapter 3 have a jagged, or

stairstep, appearance.

Displayed primitives generated by the raster algorithms

discussed in Chapter 3 have a jagged, or

stairstep, appearance.

Displayed primitivesgenerated by theraster algorithmsdiscussed in Chapter 3have a jagged, orstairstep, appearance.


font (typeface)design style for (family of) characters

Courier, Times, Arial, …serif (better readable),sans serif (better legible)

definition modelbitmap font (simple to define and display), needs more space (font cache)outline font (more costly, less space, geometric transformations)

Character Primitives

SfzrnSfzrn


Example: Newspaper


the letter B represented with an 8x8 bilevel bitmap pattern and with an outline shape defined with straight line and curve segments

Character Generation Examples


fill styleshollow, solid fill, pattern fill

fill optionsedge type, width, color

pattern specificationthrough pattern tablestiling (reference point)

Area-Fill Attributes (1)

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2

4 01 2


combination of fill pattern with background colorssoft fill

combination of colorsantialiasing at object borderssemitransparent brush simulationexample: linear soft-fill

F... foreground colorB...background color P = t·F + (1 − t)·B

Area-Fill Attributes (2)

pattern back-ground

and

or

xor

replace


Triangle and Polygon Attributes

colormaterialtransparencytexturesurface detailsreflexion properties, …

→ defined illumination produces effects


General Polygon Fill Algorithms

triangle rasterizationother polygons: what is inside?scan-line fill methodflood fill method

(α, β ,γ)

barycentriccoordinates

clean bordersbetweenadjacent triangles




“interior”, “exterior” for self-intersecting polygons?




interior pixels along a scan line passing through a polygon area




starting from a seed point: fill until you reach a border


Triangles: Barycentric Coordinates

notation: (α, β ,γ)

P0(x0, y0)

P1(x1, y1)

P2(x2, y2)

(1, 0 ,0)

(0, 0, 1)

(0, 1, 0)P = αP0 + βP1 + γP2

triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1}

( , 0.13, 0.27)0.60P


Barycentric Coordinates in 3D

P0(x0, y0, z0)

P1(x1, y1, z1)

P2(x2, y2, z2)

(1, 0 ,0)

(0, 0, 1)

(0, 1, 0)

( , 0.13, 0.27)0.60

notation: (α, β ,γ)

P = αP0 + βP1 + γP2


P

P = αP0 + βP1 + γP2

triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1}Werner Purgathofer / Einf. in Visual Computing 33

for all x

for all y /* use a bounding box!*/

{compute (α, β ,γ) for (x,y) ;

if (0<α<1) and (0<β<1) and (0<γ<1)

{c = αc0 + βc1 + γc2 ;

draw pixel (x,y) with color c

}}

Triangle Rasterization Algorithm

<1 <1 <1

c = αc0 + βc1 + γc2 ;

with color c

interpolates corner values (vertices) linearly inside the triangle(and along edges)


Computing (α, β ,γ) for P(x,y)

line through P1, P2: g12(x,y) = a12x+b12y+c12=0then α = g12(xP,yP) / g12(x0,y0)

β,γ analogous

P = αP0 + βP1 + γP2


P0(x0, y0)

P1(x1, y1)

P2(x2, y2)

P(xP, yP)(α, β ,γ) ? g12(x,y)=0

P = αP0 + βP1 + γP2

triangle = {P| α+β+γ=1, 0<α<1, 0<β<1, 0<γ<1}Werner Purgathofer / Einf. in Visual Computing 35

Barycentric Coordinates Example

1.00.00.0

0.80.20.0

0.60.40.0

0.01.00.0

0.40.60.0

0.20.80.0

0.60.20.2

0.40.40.2

0.20.60.2

0.40.20.4

0.20.40.4

0.20.20.6

0.80.00.2

0.60.00.4

0.40.00.6

0.20.00.8

0.00.80.2

0.00.60.4

0.00.40.6

0.00.20.8

0.00.01.0

-0.21.4-0.2

1.2-0.20.0

-0.21.20.0

1.20.0-0.2

1.00.2-0.2

0.80.4-0.2

0.60.6-0.2

0.40.8-0.2

0.21.0-0.2

0.01.2-0.2

1.2-0.40.2

1.0-0.20.2

-0.21.00.2

1.2-0.60.4

1.0-0.40.4

0.8-0.20.4

-0.20.80.4

1.2-0.80.6

1.0-0.60.6

0.8-0.40.6

0.6-0.20.6

-0.20.60.6

1.2-1.00.8

1.0-0.80.8

0.8-0.60.8

0.6-0.40.8

0.4-0.20.8

-0.20.40.8

1.2-1.21.0

1.0-1.01.0

0.8-0.81.0

0.6-0.61.0

0.4-0.41.0

0.2-0.21.0

-0.20.21.0

1.2-1.41.2

1.0-1.21.2

0.8-1.01.2

0.6-0.81.2

0.4-0.61.2

0.2-0.41.2

0.0-0.21.2

-0.20.01.2


Avoiding to Draw Borders Twice

don‘t draw the outline of the triangle!result would depend on rendering order


Avoiding to Draw Borders Twice

don‘t draw the outline of the triangle!result would depend on rendering order

draw only pixels that are inside exact trianglei.e. pixels with α = 0 or β = 0 or γ = 0 are not drawn

clean bordersbetweenadjacent triangles


Pixels exactly on a Border …!?

holes if both triangles leave pixels awaysimplest solution: draw both pixelsbetter: arbitrary choice based on some test

e.g. only right boundaries


odd-even rule nonzero-winding- number rule

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 number of edge intersections = outside odd number of 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 number of edges up and down = outsidedifferent number of edges up and down = inside

What is Inside?: Nonzero Winding Number

polygon classificationsconvex: no interior angle > 180°concave: not convex

concavity testvector method

all vector cross products have the same sign ⇒ convexrotational method

rotate polygon-edges onto x-axis, always same direction ⇒ convex


Polygon Fill Areas

werner purgathofer · instruction in display list (random scan) entry in frame buffer (raster scan)...

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